Introduction. —Can gravity transmit quantum information? An empirical answer to this question remains far beyond current experimental reach. Nevertheless, recent advances in quantum optomechanics [1 , 2 ] suggest that proposals to address it may no longer constitute pure Gedankenexperiments. Low-energy tests of quantum aspects of gravity, first envisioned by Feynman [3 ] and later extended and refined in various theoretical settings [4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 ] , offer one of the most promising routes for probing the interface between quantum mechanics and gravity [13 , 14 ] .
From a quantum-information perspective, this question is naturally framed in terms of separable versus non-separable dynamical maps [15 , 16 , 17 ] . Separable operations, quantum channels whose Kraus operators factorize across subsystems, cannot create entanglement from separable inputs. The observation of any dynamical signature that requires a non-separable map for its description would provide an operational way to rule out gravitational models whose effective dynamics is separable. Proposed signatures include the generation of gravitationally induced entanglement [4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 18 , 19 , 20 , 21 , 22 , 23 ] and violations of inequalities satisfied by all separable dynamics [18 , 24 , 25 , 26 , 27 , 28 ] .
It is often assumed that spontaneous-collapse noise [29 , 30 , 31 , 32 , 33 ] renders gravitational dynamics effectively separable, although recent work has suggested that non-separable features may persist even in strongly decohering regimes [34 , 35 ] . More generally, even when the dynamics remains non-separable, decoherence can suppress entanglement generation to arbitrarily low levels, making the underlying entangling resource exceedingly difficult to observe. This raises the natural question of whether the effect of noise can be reduced without simultaneously suppressing the gravitational signal.
In this Letter, we show that several physically relevant collapse models, including GRW, CSL, and Diósi–
Penrose, do not render the combined Newtonian-plus-noise dynamics separable, even when the collapse noise completely prevents entanglement generation in commonly considered initial state configurations [8 , 9 ] . In particular, appropriately chosen initial states can reveal an arbitrarily small but nonzero amount of entanglement regardless of the collapse-noise strength. We then ask whether this weak manifestation of the underlying entangling dynamics can be made detectable using quantum-information techniques without introducing an independent nonlocal channel between the masses. To this end, we construct a strictly local quantum error-correction (QEC) [36 ] protocol that suppresses relevant sources of decoherence while preserving the Newtonian interaction. This amplifies the absolute value of entanglement and sustains it for a longer time. Since all encoding, recovery, and processing operations are local, any entanglement revealed after recovery must originate from the underlying gravitational dynamics, and can be used to establish the non-separable character of the gravitational channel. We call this entangling resource locally recoverable, relative to a given entanglement-detection protocol, when local error correction renders otherwise undetectable generated entanglement detectable. Applying this framework, we find that the entangling component of the GRW and CSL dynamics is locally recoverable, whereas the Diòsi–Penrose model does not exhibit the same behavior under bipartite QEC protocols here.
The model.— We consider a bipartite quantum system consisting of two equal masses, A A A B B B m A = m B = m m_A=m_B=m m A = m B = m Figure 1 . The masses are confined in independent harmonic traps of frequency ω A = ω B = ω \omega_A=\omega_B=\omega ω A = ω B = ω d d d ( x A , p A ) (x_A,p_A) ( x A , p A ) ( x B , p B ) (x_B,p_B) ( x B , p B ) x A x_A x A x B x_B x B
Figure 1. Two masses m A m_A m A m B m_B m B d d d
d d t ρ = L ( ρ ) = − i ℏ [ H 0 − G m 2 ∣ d + x A − x B ∣ , ρ ] + L d e c ( ρ ) , (1) \frac{\mathrm{d}}{\mathrm{d}t}\rho=\mathcal{L}(\rho)=-\frac{i}{\hbar}\left[H_0-\frac{Gm^2}{\lvert d+x_A-x_B\rvert},\rho\right]+\mathcal{L}_{\mathrm{dec}}(\rho), \tag*{(1)} d t d ρ = L ( ρ ) = − ℏ i [ H 0 − ∣ d + x A − x B ∣ G m 2 , ρ ] + L dec ( ρ ) , (1) where H 0 = ∑ α = A , B ( p α 2 / 2 m + m ω 2 x α 2 / 2 ) H_0=\sum_{\alpha=A,B}(p_\alpha^2/2m+m\omega^2x_\alpha^2/2) H 0 = ∑ α = A , B ( p α 2 /2 m + m ω 2 x α 2 /2 ) L d e c \mathcal{L}_{\mathrm{dec}} L dec L d e c \mathcal{L}_{\mathrm{dec}} L dec
A wide variety of models relevant to low-energy tests of gravity lead to a dynamical equation of the form Eq. (1) . If gravity possesses quantum degrees of freedom, the Newtonian potential follows from standard low-energy arguments, while L d e c \mathcal{L}_{\mathrm{dec}} L dec [37 ] , gravitationally induced noise, as in the Anastopoulos–Hu model [38 ] , or hypothesized fundamental collapse mechanisms, such as those described in the GRW [29 ] and CSL [30 , 31 ] models. If gravity is taken to be classical and coupled to quantum matter, tracing out the classical degrees of freedom often yields L d e c \mathcal{L}_{\mathrm{dec}} L dec [39 , 40 , 41 ] and Kafri–Taylor–Milburn (KTM) models [18 ] .
Quantum-Error-Correction assisted tests— We introduce a strictly local quantum error-correction protocol into the dynamics [36 ] by defining a code space with tensor-product structure C = C A ⊗ C B \mathcal{C}=\mathcal{C}_A\otimes\mathcal{C}_B C = C A ⊗ C B C α = span { ∣ k ˉ ⟩ α } k ⊂ H α \mathcal{C}_\alpha=\operatorname{span}\{|\bar{k}\rangle_\alpha\}_k\subset\mathcal{H}_\alpha C α = span { ∣ k ˉ ⟩ α } k ⊂ H α α ∈ { A , B } \alpha\in\{A,B\} α ∈ { A , B }
R ( ρ A B ) = ( R A ⊗ R B ) ( ρ A B ) , R α ( ρ α ) = ∑ i R α i ρ α R α i † . (2) R(\rho_{AB})=(R_A\otimes R_B)(\rho_{AB}),\qquad R_\alpha(\rho_\alpha)=\sum_i R_\alpha^i\rho_\alpha R_\alpha^{i\dagger}. \tag*{(2)} R ( ρ A B ) = ( R A ⊗ R B ) ( ρ A B ) , R α ( ρ α ) = i ∑ R α i ρ α R α i † . (2) Here R α i R_\alpha^i R α i α ∈ { A , B } \alpha\in\{A,B\} α ∈ { A , B }
For a given set of error Kraus operators { E k } k \{E_k\}_k { E k } k C \mathcal{C} C [42 ]
⟨ i ˉ ∣ E k † E l ∣ j ˉ ⟩ = C k l δ i j , (3) \langle\bar{i}|E_k^\dagger E_l|\bar{j}\rangle=C_{kl}\delta_{ij}, \tag*{(3)} ⟨ i ˉ ∣ E k † E l ∣ j ˉ ⟩ = C k l δ ij , (3) where { ∣ i ˉ ⟩ λ } \{|\bar{i}\rangle_\lambda\} { ∣ i ˉ ⟩ λ } global code space C ⊂ H A ⊗ H B \mathcal{C}\subset\mathcal{H}_A\otimes\mathcal{H}_B C ⊂ H A ⊗ H B C k l C_{kl} C k l
We assume that the local recovery operation R R R L \mathcal{L} L (1) . If R R R R 2 = R R^2=R R 2 = R lim N → ∞ ( R ∘ e L t / N ) N ( ρ ) = e ( R ∘ L ) t ( ρ ) \lim_{N\to\infty}(R\circ e^{\mathcal{L}t/N})^N(\rho)=e^{(R\circ\mathcal{L})t}(\rho) lim N → ∞ ( R ∘ e L t / N ) N ( ρ ) = e ( R ∘ L ) t ( ρ ) ρ \rho ρ R ( ρ ) = ρ R(\rho)=\rho R ( ρ ) = ρ
L e f f = R ∘ L . (4) \mathcal{L}_{\mathrm{eff}}=R\circ\mathcal{L}. \tag*{(4)} L eff = R ∘ L . (4) The recovery map R R R R R R L e f f \mathcal{L}_{\mathrm{eff}} L eff L \mathcal{L} L
Our task is to construct a local R R R x x x
L G R W ( ρ ) = λ G R W ( ∫ d q 1 π r C e − ( x − q ) 2 2 r C 2 ρ e − ( x − q ) 2 2 r C 2 − ρ ) , (5) \mathcal{L}_{\mathrm{GRW}}(\rho)=\lambda_{\mathrm{GRW}}\left(\int\mathrm{d}q\frac{1}{\sqrt{\pi}r_{\mathrm{C}}}e^{-\frac{(x-q)^2}{2r_{\mathrm{C}}^2}}\rho e^{-\frac{(x-q)^2}{2r_{\mathrm{C}}^2}}-\rho\right), \tag*{(5)} L GRW ( ρ ) = λ GRW ( ∫ d q π r C 1 e − 2 r C 2 ( x − q ) 2 ρ e − 2 r C 2 ( x − q ) 2 − ρ ) , (5) where λ G R W \lambda_{\mathrm{GRW}} λ GRW r C r_{\mathrm{C}} r C Tr [ ρ ∣ x 1 ⟩ ⟨ x 2 ∣ ] \operatorname{Tr}[\rho|x_1\rangle\langle x_2|] Tr [ ρ ∣ x 1 ⟩ ⟨ x 2 ∣ ] λ G R W ( 1 − exp [ − ( x 1 − x 2 ) 2 / 4 r C 2 ] ) \lambda_{\mathrm{GRW}}\left(1-\exp[-(x_1-x_2)^2/4r_{\mathrm{C}}^2]\right) λ GRW ( 1 − exp [ − ( x 1 − x 2 ) 2 /4 r C 2 ] ) x A ⊗ x B x_A\otimes x_B x A ⊗ x B x A ⊗ I B x_A\otimes\mathbb{I}_B x A ⊗ I B I A ⊗ x B \mathbb{I}_A\otimes x_B I A ⊗ x B x A x_A x A x B x_B x B x A ⊗ x B x_A\otimes x_B x A ⊗ x B same operators that enter the interaction Hamiltonian; if the dominant local noise acted through operators orthogonal to x A x_A x A x B x_B x B p 2 p^2 p 2 [38 ] , see also discussion section and App. D.
To circumvent this limitation, we instead exploit the full functional form of both the gravitational interaction and the decoherence mechanisms beyond their leading-order contributions. Rather than attempting to preserve a first-order interaction, we design the encoding such that an effective two-qubit dynamics is retained, and logical σ x \sigma^x σ x
We encode each mass into a two-dimensional logical subspace spanned by Fock states of the motional degree of freedom,
C α = span { ∣ 0 ˉ ⟩ α = ∣ 0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣ 2 N + 1 ⟩ α } , (6) C_\alpha= \operatorname{span}\{|\bar{0}\rangle_\alpha= |0\rangle_\alpha, |\bar{1}\rangle_\alpha= |2N+1\rangle_\alpha\}, \tag*{(6)} C α = span { ∣ 0 ˉ ⟩ α = ∣0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣2 N + 1 ⟩ α } , (6) with α ∈ { A , B } \alpha\in\{A,B\} α ∈ { A , B } N N N [43 , 44 ] but employs simple superpositions in the Fock basis. An ideal GKP encoding would eliminate the logical interaction to all orders, contrary to our objective.
The key algebraic properties of this encoding, derived in detail in App. B.3, are as follows. For i = 1 , … , 2 N i=1,\ldots,2N i = 1 , … , 2 N ⟨ 0 ˉ ∣ x i ∣ 1 ˉ ⟩ = 0 \langle\bar{0}|x^i|\bar{1}\rangle=0 ⟨ 0 ˉ ∣ x i ∣ 1 ˉ ⟩ = 0 ⟨ 0 ˉ ∣ x i ∣ 0 ˉ ⟩ = ⟨ 1 ˉ ∣ x i ∣ 1 ˉ ⟩ = 0 \langle\bar{0}|x^i|\bar{0}\rangle=\langle\bar{1}|x^i|\bar{1}\rangle=0 ⟨ 0 ˉ ∣ x i ∣ 0 ˉ ⟩ = ⟨ 1 ˉ ∣ x i ∣ 1 ˉ ⟩ = 0 i i i i i i i > 2 N i>2N i > 2 N ⟨ 0 ˉ ∣ x i ∣ 1 ˉ ⟩ \langle\bar{0}|x^i|\bar{1}\rangle ⟨ 0 ˉ ∣ x i ∣ 1 ˉ ⟩
A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ G m A m B ∣ d + x A − x B ∣ ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ≠ 0 , {}_A\langle\bar{0}|{}_B\langle\bar{0}|\frac{Gm_A m_B}{|d+x_A-x_B|}|\bar{1}\rangle_A|\bar{1}\rangle_B\ne0, A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ∣ d + x A − x B ∣ G m A m B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = 0 , which, upon projection onto the code space, yields an effective σ A x ⊗ σ B x \sigma_A^x\otimes\sigma_B^x σ A x ⊗ σ B x
These properties also determine the error-correction behavior. Since ⟨ 0 ˉ ∣ x i ∣ 1 ˉ ⟩ = 0 \langle\bar{0}|x^i|\bar{1}\rangle=0 ⟨ 0 ˉ ∣ x i ∣ 1 ˉ ⟩ = 0 i ≤ 2 N i\le2N i ≤ 2 N x i ∣ 0 ˉ ⟩ x^i|\bar{0}\rangle x i ∣ 0 ˉ ⟩ x j ∣ 1 ˉ ⟩ x^j|\bar{1}\rangle x j ∣ 1 ˉ ⟩ i , j ≤ N i,j\le N i , j ≤ N σ x \sigma^x σ x x k x^k x k k ≤ N k\le N k ≤ N σ z \sigma^z σ z σ x \sigma_x σ x ⟨ 0 ˉ ∣ x i ∣ 0 ˉ ⟩ ≠ ⟨ 1 ˉ ∣ x i ∣ 1 ˉ ⟩ \langle\bar{0}|x^i|\bar{0}\rangle\ne\langle\bar{1}|x^i|\bar{1}\rangle ⟨ 0 ˉ ∣ x i ∣ 0 ˉ ⟩ = ⟨ 1 ˉ ∣ x i ∣ 1 ˉ ⟩ i i i σ z \sigma^z σ z σ z \sigma^z σ z σ A x ⊗ σ B x \sigma_A^x\otimes\sigma_B^x σ A x ⊗ σ B x § 0.3.7 .
The corresponding recovery channel is constructed by introducing auxiliary states ∣ 0 ˉ i ⟩ = ∣ i ⟩ |\bar{0}_i\rangle=|i\rangle ∣ 0 ˉ i ⟩ = ∣ i ⟩ ∣ 1 ˉ i ⟩ = c i ∣ 2 N + 1 − i ⟩ + d i ∣ 2 N + 1 + i ⟩ |\bar{1}_i\rangle=c_i|2N+1-i\rangle+d_i|2N+1+i\rangle ∣ 1 ˉ i ⟩ = c i ∣2 N + 1 − i ⟩ + d i ∣2 N + 1 + i ⟩ i = 1 , … , N i=1,\ldots,N i = 1 , … , N c i , d i c_i,d_i c i , d i
R 0 = ∑ k = 0 , 1 ∣ k ˉ ⟩ ⟨ k ˉ ∣ , R i = ∑ k = 0 , 1 ∣ k ˉ ⟩ ⟨ k ˉ i ∣ ( 1 ≤ i ≤ N ) , R^0=\sum_{k=0,1}|\bar{k}\rangle\langle\bar{k}|,\qquad R^i=\sum_{k=0,1}|\bar{k}\rangle\langle\bar{k}_i|\quad(1\le i\le N), R 0 = k = 0 , 1 ∑ ∣ k ˉ ⟩ ⟨ k ˉ ∣ , R i = k = 0 , 1 ∑ ∣ k ˉ ⟩ ⟨ k ˉ i ∣ ( 1 ≤ i ≤ N ) , and R i = ∣ 1 ˉ ⟩ ⟨ 1 ˉ i ∣ R^i=|\bar{1}\rangle\langle\bar{1}_i| R i = ∣ 1 ˉ ⟩ ⟨ 1 ˉ i ∣ i > N i>N i > N ∣ 1 i ⟩ |1_i\rangle ∣ 1 i ⟩
R α ( ρ ) = ∑ i = 0 ∞ R α i ρ R α i † , (7) R_\alpha(\rho)=\sum_{i=0}^{\infty}R_\alpha^i\rho R_\alpha^{i\dagger}, \tag*{(7)} R α ( ρ ) = i = 0 ∑ ∞ R α i ρ R α i † , (7) with identical constructions for each subsystem α ∈ { A , B } \alpha\in\{A,B\} α ∈ { A , B }
With this encoding and recovery map, the effective Lindbladian in the logical code space is of the following form
L e f f ( ρ ) = − i ℏ [ H e f f , ρ ] + ∑ α = A , B ∑ μ ∈ { x , z , + , − } Γ e f f μ D [ σ α μ ] ( ρ ) + ∑ μ , ν ∈ { x , z } Γ e f f μ ν ( D [ σ A μ − σ B ν ] − D [ σ A μ ] − D [ σ B ν ] ) ( ρ ) , \mathcal{L}_{\mathrm{eff}}(\rho)=-\frac{i}{\hbar}[H_{\mathrm{eff}},\rho]+\sum_{\alpha=A,B}\sum_{\mu\in\{x,z,+,-\}}\Gamma_{\mathrm{eff}}^{\mu}\mathcal{D}[\sigma_\alpha^\mu](\rho)
+\sum_{\mu,\nu\in\{x,z\}}\Gamma_{\mathrm{eff}}^{\mu\nu}\left(\mathcal{D}[\sigma_A^\mu-\sigma_B^\nu]-\mathcal{D}[\sigma_A^\mu]-\mathcal{D}[\sigma_B^\nu]\right)(\rho), L eff ( ρ ) = − ℏ i [ H eff , ρ ] + α = A , B ∑ μ ∈ { x , z , + , − } ∑ Γ eff μ D [ σ α μ ] ( ρ ) + μ , ν ∈ { x , z } ∑ Γ eff μν ( D [ σ A μ − σ B ν ] − D [ σ A μ ] − D [ σ B ν ] ) ( ρ ) , H e f f = H l o c + ℏ ∑ μ , ν ∈ { x , z } λ e f f μ ν σ A μ σ B ν . (8) H_{\mathrm{eff}}=H_{\mathrm{loc}}+\hbar\sum_{\mu,\nu\in\{x,z\}}\lambda_{\mathrm{eff}}^{\mu\nu}\sigma_A^\mu\sigma_B^\nu. \tag*{(8)} H eff = H loc + ℏ μ , ν ∈ { x , z } ∑ λ eff μν σ A μ σ B ν . (8) Here D [ L ] ( ρ ) = L ρ L † − { L † L , ρ } / 2 \mathcal{D}[L](\rho)=L\rho L^\dagger-\{L^\dagger L,\rho\}/2 D [ L ] ( ρ ) = L ρ L † − { L † L , ρ } /2 σ ± \sigma^\pm σ ± σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ \sigma^+=|\bar{1}\rangle\langle\bar{0}| σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ σ − = ( σ + ) † \sigma^-=(\sigma^+)^\dagger σ − = ( σ + ) † H l o c H_{\mathrm{loc}} H loc λ e f f x x = − λ e f f x z \lambda_{\mathrm{eff}}^{xx}=-\lambda_{\mathrm{eff}}^{xz} λ eff xx = − λ eff x z Γ e f f x x = − Γ e f f x z \Gamma_{\mathrm{eff}}^{xx}=-\Gamma_{\mathrm{eff}}^{xz} Γ eff xx = − Γ eff x z Γ e f f μ ν = 0 \Gamma_{\mathrm{eff}}^{\mu\nu}=0 Γ eff μν = 0
Then let us define
η = ∥ ( Γ e f f x x − λ e f f x x Γ e f f x x Γ e f f x z − λ e f f x z Γ e f f x z Γ e f f z z Γ e f f x x Γ e f f z z Γ e f f x x Γ e f f z z Γ e f f x z Γ e f f z z ) ∥ 2 , (9) \eta=\left\|\begin{pmatrix}
\frac{\Gamma_{\mathrm{eff}}^{xx}-\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^{xx}} & \frac{\Gamma_{\mathrm{eff}}^{xz}-\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{xz}\Gamma_{\mathrm{eff}}^{zz}}}\\
\sqrt{\frac{\Gamma_{\mathrm{eff}}^{xx}\Gamma_{\mathrm{eff}}^{zz}}{\Gamma_{\mathrm{eff}}^{xx}\Gamma_{\mathrm{eff}}^{zz}}} & \frac{\Gamma_{\mathrm{eff}}^{xz}}{\Gamma_{\mathrm{eff}}^{zz}}
\end{pmatrix}\right\|_2, \tag*{(9)} η = Γ eff xx Γ eff xx − λ eff xx Γ eff xx Γ eff z z Γ eff xx Γ eff z z Γ eff x z Γ eff z z Γ eff x z − λ eff x z Γ eff z z Γ eff x z 2 , (9) where Γ ~ e f f x = Γ e f f x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − \widetilde{\Gamma}_{\mathrm{eff}}^x=\Gamma_{\mathrm{eff}}^x+\frac{\Gamma_{\mathrm{eff}}^+\Gamma_{\mathrm{eff}}^-}{\Gamma_{\mathrm{eff}}^++\Gamma_{\mathrm{eff}}^-} Γ eff x = Γ eff x + Γ eff + + Γ eff − Γ eff + Γ eff −
and ∥ ⋅ ∥ 2 \|\cdot\|_2 ∥ ⋅ ∥ 2
Theorem 1 (Separability of the effective dynamics). For the effective Lindbladian L e f f \mathcal{L}_{\mathrm{eff}} L eff (8) :
If η > 1 \eta>1 η > 1
If η ≤ 1 \eta\le1 η ≤ 1
The proof is given in App. B.5. For the GRW model considered here, note that all Γ e f f μ ν = 0 \Gamma_{\mathrm{eff}}^{\mu\nu}=0 Γ eff μν = 0 η x ≡ ∣ Γ e f f x x − λ e f f x x ∣ / Γ e f f x x = ∣ λ e f f x x ∣ / Γ e f f x x ≤ η \eta^x\equiv|\Gamma_{\mathrm{eff}}^{xx}-\lambda_{\mathrm{eff}}^{xx}|/\Gamma_{\mathrm{eff}}^{xx}=|\lambda_{\mathrm{eff}}^{xx}|/\Gamma_{\mathrm{eff}}^{xx}\le\eta η x ≡ ∣ Γ eff xx − λ eff xx ∣/ Γ eff xx = ∣ λ eff xx ∣/ Γ eff xx ≤ η
( λ e f f x x ) 2 + ( Γ e f f x x ) 2 > min ( Γ e f f x x + Γ e f f + , Γ e f f x x + Γ e f f − ) (10) \sqrt{\left(\lambda_{\mathrm{eff}}^{xx}\right)^2+\left(\Gamma_{\mathrm{eff}}^{xx}\right)^2}>\min\left(\Gamma_{\mathrm{eff}}^{xx}+\Gamma_{\mathrm{eff}}^{+},\Gamma_{\mathrm{eff}}^{xx}+\Gamma_{\mathrm{eff}}^{-}\right) \tag*{(10)} ( λ eff xx ) 2 + ( Γ eff xx ) 2 > min ( Γ eff xx + Γ eff + , Γ eff xx + Γ eff − ) (10) see Prop. 4.
Recoverability of entanglement under collapse noise.— We now analyze the scaling of the effective σ x \sigma^x σ x ∣ λ e f f x ∣ / Γ ˉ e f f x \lvert\lambda_{\mathrm{eff}}^{x}\rvert/\bar{\Gamma}_{\mathrm{eff}}^{x} ∣ λ eff x ∣ / Γ ˉ eff x η \eta η
For a fixed code distance N N N σ A x σ B x \sigma_A^x\sigma_B^x σ A x σ B x 4 N + 2 4N+2 4 N + 2
− G m A m B ∣ d + x A − x B ∣ = ⋯ + G m A m B ( 4 N + 2 ) ! d 4 N + 3 ( 2 N + 1 ) ! ( 2 N + 1 ) ! x A 2 N + 1 x B 2 N + 1 + ⋯ . -\frac{Gm_A m_B}{\lvert d+x_A-x_B\rvert}
=\cdots+\frac{Gm_A m_B(4N+2)!}{d^{4N+3}(2N+1)!(2N+1)!}x_A^{2N+1}x_B^{2N+1}+\cdots. − ∣ d + x A − x B ∣ G m A m B = ⋯ + d 4 N + 3 ( 2 N + 1 )! ( 2 N + 1 )! G m A m B ( 4 N + 2 )! x A 2 N + 1 x B 2 N + 1 + ⋯ . By contrast, the leading contribution of the GRW dynamics that induces logical σ x \sigma^x σ x 2 N + 2 2N+2 2 N + 2
L G R W ( ρ ) = ⋯ + λ G R W ( 2 N + 2 ) ! ( ( N + 1 ) ! ) 3 4 N + 1 r C 2 N + 2 x N + 1 ρ x N + 1 + ⋯ . \mathcal{L}_{\mathrm{GRW}}(\rho)=\cdots+\frac{\lambda_{\mathrm{GRW}}(2N+2)!}{((N+1)!)^3 4^{N+1}r_C^{2N+2}}x^{N+1}\rho x^{N+1}+\cdots. L GRW ( ρ ) = ⋯ + (( N + 1 )! ) 3 4 N + 1 r C 2 N + 2 λ GRW ( 2 N + 2 )! x N + 1 ρ x N + 1 + ⋯ . Using Stirling’s formula, this yields a rough estimate of the scaling
η x ∝ G m A m B ( 4 N + 2 ) ! x 0 4 N + 2 d 4 N + 3 ( 2 N + 1 ) ! λ G R W ( 2 N + 2 ) ! x 0 2 N + 2 ( 2 N + 1 ) ! [ ( N + 1 ) ! ] 3 4 N + 1 r C 2 N + 2 N ! ∝ N 2 ( 16 N 2 e 2 r C 2 x 0 2 d 4 ) N (11) \eta^x\propto\frac{\dfrac{Gm_A m_B(4N+2)!x_0^{4N+2}}{d^{4N+3}(2N+1)!}}{\dfrac{\lambda_{\mathrm{GRW}}(2N+2)!x_0^{2N+2}}{(2N+1)![(N+1)!]^3 4^{N+1}r_C^{2N+2}N!}}
\propto N^2\left(\frac{16N^2}{e^2}\frac{r_C^2x_0^2}{d^4}\right)^N \tag*{(11)} η x ∝ ( 2 N + 1 )! [( N + 1 )! ] 3 4 N + 1 r C 2 N + 2 N ! λ GRW ( 2 N + 2 )! x 0 2 N + 2 d 4 N + 3 ( 2 N + 1 )! G m A m B ( 4 N + 2 )! x 0 4 N + 2 ∝ N 2 ( e 2 16 N 2 d 4 r C 2 x 0 2 ) N (11) where x 0 = ℏ / 2 m ω x_0=\sqrt{\hbar/2m\omega} x 0 = ℏ/2 mω ⟨ 0 ∣ x 2 N + 1 ∣ 2 N + 1 ⟩ = x 0 2 N + 1 ( 2 N + 1 ) ! \langle0|x^{2N+1}|2N+1\rangle=x_0^{2N+1}\sqrt{(2N+1)!} ⟨ 0∣ x 2 N + 1 ∣2 N + 1 ⟩ = x 0 2 N + 1 ( 2 N + 1 )! ∣ ⟨ N ∣ x N + 1 ∣ 2 N + 1 ⟩ ∣ 2 = x 0 2 N + 2 ( 2 N + 1 ) ! / N ! \lvert\langle N|x^{N+1}|2N+1\rangle\rvert^2=x_0^{2N+2}(2N+1)!/N! ∣⟨ N ∣ x N + 1 ∣2 N + 1 ⟩ ∣ 2 = x 0 2 N + 2 ( 2 N + 1 )! / N ! ∣ λ e f f x x ∣ / Γ ˉ e f f x \lvert\lambda_{\mathrm{eff}}^{xx}\rvert/\bar{\Gamma}_{\mathrm{eff}}^{x} ∣ λ eff xx ∣ / Γ ˉ eff x N N N λ G R W \lambda_{\mathrm{GRW}} λ GRW r C r_C r C 16 r C 2 x 0 2 N 2 > e 2 d 4 16r_C^2x_0^2N^2>e^2d^4 16 r C 2 x 0 2 N 2 > e 2 d 4 η x > 1 \eta^x>1 η x > 1 η x \eta^x η x Theorem 5 in App. C5, for GRW/CSL we further show that this asymptotic advantage can be retained when x 0 x_0 x 0 d d d N N N η ≤ 2 \eta\leq\sqrt{2} η ≤ 2
In Figure 2 we plot the separability parameter η \eta η Γ e f f μ ν ≠ 0 \Gamma_{\mathrm{eff}}^{\mu\nu}\neq0 Γ eff μν = 0 (2) , construct the corresponding effective Lindbladian of the form in Eq. (8) , and extract numerically the effective interaction rate λ e f f μ ν \lambda_{\mathrm{eff}}^{\mu\nu} λ eff μν Γ e f f μ ν \Gamma_{\mathrm{eff}}^{\mu\nu} Γ eff μν Γ e f f μ ν \Gamma_{\mathrm{eff}}^{\mu\nu} Γ eff μν η \eta η
Figure 2. Separability parameter η \eta η N N N N = 0 N = 0 N = 0 N = 0 N = 0 N = 0 η > 1 \eta> 1 η > 1 η ≤ 1 \eta\le1 η ≤ 1
For the numerical analysis in Figure 2 we choose the following parameters: m A = m B = 10 − 21 k g m_A=m_B=10^{-21}\,\mathrm{kg} m A = m B = 1 0 − 21 kg x 0 = ℏ / 2 m ω = 2 × 10 − 8 m x_0=\sqrt{\hbar/2m\omega}=2\times10^{-8}\,\mathrm{m} x 0 = ℏ/2 mω = 2 × 1 0 − 8 m ρ ≃ 2.2 × 10 3 k g / m 3 \rho\simeq2.2\times10^3\,\mathrm{kg}/\mathrm{m}^3 ρ ≃ 2.2 × 1 0 3 kg / m 3 r 0 = ( 3 m A / 4 π ρ ) 1 / 3 ≃ 4.8 × 10 − 9 m r_0=(3m_A/4\pi\rho)^{1/3}\simeq4.8\times10^{-9}\,\mathrm{m} r 0 = ( 3 m A /4 π ρ ) 1/3 ≃ 4.8 × 1 0 − 9 m d = 4 × 10 − 7 m d=4\times10^{-7}\,\mathrm{m} d = 4 × 1 0 − 7 m r C = 2 × 10 − 6 m r_C=2\times10^{-6}\,\mathrm{m} r C = 2 × 1 0 − 6 m σ = 2 × 10 − 6 m \sigma=2\times10^{-6}\,\mathrm{m} σ = 2 × 1 0 − 6 m
For each set of parameters and each code distance N N N η \eta η
Notably, for the GRW/CSL and DP models the onset of non-separability does not require macroscopic masses. This contrasts with the case without QEC, where observing entanglement under increasingly strong collapse noise requires correspondingly larger particle sizes, see App. A3. Our results highlight an important distinction between non-separability and the practical observability of entanglement. Collapse noise can completely suppress entanglement generation for commonly considered spatial-superposition states without rendering the underlying dynamics separable. For suitably chosen code states, entanglement remains nonzero for arbitrary collapse strength, but in the absence of error correction both its magnitude and lifetime can become arbitrarily small, making it effectively inaccessible.
The role of local QEC is therefore not to make an otherwise separable dynamics non-separable, but to render this latent entangling resource observable. For GRW and CSL noise, error correction suppresses decoherence relative to the gravitational interaction, allowing the generated entanglement to reach larger values and persist for longer times within the code subspace. In this sense, QEC makes otherwise inaccessible gravitational entanglement locally recoverable. By contrast, the DP model shows no comparable improvement under the bipartite QEC protocol. Nevertheless, the DP model remains non-separable already at zeroth order, in accordance with Ref. [34 ] .
Discussion and Conclusions. — Our results show that collapse-induced decoherence need not render Newtonian-plus-noise dynamics separable even when observable entanglement is strongly suppressed. Local QEC reveals a further distinction: for GRW and CSL, the effective entangling scale is suppressed more slowly than the relevant logical noise, rendering the entangling resource locally recoverable. By contrast, the DP model shows no comparable enhancement under bipartite QEC protocol because its interaction and dominant noise arise from the same kernel and scale in parallel. Recoverability therefore refines the separable/non-separable classification of noisy gravitational dynamics. Related QEC ideas have also been explored in quantum metrology [45 , 46 , 47 , 48 ] .
The specific structure of the decoherence channel plays a central role in determining recoverability. In the collapse models studied in the main text, both the interaction and the dominant noise act through position operators, making local discrimination between signal and noise particularly challenging. In contrast, other physically relevant scenarios, including the Anastopoulos–Hu model and several forms of environmental decoherence, involve momentum-type noise generated by operators distinct from those responsible for the gravitational interaction. As shown in App. D, such noise can be suppressed by local QEC without simultaneously reducing the coherent gravitational coupling. This demonstrates that the limitations encountered for position-type noise are not generic, but rather arise from the particular operator structure shared by the interaction and decoherence. Nevertheless, position-type noise is a generic feature of universal collapse models [49 ] . It is also noteworthy that, above, we have exploited local imperfect error-correction; if one allows non-local imperfect error-correction, the interaction-to-noise ratio can be made arbitrarily large even at leading order, see App. B6.
The present work is not intended as a near-term experimental proposal. Increasing the code distance generally suppresses not only the effective noise but also the effective interaction strength, and the resulting recovery procedures may therefore be impractical for realistic implementations even when the required motional states can be prepared. Nevertheless, our results establish a more fundamental point: collapse-induced decoherence does not necessarily destroy the underlying entangling resource. For physically relevant models such as GRW and CSL, this resource remains locally recoverable in principle. More broadly, our results suggest that recoverability may provide a useful organizing principle for noisy gravitational dynamics, complementing the traditional separable/non-separable classification and motivating the search for more efficient local protocols capable of exploiting this recoverability in future experiments. A possible route would be to relax the requirement that the QEC protocol recover the ideal, error-free dynamics, and instead consider error-filtering protocols (possibly probabilistic) designed to produce noisy yet entangling dynamics within the code space.
Acknowledgments. — We thank Kirill Streltsov for useful discussions. OpenAI’s GPT-5.6 Sol was used to assist in developing and checking the proof of Theorem 5 in Appendix C5. The resulting proof was formulated and independently verified by the authors, who take the full responsibility for its correctness. This work was supported by the ERC Synergy Grant HyperQ (Grant No. 856432), the BMBF via project PhoQuant (Grant No. 13N16110) and the EU project SPINUS (Grant No. 101135699).
References
Overview of the decoherence models
This appendix summarizes the collapse and decoherence models considered throughout the paper and collects the corresponding master equations for reference. Since the analysis presented in the main text and subsequent appendices applies the same quantum error-correction framework to several different physical mechanisms, we provide here a concise overview of their defining features and effective dynamics.
Fundamental decoherence models
We begin by reviewing the fundamental collapse and decoherence models that will be analyzed throughout the paper. For each model, we summarize the underlying physical motivation and present the corresponding master equation used in our calculations.
Ghirardi–Rimini–Weber (GRW) Model
The GRW model provides the simplest phenomenological realization of spontaneous wavefunction collapse through random localization events acting independently on each constituent particle [29 ] . In this model, the decoherence rate is negligible for small spatial superpositions and approaches a constant for sufficiently large superposition sizes. A key feature of the model is the amplification mechanism, whereby the effective decoherence rate increases with the mass (or number of constituents) of the system. As a consequence, standard Schrödinger dynamics is recovered for microscopic systems and small superpositions, while macroscopic superpositions and separations are strongly suppressed. At the level of pure-state dynamics, the GRW modification introduces stochastic and nonlinear corrections to the Schrödinger equation in the form of random, instantaneous localization events that interrupt the otherwise unitary evolution; however, the resulting ensemble-averaged evolution of the density matrix remains linear and can be expressed in Lindblad form. For a single particle, the corresponding one-dimensional GRW decoherence is given by the following Lindbladian, which we simplify in the position representation as follows:
L G R W ( ρ ) = − λ G R W ( ρ − ∫ d q ( 1 π r C 2 ) 1 2 e − ( x − q ) 2 2 r C 2 ρ e − ( x − q ) 2 2 r C 2 ) = − λ G R W ( ρ − ∫ d q d x 1 d x 2 ( 1 π r C 2 ) 1 2 e − ( x 1 − q ) 2 2 r C 2 ∣ x 1 ⟩ ⟨ x 1 ∣ ρ ∣ x 2 ⟩ ⟨ x 2 ∣ e − ( x 2 − q ) 2 2 r C 2 ) = − λ G R W ( ρ − ∫ d x 1 d x 2 e − ( x 1 − x 2 ) 2 4 r C 2 ∣ x 1 ⟩ ⟨ x 1 ∣ ρ ∣ x 2 ⟩ ⟨ x 2 ∣ ) . (12) \begin{aligned}
\mathcal{L}_{\mathrm{GRW}}(\rho)=-\lambda_{\mathrm{GRW}}\left(\rho-\int dq\left(\frac{1}{\pi r_{\mathrm{C}}^2}\right)^{\frac{1}{2}}e^{-\frac{(x-q)^2}{2r_{\mathrm{C}}^2}}\rho e^{-\frac{(x-q)^2}{2r_{\mathrm{C}}^2}}\right) \\
&=-\lambda_{\mathrm{GRW}}\left(\rho-\int dqdx_1dx_2\left(\frac{1}{\pi r_{\mathrm{C}}^2}\right)^{\frac{1}{2}}e^{-\frac{(x_1-q)^2}{2r_{\mathrm{C}}^2}}|x_1\rangle\langle x_1|\rho|x_2\rangle\langle x_2|e^{-\frac{(x_2-q)^2}{2r_{\mathrm{C}}^2}}\right) \\
&=-\lambda_{\mathrm{GRW}}\left(\rho-\int dx_1dx_2e^{-\frac{(x_1-x_2)^2}{4r_{\mathrm{C}}^2}}|x_1\rangle\langle x_1|\rho|x_2\rangle\langle x_2|\right). \tag*{(12)}
\end{aligned} L GRW ( ρ ) = − λ GRW ( ρ − ∫ d q ( π r C 2 1 ) 2 1 e − 2 r C 2 ( x − q ) 2 ρ e − 2 r C 2 ( x − q ) 2 ) = − λ GRW ( ρ − ∫ d q d x 1 d x 2 ( π r C 2 1 ) 2 1 e − 2 r C 2 ( x 1 − q ) 2 ∣ x 1 ⟩ ⟨ x 1 ∣ ρ ∣ x 2 ⟩ ⟨ x 2 ∣ e − 2 r C 2 ( x 2 − q ) 2 ) = − λ GRW ( ρ − ∫ d x 1 d x 2 e − 4 r C 2 ( x 1 − x 2 ) 2 ∣ x 1 ⟩ ⟨ x 1 ∣ ρ ∣ x 2 ⟩ ⟨ x 2 ∣ ) . (12) The model depends on two phenomenological parameters, λ G R W \lambda_{\mathrm{GRW}} λ GRW r C r_{\mathrm{C}} r C Tr ( ρ ∣ x 1 ⟩ ⟨ x 2 ∣ ) \operatorname{Tr}(\rho|x_1\rangle\langle x_2|) Tr ( ρ ∣ x 1 ⟩ ⟨ x 2 ∣ ) λ G R W [ 1 − exp ( − ( x 1 − x 2 ) 2 / 4 r C 2 ) ] \lambda_{\mathrm{GRW}}\left[1-\exp\left(-(x_1-x_2)^2/4r_{\mathrm{C}}^2\right)\right] λ GRW [ 1 − exp ( − ( x 1 − x 2 ) 2 /4 r C 2 ) ]
For a body composed of many constituent particles, we focus on the center-of-mass (COM) position X ^ C M \hat{X}_{\mathrm{CM}} X ^ CM x ^ i = X ^ C M + r ^ i \hat{x}_i=\hat{X}_{\mathrm{CM}}+\hat{r}_i x ^ i = X ^ CM + r ^ i r ^ i \hat{r}_i r ^ i r ^ i ≈ r i \hat{r}_i\approx r_i r ^ i ≈ r i X ^ C M \hat{X}_{\mathrm{CM}} X ^ CM n n n
L G R W ( ρ ) = n × λ G R W ( ∫ d 3 x 1 d 3 x 2 e − ∣ x 1 − x 2 ∣ 2 4 r C 2 ∣ x 1 ⟩ ⟨ x 1 ∣ ρ ∣ x 2 ⟩ ⟨ x 2 ∣ − ρ ) . (13) \mathcal{L}_{\mathrm{GRW}}(\rho)=n\times\lambda_{\mathrm{GRW}}\left(\int d^3x_1d^3x_2e^{-\frac{|\mathbf{x}_1-\mathbf{x}_2|^2}{4r_{\mathrm{C}}^2}}|x_1\rangle\langle x_1|\rho|x_2\rangle\langle x_2|-\rho\right). \tag*{(13)} L GRW ( ρ ) = n × λ GRW ( ∫ d 3 x 1 d 3 x 2 e − 4 r C 2 ∣ x 1 − x 2 ∣ 2 ∣ x 1 ⟩ ⟨ x 1 ∣ ρ ∣ x 2 ⟩ ⟨ x 2 ∣ − ρ ) . (13) Generally, we take n n n
Continuous Spontaneous Localization (CSL) Model
The Continuous Spontaneous Localization (CSL) model provides a continuous-in-time realization of the GRW collapse mechanism and constitutes a widely used phenomenological framework for spontaneous wavefunction collapse [30 , 31 ] . As in the GRW model, CSL introduces stochastic and non-linear modifications of the Schrödinger equation at the level of pure states, while the corresponding ensemble-averaged evolution of the density matrix remains linear and completely positive. The key distinction with respect to GRW is that, in CSL, localization occurs continuously in time rather than through discrete and instantaneous collapse events.
This continuous dynamics is commonly modeled by coupling the system to a postulated classical stochastic field, which acts as a fundamental source of noise and drives the collapse process even in the absence of environmental interactions. In contrast to the GRW model, where localization events act independently on individual particles, in CSL all constituents couple to the same stochastic field. As a result, the collapse noise can be spatially correlated over length scales set by the characteristic localization length r C r_{\mathrm{C}} r C
The CSL dynamics is most naturally formulated in terms of the mass density operator,
M ( x ) = ∑ j , s m j a j † ( x , s ) a j ( x , s ) , (14) M(x)=\sum_{j,s}m_j a^\dagger_{j}(x,s)a_j(x,s), \tag*{(14)} M ( x ) = j , s ∑ m j a j † ( x , s ) a j ( x , s ) , (14) where a j † ( x , s ) a^\dagger_j(x,s) a j † ( x , s ) m j m_j m j s s s x x x
L C S L ( ρ ) = − λ C S L 2 m 0 2 ∫ d 3 x d 3 y g ( x − y ) [ M ( x ) , [ M ( y ) , ρ ] ] . (15) \mathcal{L}_{\mathrm{CSL}}(\rho)=-\frac{\lambda_{\mathrm{CSL}}}{2m_0^2}\int d^3x\,d^3y\,g(x-y)[M(x),[M(y),\rho]]. \tag*{(15)} L CSL ( ρ ) = − 2 m 0 2 λ CSL ∫ d 3 x d 3 y g ( x − y ) [ M ( x ) , [ M ( y ) , ρ ]] . (15) where λ C S L \lambda_{\mathrm{CSL}} λ CSL m 0 m_0 m 0
g ( x − y ) = exp ( − ∣ x − y ∣ 2 4 r C 2 ) (16) g(\mathbf{x}-\mathbf{y})=\exp\left(-\frac{|\mathbf{x}-\mathbf{y}|^2}{4r_{\mathrm{C}}^2}\right) \tag*{(16)} g ( x − y ) = exp ( − 4 r C 2 ∣ x − y ∣ 2 ) (16) is a spatial correlation function with characteristic localization length r C r_{\mathrm{C}} r C
x ^ i = X ^ C M + r ^ i , (17) \hat{\mathbf{x}}_i=\hat{\mathbf{X}}_{\mathrm{CM}}+\hat{\mathbf{r}}_i, \tag*{(17)} x ^ i = X ^ CM + r ^ i , (17) we assume that the internal coordinates remain sharply localized around their equilibrium values, such that r ^ i ≈ r i \hat{\mathbf{r}}_i\approx\mathbf{r}_i r ^ i ≈ r i
M ( x ) = ∑ i m i δ ( x − X ^ C M − r i ) , (18) M(\mathbf{x})=\sum_i m_i\delta\left(\mathbf{x}-\hat{\mathbf{X}}_{\mathrm{CM}}-\mathbf{r}_i\right), \tag*{(18)} M ( x ) = i ∑ m i δ ( x − X ^ CM − r i ) , (18) or, equivalently, M ( x ) = μ ( x − X ^ C M ) M(\mathbf{x})=\mu(\mathbf{x}-\hat{\mathbf{X}}_{\mathrm{CM}}) M ( x ) = μ ( x − X ^ CM ) μ \mu μ
Unlike the GRW model, where localization events act independently on each constituent particle, all constituents in CSL couple to the same stochastic field. As a consequence, the collapse noise can be spatially correlated over distances of order r C r_{\mathrm{C}} r C Δ X \Delta\mathbf{X} Δ X
Γ C S L ( Δ X ) = λ C S L m 0 2 ( 4 π r C 2 ) 3 / 2 ∫ d 3 k ( 2 π ) 3 e − r C 2 k 2 ∣ μ ~ ( k ) ∣ 2 [ 1 − cos ( k ⋅ Δ X ) ] , (19) \Gamma_{\mathrm{CSL}}(\Delta\mathbf{X})=\frac{\lambda_{\mathrm{CSL}}}{m_0^2}(4\pi r_{\mathrm{C}}^2)^{3/2}\int\frac{\mathrm{d}^3\mathbf{k}}{(2\pi)^3}e^{-r_{\mathrm{C}}^2 k^2}|\tilde{\mu}(\mathbf{k})|^2\left[1-\cos(\mathbf{k}\cdot\Delta\mathbf{X})\right], \tag*{(19)} Γ CSL ( Δ X ) = m 0 2 λ CSL ( 4 π r C 2 ) 3/2 ∫ ( 2 π ) 3 d 3 k e − r C 2 k 2 ∣ μ ~ ( k ) ∣ 2 [ 1 − cos ( k ⋅ Δ X ) ] , (19) where μ ~ ( k ) \tilde{\mu}(\mathbf{k}) μ ~ ( k ) ∣ μ ~ ( k ) ∣ 2 |\tilde{\mu}(\mathbf{k})|^2 ∣ μ ~ ( k ) ∣ 2 r C r_{\mathrm{C}} r C μ ~ ( k ) ≃ M \tilde{\mu}(\mathbf{k})\simeq M μ ~ ( k ) ≃ M
Γ C S L ∝ λ C S L M 2 m 0 2 , (20) \Gamma_{\mathrm{CSL}}\propto\lambda_{\mathrm{CSL}}\frac{M^2}{m_0^2}, \tag*{(20)} Γ CSL ∝ λ CSL m 0 2 M 2 , (20) which is quadratic in the total mass. For larger objects, the amplification is reduced because different regions of the body no longer experience the same realization of the collapse noise.
The same spatial correlations imply that two distinct rigid bodies can experience correlated noise whenever their separation is comparable to or smaller than r C r_{\mathrm{C}} r C
Diósi–Penrose (DP) Model
The DP model attributes decoherence to fluctuations associated with the gravitational field itself, leading simultaneously to gravitational interactions and gravity-induced decoherence [39 , 40 ] . In this framework, quantum matter probes the gravitational field, and the uncertainty principle motivates an intrinsic uncertainty in the spacetime metric experienced by a delocalized mass distribution. This can be modeled as a stochastic perturbation of the metric, which in turn induces effective decoherence of the matter degrees of freedom.
As a representative setting, consider two masses m A m_A m A m B m_B m B
L D P ( ρ ) = i G ℏ ∫ d 3 x d 3 y M ( x ) M ( y ) 2 ∣ x − y ∣ , ρ + G ℏ ∫ d 3 x d 3 y ∣ x − y ∣ ( M ( x ) ρ M ( y ) − 1 2 { M ( x ) M ( y ) , ρ } ) . (21) \mathcal{L}_{\mathrm{DP}}(\rho)=\frac{iG}{\hbar}\int\mathrm{d}^3\mathbf{x}\mathrm{d}^3\mathbf{y}\frac{M(\mathbf{x})M(\mathbf{y})}{2|\mathbf{x}-\mathbf{y}|},\rho+\frac{G}{\hbar}\int\frac{\mathrm{d}^3\mathbf{x}\mathrm{d}^3\mathbf{y}}{|\mathbf{x}-\mathbf{y}|}\left(M(\mathbf{x})\rho M(\mathbf{y})-\frac{1}{2}\{M(\mathbf{x})M(\mathbf{y}),\rho\}\right). \tag*{(21)} L DP ( ρ ) = ℏ i G ∫ d 3 x d 3 y 2∣ x − y ∣ M ( x ) M ( y ) , ρ + ℏ G ∫ ∣ x − y ∣ d 3 x d 3 y ( M ( x ) ρM ( y ) − 2 1 { M ( x ) M ( y ) , ρ } ) . (21) Here M ( x ) = ∑ α = A , B m α μ σ ( x − x α ) M(\mathbf{x})=\sum_{\alpha=A,B}m_\alpha\mu_\sigma(\mathbf{x}-\mathbf{x}_\alpha) M ( x ) = ∑ α = A , B m α μ σ ( x − x α ) x α \mathbf{x}_\alpha x α m α m_\alpha m α
μ σ ( x − x α ) = 1 ( 2 π σ 2 ) 3 / 2 exp ( − ∣ x − x α ∣ 2 2 σ 2 ) , (22) \mu_\sigma(\mathbf{x}-\mathbf{x}_\alpha)=\frac{1}{(2\pi\sigma^2)^{3/2}}\exp\left(-\frac{|\mathbf{x}-\mathbf{x}_\alpha|^2}{2\sigma^2}\right), \tag*{(22)} μ σ ( x − x α ) = ( 2 π σ 2 ) 3/2 1 exp ( − 2 σ 2 ∣ x − x α ∣ 2 ) , (22) where σ \sigma σ
Note that the DP model can also be obtained as an effective dynamics describing interactions between quantum systems mediated by a classical gravitational field [50 ] .
As in the CSL model, the DP generator is nonlocal in space, as it couples mass-density operators evaluated at different positions through a two-point kernel. In fact, the mathematical structure of the DP model is closely related to that of CSL. For example, after introducing the smeared mass density, the DP decoherence rate associated with a rigid body can be expressed in Fourier space as
Γ D P ( Δ X ) ∝ G ∫ d 3 k ( 2 π ) 3 e − σ 2 k 2 k 2 ∣ μ ~ ( k ) ∣ 2 [ 1 − cos ( k ⋅ Δ X ) ] , (23) \Gamma_{\mathrm{DP}}(\Delta X) \propto G \int\frac{d^{3}k}{(2\pi)^{3}} \frac{e^{-\sigma^{2}k^{2}}}{k^{2}} \lvert\tilde{\mu}(k)\rvert^{2} [1-\cos(k\cdot\Delta X)], \tag*{(23)} Γ DP ( Δ X ) ∝ G ∫ ( 2 π ) 3 d 3 k k 2 e − σ 2 k 2 ∣ μ ~ ( k ) ∣ 2 [ 1 − cos ( k ⋅ Δ X )] , (23) where μ ~ ( k ) \tilde{\mu}(k) μ ~ ( k ) (19) . In both cases, the decoherence rate is governed by the form factor ∣ μ ~ ( k ) ∣ 2 \lvert\tilde{\mu}(k)\rvert^{2} ∣ μ ~ ( k ) ∣ 2 r C r_{\mathrm{C}} r C σ \sigma σ μ ~ ( k ) ≃ M \tilde{\mu}(k) \simeq M μ ~ ( k ) ≃ M M 2 M^{2} M 2 e − σ 2 k 2 e^{-\sigma^{2}k^{2}} e − σ 2 k 2 μ ~ ( k ) \tilde{\mu}(k) μ ~ ( k ) M M M ∣ μ ~ ( k ) ∣ 2 \lvert\tilde{\mu}(k)\rvert^{2} ∣ μ ~ ( k ) ∣ 2
Despite these similarities, there are important differences between the DP and the CSL model. In CSL, the Gaussian kernel itself determines the spatial correlations of the collapse noise, which become exponentially suppressed beyond the length scale r C r_{\mathrm{C}} r C 1 / ∣ x − y ∣ 1/\lvert x-y\rvert 1/ ∣ x − y ∣ e − σ 2 k 2 e^{-\sigma^{2}k^{2}} e − σ 2 k 2 1 / r 1/r 1/ r erf ( r / 2 σ ) r \frac{\operatorname{erf}(r/2\sigma)}{r} r erf ( r /2 σ ) r ≫ σ r\gg\sigma r ≫ σ
The introduction of the smearing length σ \sigma σ σ \sigma σ σ \sigma σ σ \sigma σ σ \sigma σ σ \sigma σ
A final distinction from CSL, which will play a central role in the QEC analysis of App. C.3, is that the same Newtonian kernel generates both the decoherence and the coherent gravitational interaction. As a consequence, the interaction and the dominant error channels remain intrinsically linked, in contrast to CSL where the collapse noise and gravitational interaction originate from distinct kernels.
Kafri–Taylor–Milburn (KTM) Model
The KTM model provides a representative example of a classical mediator theory in which Newtonian interactions arise from local measurements and classical feedback and are therefore separable by construction [18 ] . In this framework, the overall dynamics is LOCC (and hence separable) by construction.
To state the general structure, consider a bipartite system with subsystems A A A B B B A A A M A M_{A} M A B B B F B F_{B} F B [51 ]
d ρ d t = − i 2 [ M A † F B + M A F B , ρ ] + D [ M A − i F B ] ( ρ ) , (24) \frac{d\rho}{dt}=-\frac{i}{2}[M_{A}^{\dagger}F_{B}+M_{A}F_{B},\rho]+\mathcal{D}[M_{A}-iF_{B}](\rho), \tag*{(24)} d t d ρ = − 2 i [ M A † F B + M A F B , ρ ] + D [ M A − i F B ] ( ρ ) , (24) where D [ A ] ( ρ ) = A ρ A † − { A † A / 2 , ρ } \mathcal{D}[A](\rho)=A\rho A^{\dagger}-\{A^{\dagger}A/2,\rho\} D [ A ] ( ρ ) = A ρ A † − { A † A /2 , ρ } O A O_{A} O A O B O_{B} O B H A \mathcal{H}_{A} H A H B \mathcal{H}_{B} H B
d ρ d t = − i [ a A O A ⊗ a B O B , ρ ] + D [ a A O A − i a B O B ] ( ρ ) − i [ b A O A ⊗ b B O B , ρ ] + D [ b B O B − i b A O A ] ( ρ ) = − i ( a A a B + b A b B ) [ O A ⊗ O B , ρ ] + ( a A 2 + b A 2 ) D [ O A ] ( ρ ) + ( a B 2 + b B 2 ) D [ O B ] ( ρ ) + i ( a A a B − b A b B ) ( O A ρ O B − O B ρ O A ) . (25) \begin{aligned}
\frac{d\rho}{dt}={}&-i[a_{A}O_{A}\otimes a_{B}O_{B},\rho]+\mathcal{D}[a_{A}O_{A}-ia_{B}O_{B}](\rho)-i[b_{A}O_{A}\otimes b_{B}O_{B},\rho]+\mathcal{D}[b_{B}O_{B}-ib_{A}O_{A}](\rho)\\
={}&-i(a_{A}a_{B}+b_{A}b_{B})[O_{A}\otimes O_{B},\rho]+(a_{A}^{2}+b_{A}^{2})\mathcal{D}[O_{A}](\rho)+(a_{B}^{2}+b_{B}^{2})\mathcal{D}[O_{B}](\rho)+i(a_{A}a_{B}-b_{A}b_{B})(O_{A}\rho O_{B}-O_{B}\rho O_{A}). \tag*{(25)}
\end{aligned} d t d ρ = = − i [ a A O A ⊗ a B O B , ρ ] + D [ a A O A − i a B O B ] ( ρ ) − i [ b A O A ⊗ b B O B , ρ ] + D [ b B O B − i b A O A ] ( ρ ) − i ( a A a B + b A b B ) [ O A ⊗ O B , ρ ] + ( a A 2 + b A 2 ) D [ O A ] ( ρ ) + ( a B 2 + b B 2 ) D [ O B ] ( ρ ) + i ( a A a B − b A b B ) ( O A ρ O B − O B ρ O A ) . (25) When the products of measurement and feedback rates are symmetric, i.e. a A a B = b A b B a_A a_B = b_A b_B a A a B = b A b B
( a A 2 + b A 2 ) ( a B 2 + b B 2 ) ≥ ( a A 2 a B 2 + b A 2 b B 2 ) = a A a B + b A b B . (26) \sqrt{(a_A^2+b_A^2)(a_B^2+b_B^2)} \geq\left(\sqrt{a_A^2a_B^2}+\sqrt{b_A^2b_B^2}\right)=a_Aa_B+b_Ab_B. \tag*{(26)} ( a A 2 + b A 2 ) ( a B 2 + b B 2 ) ≥ ( a A 2 a B 2 + b A 2 b B 2 ) = a A a B + b A b B . (26) In this way, a Newtonian interaction can be synthesized in an LOCC fashion. By setting O A = x A O_A=x_A O A = x A O B = x B O_B=x_B O B = x B a A = b B = Γ / ℏ a_A=b_B=\sqrt{\Gamma/\hbar} a A = b B = Γ/ℏ a B = b A = G m A m B / ( d 3 Γ ℏ ) a_B=b_A=Gm_Am_B/(d^3\sqrt{\Gamma\hbar}) a B = b A = G m A m B / ( d 3 Γℏ )
L K T M ( ρ ) = − 2 i G m A m B ℏ d 3 [ x A x B , ρ ] + ( Γ ℏ + G 2 m A 2 m B 2 Γ ℏ d 6 ) ( D [ x A ] + D [ x B ] ) ( ρ ) , (27) \mathcal{L}_{\mathrm{KTM}}(\rho)=-\frac{2iGm_Am_B}{\hbar d^3}[x_Ax_B,\rho]+\left(\frac{\Gamma}{\hbar}+\frac{G^2m_A^2m_B^2}{\Gamma\hbar d^6}\right)(\mathcal{D}[x_A]+\mathcal{D}[x_B])(\rho), \tag*{(27)} L KTM ( ρ ) = − ℏ d 3 2 i G m A m B [ x A x B , ρ ] + ( ℏ Γ + Γℏ d 6 G 2 m A 2 m B 2 ) ( D [ x A ] + D [ x B ]) ( ρ ) , (27) which is LOCC and hence separable by construction. The local decoherence rate is minimized when Γ / ℏ = G 2 m A 2 m B 2 / Γ ℏ d 6 \Gamma/\hbar=G^2m_A^2m_B^2/\Gamma\hbar d^6 Γ/ℏ = G 2 m A 2 m B 2 /Γℏ d 6 2 G m A m B / ℏ d 3 2Gm_Am_B/\hbar d^3 2 G m A m B /ℏ d 3
− G m A m B ∣ d + x A − x B ∣ = − G m A m B d + G m A m B ( x A − x B ) d 2 − G m A m B ( x A 2 + x B 2 − 2 x A x B ) d 3 + ⋯ . -\frac{Gm_Am_B}{\lvert d+x_A-x_B\rvert}=-\frac{Gm_Am_B}{d}+\frac{Gm_Am_B(x_A-x_B)}{d^2}-\frac{Gm_Am_B(x_A^2+x_B^2-2x_Ax_B)}{d^3}+\cdots. − ∣ d + x A − x B ∣ G m A m B = − d G m A m B + d 2 G m A m B ( x A − x B ) − d 3 G m A m B ( x A 2 + x B 2 − 2 x A x B ) + ⋯ . At the same time, the accompanying local x A , x B x_A,x_B x A , x B
Anastopoulos–Hu Model
The Anastopoulos–Hu model describes decoherence induced by fluctuations of the quantized gravitational field and predicts a qualitatively different momentum-basis decoherence mechanism [38 ] . In this framework, quantum matter is coupled to linearized general relativity around Minkowski spacetime (derived from the Einstein–Hilbert action in the weak-field regime). Assuming the gravitons are in a Gaussian thermal state characterized by an effective “temperature” Θ \Theta Θ s − 1 \mathrm{s}^{-1} s − 1
For a single particle in the non-relativistic limit in one dimension, ∣ p ∣ ≪ m c \lvert p\rvert\ll mc ∣ p ∣ ≪ m c
L A H ( ρ ) = − 4 π G Θ 9 ℏ m 2 c 5 [ p 2 , [ p 2 , ρ ] ] . (28) \mathcal{L}_{\mathrm{AH}}(\rho)=-\frac{4\pi G\Theta}{9\hbar m^2c^5}[p^2,[p^2,\rho]]. \tag*{(28)} L AH ( ρ ) = − 9ℏ m 2 c 5 4 π G Θ [ p 2 , [ p 2 , ρ ]] . (28) This term acts in the momentum basis, in contrast to the position-basis decoherence mechanisms discussed above and below.
Environmental decoherence
In addition to the fundamental mechanisms discussed above, realistic experiments are subject to a variety of environmental decoherence processes. Many of these mechanisms lead to effective master equations of a form closely related to the GRW model, which motivates the focus on GRW-type dynamics throughout the paper. For instance, a levitated mass in an optical cavity is never in a perfect vacuum: residual gas molecules can collide with the object and impart momentum kicks. Moreover, the cavity contains thermal photons (e.g., from black-body radiation), and the levitated particle may scatter, emit, or absorb such photons [37 ] .
For a particle prepared in a spatial superposition, let ρ \rho ρ
d d t ⟨ x ∣ ρ ∣ y ⟩ = − Γ ( 1 − exp ( − ( x − y ) 2 4 a 2 ) ) ⟨ x ∣ ρ ∣ y ⟩ , (29) \frac{\mathrm{d}}{\mathrm{d}t}\langle x\lvert\rho\lvert y\rangle=-\Gamma\left(1-\exp\left(-\frac{(x-y)^2}{4a^2}\right)\right)\langle x\lvert\rho\lvert y\rangle, \tag*{(29)} d t d ⟨ x ∣ ρ ∣ y ⟩ = − Γ ( 1 − exp ( − 4 a 2 ( x − y ) 2 ) ) ⟨ x ∣ ρ ∣ y ⟩ , (29) where the decoherence rate Γ \Gamma Γ a a a
Finally, note that this functional form is closely analogous to the GRW model discussed above. For this reason, in the following discussions we focus on the GRW model; similar arguments apply to the environmental case.
Gravity-induced entanglement with spatially-superposed states
Before proceeding, let us revisit the gravity-induced entanglement tests in the presence of collapse noises. We will consider the BMV protocol [8 , 9 ] in the presence of collapse noise, and show that the attainable entanglement and the lifetime of entanglement are restricted by collapse noises. In the case of GRW noise, one may need to increase the mass macroscopically to observe entanglement.
Without loss of generality, we consider two masses A A A B B B m A = m B = m m_A=m_B=m m A = m B = m ∣ x 1 ⟩ \lvert x_1\rangle ∣ x 1 ⟩ ∣ x 2 ⟩ \lvert x_2\rangle ∣ x 2 ⟩ d d d Δ x = ∣ x 2 − x 1 ∣ \Delta x=\lvert x_2-x_1\rvert Δ x = ∣ x 2 − x 1 ∣
x A , 1 = − Δ x 2 , x A , 2 = Δ x 2 , x B , 1 = d − Δ x 2 , x B , 2 = d + Δ x 2 . (30) x_{A,1}=-\frac{\Delta x}{2},\qquad x_{A,2}=\frac{\Delta x}{2},\qquad x_{B,1}=d-\frac{\Delta x}{2},\qquad x_{B,2}=d+\frac{\Delta x}{2}. \tag*{(30)} x A , 1 = − 2 Δ x , x A , 2 = 2 Δ x , x B , 1 = d − 2 Δ x , x B , 2 = d + 2 Δ x . (30) We assume d > Δ x d>\Delta x d > Δ x σ A z ∣ x 1 ⟩ A = ∣ x 1 ⟩ A \sigma_A^z\lvert x_1\rangle_A=\lvert x_1\rangle_A σ A z ∣ x 1 ⟩ A = ∣ x 1 ⟩ A σ A z ∣ x 2 ⟩ A = − ∣ x 2 ⟩ A \sigma_A^z\lvert x_2\rangle_A=-\lvert x_2\rangle_A σ A z ∣ x 2 ⟩ A = − ∣ x 2 ⟩ A B B B
H N G ℏ = E 0 + E A σ A z + E B σ B z + λ N G σ A z σ B z , (31) \frac{H_{\mathrm{NG}}}{\hbar}=E_0+E_A\sigma_A^z+E_B\sigma_B^z+\lambda_{\mathrm{NG}}\sigma_A^z\sigma_B^z, \tag*{(31)} ℏ H NG = E 0 + E A σ A z + E B σ B z + λ NG σ A z σ B z , (31) where
λ N G = G m A m B 4 ℏ ( 1 d + Δ x + 1 d − Δ x − 2 d ) = G m A m B Δ x 2 2 ℏ d ( d 2 − Δ x 2 ) . (32) \lambda_{\mathrm{NG}}=\frac{Gm_A m_B}{4\hbar}\left(\frac{1}{d+\Delta x}+\frac{1}{d-\Delta x}-\frac{2}{d}\right)=\frac{Gm_A m_B\Delta x^2}{2\hbar d(d^2-\Delta x^2)}. \tag*{(32)} λ NG = 4ℏ G m A m B ( d + Δ x 1 + d − Δ x 1 − d 2 ) = 2ℏ d ( d 2 − Δ x 2 ) G m A m B Δ x 2 . (32) The GRW collapse induces pure dephasing in the position basis. For mass α = A , B \alpha=A,B α = A , B ∣ x 1 ⟩ α \lvert x_1\rangle_\alpha ∣ x 1 ⟩ α ∣ x 2 ⟩ α \lvert x_2\rangle_\alpha ∣ x 2 ⟩ α
Γ α d e p = λ G R W n [ 1 − exp ( − Δ x 2 4 r C 2 ) ] . (33) \Gamma_\alpha^{\mathrm{dep}}=\lambda_{\mathrm{GRW}}n\left[1-\exp\left(-\frac{\Delta x^2}{4r_C^2}\right)\right]. \tag*{(33)} Γ α dep = λ GRW n [ 1 − exp ( − 4 r C 2 Δ x 2 ) ] . (33) Dropping the irrelevant global energy and local Hamiltonian terms, the effective two-qubit Lindbladian is then
d d t ρ = − i λ N G [ σ A z σ B z , ρ ] + Γ D [ σ A z ] ( ρ ) + Γ D [ σ B z ] ( ρ ) , (34) \frac{\mathrm{d}}{\mathrm{d}t}\rho=-i\lambda_{\mathrm{NG}}[\sigma_A^z\sigma_B^z,\rho]+\Gamma\mathcal{D}[\sigma_A^z](\rho)+\Gamma\mathcal{D}[\sigma_B^z](\rho), \tag*{(34)} d t d ρ = − i λ NG [ σ A z σ B z , ρ ] + Γ D [ σ A z ] ( ρ ) + Γ D [ σ B z ] ( ρ ) , (34) where D [ O ] ( ρ ) = O ρ O † − 1 2 { O † O , ρ } \mathcal{D}[O](\rho)=O\rho O^\dagger-\frac{1}{2}\{O^\dagger O,\rho\} D [ O ] ( ρ ) = O ρ O † − 2 1 { O † O , ρ } Γ = Γ A d e p / 2 = Γ B d e p / 2 \Gamma=\Gamma_A^{\mathrm{dep}}/2=\Gamma_B^{\mathrm{dep}}/2 Γ = Γ A dep /2 = Γ B dep /2
Let us then analyze entanglement generation in this setting. We take the initial state to be ∣ ψ ( 0 ) ⟩ = ∣ + ⟩ A ∣ + ⟩ B \lvert\psi(0)\rangle=\lvert+\rangle_A\lvert+\rangle_B ∣ ψ ( 0 )⟩ = ∣ + ⟩ A ∣ + ⟩ B t t t
C ( t ) = e − 2 Γ t max { 0 , ∣ sin ( 2 λ N G t ) ∣ − sinh ( 2 Γ t ) } . (35) C(t)=e^{-2\Gamma t}\max\{0,\lvert\sin(2\lambda_{\mathrm{NG}}t)\rvert-\sinh(2\Gamma t)\}. \tag*{(35)} C ( t ) = e − 2Γ t max { 0 , ∣ sin ( 2 λ NG t )∣ − sinh ( 2Γ t )} . (35) Thus entanglement exists only when ∣ sin ( 2 λ N G t ) ∣ > sinh ( 2 Γ t ) \lvert\sin(2\lambda_{\mathrm{NG}}t)\rvert>\sinh(2\Gamma t) ∣ sin ( 2 λ NG t )∣ > sinh ( 2Γ t ) ∣ λ N G ∣ > Γ \lvert\lambda_{\mathrm{NG}}\rvert>\Gamma ∣ λ NG ∣ > Γ t E S D t_{\mathrm{ESD}} t ESD
∣ sin ( 2 λ N G t E S D ) ∣ = sinh ( 2 Γ t E S D ) . (36) \lvert\sin(2\lambda_{\mathrm{NG}}t_{\mathrm{ESD}})\rvert=\sinh(2\Gamma t_{\mathrm{ESD}}). \tag*{(36)} ∣ sin ( 2 λ NG t ESD )∣ = sinh ( 2Γ t ESD ) . (36) Since ∣ sin ( 2 λ N G t ) ∣ ≤ 1 \lvert\sin(2\lambda_{\mathrm{NG}}t)\rvert\le1 ∣ sin ( 2 λ NG t )∣ ≤ 1 sinh ( 2 Γ t ) ≥ 1 \sinh(2\Gamma t)\ge1 sinh ( 2Γ t ) ≥ 1
t E S D ≤ arsinh ( 1 ) 2 Γ ≃ 0.44 Γ . (37) t_{\mathrm{ESD}}\le\frac{\operatorname{arsinh}(1)}{2\Gamma}\simeq\frac{0.44}{\Gamma}. \tag*{(37)} t ESD ≤ 2Γ arsinh ( 1 ) ≃ Γ 0.44 . (37) This upper bound can be achieved in the strong-coupling limit ∣ λ N G ∣ / Γ → ∞ \lvert\lambda_{\mathrm{NG}}\rvert/\Gamma\to\infty ∣ λ NG ∣ /Γ → ∞ C ( t ) C(t) C ( t ) C ( t ) C(t) C ( t ) λ N G / Γ \lambda_{\mathrm{NG}}/\Gamma λ NG /Γ Figure 3 .
Figure 3. The concurrence C ( t ) C(t) C ( t ) Γ t \Gamma t Γ t λ N G / Γ \lambda_{\mathrm{NG}}/\Gamma λ NG /Γ λ N G / Γ ≤ 1 \lambda_{\mathrm{NG}}/\Gamma\le1 λ NG /Γ ≤ 1 ∣ + ⟩ A ∣ + ⟩ B \lvert+\rangle_A\lvert+\rangle_B ∣ + ⟩ A ∣ + ⟩ B λ N G / Γ \lambda_{\mathrm{NG}}/\Gamma λ NG /Γ e − 2 Γ t [ 1 − sinh ( 2 Γ t ) ] e^{-2\Gamma t}[1-\sinh(2\Gamma t)] e − 2Γ t [ 1 − sinh ( 2Γ t )]
On the other hand, for fixed λ N G \lambda_{\mathrm{NG}} λ NG Γ \Gamma Γ C max = max t ≥ 0 C ( t ) C_{\max}=\max_{t\ge0}C(t) C m a x = max t ≥ 0 C ( t ) C max → 0 C_{\max}\to0 C m a x → 0 ∣ λ N G ∣ / Γ → 1 + \lvert\lambda_{\mathrm{NG}}\rvert/\Gamma\to1^+ ∣ λ NG ∣ /Γ → 1 +
C max ≃ 2 3 ( ∣ λ N G ∣ Γ − 1 ) 3 / 2 . (38) C_{\max}\simeq\frac{2}{3}\left(\frac{\lvert\lambda_{\mathrm{NG}}\rvert}{\Gamma}-1\right)^{3/2}. \tag*{(38)} C m a x ≃ 3 2 ( Γ ∣ λ NG ∣ − 1 ) 3/2 . (38) Then C max C_{\max} C m a x ∣ λ N G ∣ / Γ \lvert\lambda_{\mathrm{NG}}\rvert/\Gamma ∣ λ NG ∣ /Γ ∣ λ N G ∣ / Γ \lvert\lambda_{\mathrm{NG}}\rvert/\Gamma ∣ λ NG ∣ /Γ
Let us then study the limit of this scheme by imposing an upper bound on ∣ λ N G ∣ / Γ \lvert\lambda_{\mathrm{NG}}\rvert/\Gamma ∣ λ NG ∣ /Γ
∣ λ N G ∣ Γ = G m A m B Δ x 2 ℏ λ G R W n d ( d 2 − Δ x 2 ) [ 1 − exp ( − Δ x 2 4 r C 2 ) ] (39) \frac{\lvert\lambda_{\mathrm{NG}}\rvert}{\Gamma}=\frac{Gm_A m_B\Delta x^2}{\hbar\lambda_{\mathrm{GRW}}nd(d^2-\Delta x^2)\left[1-\exp\left(-\frac{\Delta x^2}{4r_C^2}\right)\right]} \tag*{(39)} Γ ∣ λ NG ∣ = ℏ λ GRW n d ( d 2 − Δ x 2 ) [ 1 − exp ( − 4 r C 2 Δ x 2 ) ] G m A m B Δ x 2 (39) Since this model should only apply if the two spheres don’t touch each other, let us say there is a minimum d − Δ x = d min d-\Delta x=d_{\min} d − Δ x = d m i n
∣ λ N G ∣ Γ ≤ G m A m B ℏ λ G R W n d min max ( 4 r C 2 d min 2 , 1 2 ) (40) \frac{\lvert\lambda_{\mathrm{NG}}\rvert}{\Gamma}\le\frac{Gm_A m_B}{\hbar\lambda_{\mathrm{GRW}}nd_{\min}}\max\left(\frac{4r_C^2}{d_{\min}^2},\frac{1}{2}\right) \tag*{(40)} Γ ∣ λ NG ∣ ≤ ℏ λ GRW n d m i n G m A m B max ( d m i n 2 4 r C 2 , 2 1 ) (40) This maximum serves as an upper bound on ∣ λ N G ∣ / Γ \lvert\lambda_{\mathrm{NG}}\rvert/\Gamma ∣ λ NG ∣ /Γ
As a comparison, let us also study the CSL model and the Diósi–Penrose model. In this case there will be an additional correlated noise in the master equation as follows:
d d t ρ = − i λ N G [ σ A z σ B z , ρ ] + Γ D [ σ A z ] ( ρ ) + Γ D [ σ B z ] ( ρ ) + Γ A B ( σ A z ρ σ B z + σ B z ρ σ A z − { σ A z σ B z , ρ } ) , (41) \frac{\mathrm{d}}{\mathrm{d}t}\rho=-i\lambda_{\mathrm{NG}}[\sigma_A^z\sigma_B^z,\rho]+\Gamma\mathcal{D}[\sigma_A^z](\rho)+\Gamma\mathcal{D}[\sigma_B^z](\rho)+\Gamma_{AB}\left(\sigma_A^z\rho\sigma_B^z+\sigma_B^z\rho\sigma_A^z-\{\sigma_A^z\sigma_B^z,\rho\}\right), \tag*{(41)} d t d ρ = − i λ NG [ σ A z σ B z , ρ ] + Γ D [ σ A z ] ( ρ ) + Γ D [ σ B z ] ( ρ ) + Γ A B ( σ A z ρ σ B z + σ B z ρ σ A z − { σ A z σ B z , ρ } ) , (41) where Γ A B \Gamma_{AB} Γ A B ∣ + ⟩ A ∣ + ⟩ B \lvert+\rangle_A\lvert+\rangle_B ∣ + ⟩ A ∣ + ⟩ B t t t
C ( t ) = 1 2 max { 0 , e − 2 Γ t 4 sin 2 ( 2 λ N G t ) + e − 4 Γ t sinh 2 ( 4 Γ A B t ) − ∣ 1 − e − 4 Γ t cosh 4 Γ A B t ∣ } . (42) C(t)=\frac{1}{2}\max\left\{0,e^{-2\Gamma t}\sqrt{4\sin^2(2\lambda_{\mathrm{NG}}t)+e^{-4\Gamma t}\sinh^2(4\Gamma_{AB}t)-\left\lvert1-e^{-4\Gamma t}\cosh4\Gamma_{AB}t\right\rvert}\right\}. \tag*{(42)} C ( t ) = 2 1 max { 0 , e − 2Γ t 4 sin 2 ( 2 λ NG t ) + e − 4Γ t sinh 2 ( 4 Γ A B t ) − ∣ 1 − e − 4Γ t cosh 4 Γ A B t ∣ } . (42) Then one can show that in the weak interaction limit λ N G 2 + Γ A B 2 / Γ → 1 + \sqrt{\lambda_{\mathrm{NG}}^2+\Gamma_{AB}^2}/\Gamma\to1^+ λ NG 2 + Γ A B 2 /Γ → 1 + ∣ Γ A B ∣ ≠ Γ \lvert\Gamma_{AB}\rvert\ne\Gamma ∣ Γ A B ∣ = Γ ∣ Γ A B ∣ ≤ Γ \lvert\Gamma_{AB}\rvert\le\Gamma ∣ Γ A B ∣ ≤ Γ
C max ≃ 2 3 1 − Γ A B 2 Γ 2 ( λ N G 2 + Γ A B 2 Γ − 1 ) 3 / 2 . (43) C_{\max}\simeq\frac{2}{3\sqrt{1-\frac{\Gamma_{AB}^2}{\Gamma^2}}}\left(\frac{\sqrt{\lambda_{\mathrm{NG}}^2+\Gamma_{AB}^2}}{\Gamma}-1\right)^{3/2}. \tag*{(43)} C m a x ≃ 3 1 − Γ 2 Γ A B 2 2 ( Γ λ NG 2 + Γ A B 2 − 1 ) 3/2 . (43) Therefore, similarly the ratio λ N G 2 + Γ A B 2 / Γ \sqrt{\lambda_{\mathrm{NG}}^2+\Gamma_{AB}^2}/\Gamma λ NG 2 + Γ A B 2 /Γ r 0 ≪ r C r_0\ll r_C r 0 ≪ r C
Γ = λ C S L 2 ( m m 0 ) 2 [ 1 − exp ( − Δ x 2 4 r C 2 ) ] , Γ A B = λ C S L 4 ( m m 0 ) 2 [ 2 e − d 2 4 r C 2 − e − ( d + Δ x ) 2 4 r C 2 − e − ( d − Δ x ) 2 4 r C 2 ] (44) \Gamma=\frac{\lambda_{\mathrm{CSL}}}{2}\left(\frac{m}{m_0}\right)^2\left[1-\exp\left(-\frac{\Delta x^2}{4r_C^2}\right)\right],\qquad
\Gamma_{AB}=\frac{\lambda_{\mathrm{CSL}}}{4}\left(\frac{m}{m_0}\right)^2\left[2e^{-\frac{d^2}{4r_C^2}}-e^{-\frac{(d+\Delta x)^2}{4r_C^2}}-e^{-\frac{(d-\Delta x)^2}{4r_C^2}}\right] \tag*{(44)} Γ = 2 λ CSL ( m 0 m ) 2 [ 1 − exp ( − 4 r C 2 Δ x 2 ) ] , Γ A B = 4 λ CSL ( m 0 m ) 2 [ 2 e − 4 r C 2 d 2 − e − 4 r C 2 ( d + Δ x ) 2 − e − 4 r C 2 ( d − Δ x ) 2 ] (44) Similar analysis as GRW model would also indicate that λ N G 2 + Γ A B 2 / Γ \sqrt{\lambda_{\mathrm{NG}}^{2}+\Gamma_{\mathrm{AB}}^{2}}/\Gamma λ NG 2 + Γ AB 2 /Γ
As for the Diósi–Penrose model, let us also consider the limit r 0 ≪ σ r_{0}\ll\sigma r 0 ≪ σ
Γ = G m 2 2 ℏ [ 1 π σ − Erf ( Δ x 2 σ ) Δ x ] , − λ N G = Γ A B = G m 2 4 ℏ [ 2 Erf ( d σ ) d − Erf ( d + Δ x 2 σ ) d + Δ x − Erf ( d − Δ x 2 σ ) d − Δ x ] . (45) \Gamma=\frac{Gm^{2}}{2\hbar}\left[\frac{1}{\sqrt{\pi}\sigma}-\frac{\operatorname{Erf}\left(\frac{\Delta x}{2\sigma}\right)}{\Delta x}\right],\qquad-\lambda_{\mathrm{NG}}=\Gamma_{\mathrm{AB}}=\frac{Gm^{2}}{4\hbar}\left[2\frac{\operatorname{Erf}\left(\frac{d}{\sigma}\right)}{d}-\frac{\operatorname{Erf}\left(\frac{d+\Delta x}{2\sigma}\right)}{d+\Delta x}-\frac{\operatorname{Erf}\left(\frac{d-\Delta x}{2\sigma}\right)}{d-\Delta x}\right]. \tag*{(45)} Γ = 2ℏ G m 2 [ π σ 1 − Δ x Erf ( 2 σ Δ x ) ] , − λ NG = Γ AB = 4ℏ G m 2 [ 2 d Erf ( σ d ) − d + Δ x Erf ( 2 σ d + Δ x ) − d − Δ x Erf ( 2 σ d − Δ x ) ] . (45) In this situation, note that the critical ratio λ N G 2 + Γ A B 2 / Γ \sqrt{\lambda_{\mathrm{NG}}^{2}+\Gamma_{\mathrm{AB}}^{2}}/\Gamma λ NG 2 + Γ AB 2 /Γ d min / σ d_{\min}/\sigma d m i n / σ [34 ] . In the limit that d min / σ , Δ x / σ → 0 + d_{\min}/\sigma,\Delta x/\sigma\to0^{+} d m i n / σ , Δ x / σ → 0 + λ N G 2 + Γ A B 2 / Γ = 2 \sqrt{\lambda_{\mathrm{NG}}^{2}+\Gamma_{\mathrm{AB}}^{2}}/\Gamma=\sqrt{2} λ NG 2 + Γ AB 2 /Γ = 2
In summary, we show that the current protocols of gravity-induced entanglement tests may be quite limited by fundamental collapse models.
Technical details of the quantum error-correction protocol
This appendix provides additional technical details and supporting results for the quantum error-correction framework introduced in the main text. We begin by clarifying the meaning of error correction in the continuous-recovery (Zeno) limit and showing that perfect error correction in this regime necessarily removes the same interactions one aims to preserve. We then introduce an imperfect local QEC strategy that, although it does not completely remove the noise, can improve the signal-to-noise ratio by retaining the nonlocal interaction while suppressing part of the decoherence. We also derive the corresponding logical error channels, and establish the separability criterion for the resulting effective dynamics. Finally, we discuss possible extensions beyond strictly local codes and alternative recovery strategies.
Error correction in the Zeno limit
In the main text we make use of the continuous-recovery (Zeno) limit as a convenient theoretical description of error correction. This subsection briefly reviews this limit and clarifies its relation to more realistic finite-rate recovery protocols.
Consider a total evolution time t t t R R R N N N
( R ∘ e L t / N ) N . (46) \left(R \circ e^{\mathcal{L}t/N}\right)^N. \tag*{(46)} ( R ∘ e L t / N ) N . (46) Restricting to initial states within the code space, for which R ( ρ ) = ρ R(\rho)=\rho R ( ρ ) = ρ R R R R 2 = R R^2=R R 2 = R
( R ∘ e L t / N ) N ( ρ ) = ( 1 + R ∘ L t N + t 2 2 N 2 R ∘ L 2 + ⋯ ) N ( ρ ) = exp ( R ∘ L t + t 2 2 N [ R ∘ L 2 − ( R ∘ L ) 2 ] + O ( 1 N 2 ) ) ( ρ ) , (47) \begin{aligned}
\left(R \circ e^{\mathcal{L}t/N}\right)^N(\rho)
=
\left(1+R\circ\mathcal{L}\frac{t}{N}+\frac{t^2}{2N^2}R\circ\mathcal{L}^2+\cdots\right)^N(\rho) \\
&=
\exp\left(R\circ\mathcal{L}t+\frac{t^2}{2N}\left[R\circ\mathcal{L}^2-(R\circ\mathcal{L})^2\right]+O\left(\frac{1}{N^2}\right)\right)(\rho), \tag*{(47)}
\end{aligned} ( R ∘ e L t / N ) N ( ρ ) = ( 1 + R ∘ L N t + 2 N 2 t 2 R ∘ L 2 + ⋯ ) N ( ρ ) = exp ( R ∘ L t + 2 N t 2 [ R ∘ L 2 − ( R ∘ L ) 2 ] + O ( N 2 1 ) ) ( ρ ) , (47) where higher-order terms arise from the noncommutativity of the recovery and dynamical maps. Note that we use the identity operator as just defined in the code subspace.
In the limit N → ∞ N\to\infty N → ∞
L e f f = R ∘ L . (48) \mathcal{L}_{\mathrm{eff}}=R\circ\mathcal{L}. \tag*{(48)} L eff = R ∘ L . (48) In this regime, recovery acts after every infinitesimal time step, continuously projecting the dynamics onto the code space. As a result, any component of the generator that is perfectly corrected by R R R perfect dynamical error correction .
At finite N N N N / t N/t N / t t t t
R ∘ e L t / N ∘ R ∘ e L t / N ∘ ⋯ ∘ R ∘ e L t / N , (49) R\circ e^{\mathcal{L}t/N}\circ R\circ e^{\mathcal{L}t/N}\circ\cdots\circ R\circ e^{\mathcal{L}t/N}, \tag*{(49)} R ∘ e L t / N ∘ R ∘ e L t / N ∘ ⋯ ∘ R ∘ e L t / N , (49) rather than by a Lindbladian generator of the form R ∘ L R\circ\mathcal{L} R ∘ L
While the Zeno limit represents an idealized form of error correction, it serves here as a useful theoretical tool for classifying the separability properties of the underlying generator L \mathcal{L} L R R R L \mathcal{L} L R ∘ e L t R\circ e^{\mathcal{L}t} R ∘ e L t L \mathcal{L} L
Limitations of perfect error correction in the Zeno regime
The main text argues that perfect correction of the relevant local errors is incompatible with preserving the interaction responsible for entanglement generation. For completeness, we provide here a formal proof of this statement and clarify its relation to standard fault-tolerant quantum-computing protocols.
We consider dynamics in which the coherent interaction and the local errors are generated by the same operators,
L ( ρ ) = − i λ [ O A ⊗ O B , ρ ] + Γ a D [ O A ] ( ρ ) + Γ b D [ O B ] ( ρ ) . (50) \mathcal{L}(\rho)=-i\lambda[O_A\otimes O_B,\rho]+\Gamma_a\mathcal{D}[O_A](\rho)+\Gamma_b\mathcal{D}[O_B](\rho). \tag*{(50)} L ( ρ ) = − iλ [ O A ⊗ O B , ρ ] + Γ a D [ O A ] ( ρ ) + Γ b D [ O B ] ( ρ ) . (50) As explained above, in the Zeno limit of continuous recovery, the effective error-corrected dynamics is generated by L e f f = R ∘ L \mathcal{L}_{\mathrm{eff}}=\mathcal{R}\circ\mathcal{L} L eff = R ∘ L perfectly corrects the error set generated by O A O_A O A O B O_B O B R ( D [ O α ] ( ρ ) ) = 0 \mathcal{R}(\mathcal{D}[O_\alpha](\rho))=0 R ( D [ O α ] ( ρ )) = 0 R ( [ O A ⊗ O B , ρ ] ) = 0 \mathcal{R}([O_A\otimes O_B,\rho])=0 R ([ O A ⊗ O B , ρ ]) = 0
At first sight, this conclusion may appear to contradict fault-tolerant quantum computation, where noisy entangling interactions can be used to implement protected logical gates. There is, however, no contradiction. In fact, this conclusion is quite similar to the Eastin–Knill theorem in the infinitesimal limit [52 ] . The present result applies to protocols that attempt to preserve the interaction under instantaneous (continuous) recovery, whereas fault-tolerant schemes treat the interaction as a gate primitive and apply error correction only to the accumulated noise. In such schemes, logical entangling operations are synthesized from sequences of short interaction pulses with fixed interaction time (in contrast to the continuous recovery) interleaved with recovery and succeed only when the noise per primitive remains below a finite threshold, thereby restricting attention to a limited noise regime. Since our aim is to assess, in principle, whether error correction can reveal gravitationally mediated entanglement across the full parameter space of the hybrid gravity model, we do not pursue this threshold-dependent approach here.
We also stress that this limitation is specific to perfect error correction in the Zeno limit. If one instead allows for imperfect recovery, the interaction need not be completely projected out, and the signal-to-noise ratio can in principle be improved arbitrarily by suitably tailoring the recovery map. This alternative strategy, which does not rely on threshold assumptions but abandons perfect correction, is discussed in App. B 6.
We formalize the Zeno-limit statement in the following proposition.
Proposition 2. Given a quantum code defined on a linear subspace C ⊂ H A ⊗ H B \mathcal{C}\subset\mathcal{H}_A\otimes\mathcal{H}_B C ⊂ H A ⊗ H B O A , O B O_A,O_B O A , O B R \mathcal{R} R
the error-correction channel R \mathcal{R} R R ( ρ ) = ρ \mathcal{R}(\rho)=\rho R ( ρ ) = ρ ρ ∈ L ( C ) \rho\in\mathcal{L}(\mathcal{C}) ρ ∈ L ( C )
the local errors are fully corrected, i.e. R ( D [ O A ] ( ρ ) ) = R ( D [ O B ] ( ρ ) ) = 0 \mathcal{R}(\mathcal{D}[O_A](\rho))=\mathcal{R}(\mathcal{D}[O_B](\rho))=0 R ( D [ O A ] ( ρ )) = R ( D [ O B ] ( ρ )) = 0
the error-correction channel R \mathcal{R} R R ∘ L ( ρ ) ∈ L ( C ) \mathcal{R}\circ\mathcal{L}(\rho)\in\mathcal{L}(\mathcal{C}) R ∘ L ( ρ ) ∈ L ( C ) ρ ∈ L ( C ) \rho\in\mathcal{L}(\mathcal{C}) ρ ∈ L ( C )
Then we conclude (i) the Knill–Laflamme conditions
A B ⟨ i ˉ ∣ O A ∣ j ˉ ⟩ A B ∝ δ i j , A B ⟨ i ˉ ∣ O A 2 ∣ j ˉ ⟩ A B ∝ δ i j , A B ⟨ i ˉ ∣ O B ∣ j ˉ ⟩ A B ∝ δ i j {}_{AB}\langle\bar{i}|O_A|\bar{j}\rangle_{AB}\propto\delta_{ij},\qquad
{}_{AB}\langle\bar{i}|O_A^2|\bar{j}\rangle_{AB}\propto\delta_{ij},\qquad
{}_{AB}\langle\bar{i}|O_B|\bar{j}\rangle_{AB}\propto\delta_{ij} A B ⟨ i ˉ ∣ O A ∣ j ˉ ⟩ A B ∝ δ ij , A B ⟨ i ˉ ∣ O A 2 ∣ j ˉ ⟩ A B ∝ δ ij , A B ⟨ i ˉ ∣ O B ∣ j ˉ ⟩ A B ∝ δ ij A B ⟨ i ˉ ∣ O B 2 ∣ j ˉ ⟩ A B ∝ δ i j , A B ⟨ i ˉ ∣ O A ⊗ O B ∣ j ˉ ⟩ A B ∝ δ i j , {}_{AB}\langle\bar{i}|O_B^2|\bar{j}\rangle_{AB}\propto\delta_{ij},\qquad
{}_{AB}\langle\bar{i}|O_A\otimes O_B|\bar{j}\rangle_{AB}\propto\delta_{ij}, A B ⟨ i ˉ ∣ O B 2 ∣ j ˉ ⟩ A B ∝ δ ij , A B ⟨ i ˉ ∣ O A ⊗ O B ∣ j ˉ ⟩ A B ∝ δ ij , and (ii) the unitary dynamics is trivial on the code after recovery, i.e. R ( [ O A ⊗ O B , ρ ] ) = 0 \mathcal{R}([O_A\otimes O_B,\rho])=0 R ([ O A ⊗ O B , ρ ]) = 0
The third condition R ∘ L ( ρ ) ∈ L ( C ) \mathcal{R}\circ\mathcal{L}(\rho)\in\mathcal{L}(\mathcal{C}) R ∘ L ( ρ ) ∈ L ( C ) R ∘ ( I + L d t ) \mathcal{R}\circ(I+\mathcal{L}dt) R ∘ ( I + L d t ) C \mathcal{C} C
Proof. We first use the condition R ( ρ ) = ρ \mathcal{R}(\rho)=\rho R ( ρ ) = ρ R ( ρ ) = ∑ i R i ρ R i † \mathcal{R}(\rho)=\sum_iR_i\rho R_i^\dagger R ( ρ ) = ∑ i R i ρ R i † Π A B \Pi_{AB} Π A B ∣ ψ ⟩ A B |\psi\rangle_{AB} ∣ ψ ⟩ A B
∣ ψ ⟩ A B ⊗ ∣ 0 ⟩ a u x → ∑ i R i ∣ ψ ⟩ A B ⊗ ∣ i ⟩ a u x = ∣ ψ ⟩ A B ⊗ ∣ ϕ ( ψ ) ⟩ a u x , |\psi\rangle_{AB}\otimes|0\rangle_{\mathrm{aux}}\to\sum_iR_i|\psi\rangle_{AB}\otimes|i\rangle_{\mathrm{aux}}=|\psi\rangle_{AB}\otimes|\phi(\psi)\rangle_{\mathrm{aux}}, ∣ ψ ⟩ A B ⊗ ∣0 ⟩ aux → i ∑ R i ∣ ψ ⟩ A B ⊗ ∣ i ⟩ aux = ∣ ψ ⟩ A B ⊗ ∣ ϕ ( ψ ) ⟩ aux , where the last equality holds since R ( ∣ ψ ⟩ ⟨ ψ ∣ ) = ∣ ψ ⟩ ⟨ ψ ∣ \mathcal{R}(|\psi\rangle\langle\psi|)=|\psi\rangle\langle\psi| R ( ∣ ψ ⟩ ⟨ ψ ∣ ) = ∣ ψ ⟩ ⟨ ψ ∣ C \mathcal{C} C ∣ ψ ⟩ A B ∈ C |\psi\rangle_{AB}\in\mathcal{C} ∣ ψ ⟩ A B ∈ C ∣ ψ ⟩ A B |\psi\rangle_{AB} ∣ ψ ⟩ A B ∣ ϕ ( ψ ) ⟩ a u x = ∣ ϕ ⟩ a u x |\phi(\psi)\rangle_{\mathrm{aux}}=|\phi\rangle_{\mathrm{aux}} ∣ ϕ ( ψ ) ⟩ aux = ∣ ϕ ⟩ aux
R i Π A B ∝ Π A B , ∀ i . R_i\Pi_{AB}\propto\Pi_{AB},\qquad\forall i. R i Π A B ∝ Π A B , ∀ i . Next, the infinitesimal Lindbladian evolution can be cast in Kraus form with Kraus operators
E 0 = I − d t 2 ( O A 2 + O B 2 ) , E 1 = d t O A , E 2 = d t O B , E_0=I-\frac{dt}{2}(O_A^2+O_B^2),\qquad E_1=\sqrt{dt}O_A,\qquad E_2=\sqrt{dt}O_B, E 0 = I − 2 d t ( O A 2 + O B 2 ) , E 1 = d t O A , E 2 = d t O B , such that
∑ i = 0 2 E i ρ E i † = ρ + d t ( D [ O A ] ( ρ ) + D [ O B ] ( ρ ) + O ( d t 2 ) ) . \sum_{i=0}^{2} E_i \rho E_i^\dagger= \rho+ dt\bigl(\mathcal{D}[O_A](\rho) + \mathcal{D}[O_B](\rho) + O(dt^2)\bigr). i = 0 ∑ 2 E i ρ E i † = ρ + d t ( D [ O A ] ( ρ ) + D [ O B ] ( ρ ) + O ( d t 2 ) ) . Therefore, the two conditions are translated into
R ( ∑ i = 0 2 E i ρ E i † ) = ρ + O ( d t 2 ) . R\left(\sum_{i=0}^{2} E_i \rho E_i^\dagger\right) = \rho+ O(dt^2). R ( i = 0 ∑ 2 E i ρ E i † ) = ρ + O ( d t 2 ) . Using the discussion above, this implies the following equation up to O ( d t ) O(dt) O ( d t )
R i E j Π A B ∝ Π A B ⇒ Π A B E j † E k Π A B = ∑ i Π A B E j † R i † R i E k Π A B ∝ Π A B , R_i E_j \Pi_{AB} \propto\Pi_{AB} \Rightarrow\Pi_{AB} E_j^\dagger E_k \Pi_{AB} = \sum_i \Pi_{AB} E_j^\dagger R_i^\dagger R_i E_k \Pi_{AB} \propto\Pi_{AB}, R i E j Π A B ∝ Π A B ⇒ Π A B E j † E k Π A B = i ∑ Π A B E j † R i † R i E k Π A B ∝ Π A B , i.e. the Knill–Laflamme condition. At order O ( d t ) O(dt) O ( d t )
A B ⟨ i ˉ ∣ O A ∣ j ˉ ⟩ A B ∝ δ i j , A B ⟨ i ˉ ∣ O A 2 ∣ j ˉ ⟩ A B ∝ δ i j , A B ⟨ i ˉ ∣ O B ∣ j ˉ ⟩ A B ∝ δ i j , {}_{AB}\langle\bar{i}|O_A|\bar{j}\rangle_{AB} \propto\delta_{ij}, \qquad{}_{AB}\langle\bar{i}|O_A^2|\bar{j}\rangle_{AB} \propto\delta_{ij}, \qquad{}_{AB}\langle\bar{i}|O_B|\bar{j}\rangle_{AB} \propto\delta_{ij}, A B ⟨ i ˉ ∣ O A ∣ j ˉ ⟩ A B ∝ δ ij , A B ⟨ i ˉ ∣ O A 2 ∣ j ˉ ⟩ A B ∝ δ ij , A B ⟨ i ˉ ∣ O B ∣ j ˉ ⟩ A B ∝ δ ij , A B ⟨ i ˉ ∣ O B 2 ∣ j ˉ ⟩ A B ∝ δ i j , A B ⟨ i ˉ ∣ O A ⊗ O B ∣ j ˉ ⟩ A B ∝ δ i j . {}_{AB}\langle\bar{i}|O_B^2|\bar{j}\rangle_{AB} \propto\delta_{ij}, \qquad{}_{AB}\langle\bar{i}|O_A \otimes O_B|\bar{j}\rangle_{AB} \propto\delta_{ij}. A B ⟨ i ˉ ∣ O B 2 ∣ j ˉ ⟩ A B ∝ δ ij , A B ⟨ i ˉ ∣ O A ⊗ O B ∣ j ˉ ⟩ A B ∝ δ ij . We now study the error-corrected unitary dynamics generated by [ O A ⊗ O B , ρ ] [O_A \otimes O_B,\rho] [ O A ⊗ O B , ρ ] Π A B ρ Π A B = ρ \Pi_{AB}\rho\Pi_{AB}=\rho Π A B ρ Π A B = ρ ρ ( I − Π A B ) = 0 \rho(I-\Pi_{AB})=0 ρ ( I − Π A B ) = 0
Π A B [ O A ⊗ O B , ρ ] Π A B = [ Π A B O A ⊗ O B Π A B , ρ ] ∝ [ Π A B , ρ ] = 0 , \Pi_{AB}[O_A \otimes O_B,\rho]\Pi_{AB} = [\Pi_{AB}O_A \otimes O_B\Pi_{AB},\rho] \propto[\Pi_{AB},\rho] = 0, Π A B [ O A ⊗ O B , ρ ] Π A B = [ Π A B O A ⊗ O B Π A B , ρ ] ∝ [ Π A B , ρ ] = 0 , ( I − Π A B ) [ O A ⊗ O B , ρ ] ( I − Π A B ) = 0. (I-\Pi_{AB})[O_A \otimes O_B,\rho](I-\Pi_{AB}) = 0. ( I − Π A B ) [ O A ⊗ O B , ρ ] ( I − Π A B ) = 0. The first equation implies that within the code space O A ⊗ O B O_A \otimes O_B O A ⊗ O B [ O A ⊗ O B , ρ ] [O_A \otimes O_B,\rho] [ O A ⊗ O B , ρ ] Π A B ⊕ ( I − Π A B ) \Pi_{AB} \oplus(I-\Pi_{AB}) Π A B ⊕ ( I − Π A B ) R ( D [ O A ] ( ρ ) ) = R ( D [ O B ] ( ρ ) ) = 0 R(\mathcal{D}[O_A](\rho))=R(\mathcal{D}[O_B](\rho))=0 R ( D [ O A ] ( ρ )) = R ( D [ O B ] ( ρ )) = 0 R ∘ L ( ρ ) ∈ L ( C ) R \circ\mathcal{L}(\rho) \in\mathcal{L}(\mathcal{C}) R ∘ L ( ρ ) ∈ L ( C ) Π A B R ( [ O A ⊗ O B , ρ ] ) Π A B = R ( [ O A ⊗ O B , ρ ] ) \Pi_{AB}R([O_A \otimes O_B,\rho])\Pi_{AB}=R([O_A \otimes O_B,\rho]) Π A B R ([ O A ⊗ O B , ρ ]) Π A B = R ([ O A ⊗ O B , ρ ])
Assume without loss of generality that R i Π A B = r i Π A B R_i\Pi_{AB}=r_i\Pi_{AB} R i Π A B = r i Π A B r i r_i r i
0 = Π A B ( I − Π A B ) = Π A B ∑ i R i † R i ( I − Π A B ) = ∑ i r i ∗ Π A B R i ( I − Π A B ) , 0=\Pi_{AB}(I-\Pi_{AB})=\Pi_{AB}\sum_i R_i^\dagger R_i(I-\Pi_{AB})=\sum_i r_i^*\Pi_{AB}R_i(I-\Pi_{AB}), 0 = Π A B ( I − Π A B ) = Π A B i ∑ R i † R i ( I − Π A B ) = i ∑ r i ∗ Π A B R i ( I − Π A B ) , and therefore
R ( [ O A ⊗ O B , ρ ] ) = Π A B R ( [ O A ⊗ O B , ρ ] ) Π A B = ∑ i Π A B R i Π A B [ O A ⊗ O B , ρ ] ( I − Π A B ) R i † Π A B + Π A B R i ( I − Π A B ) [ O A ⊗ O B , ρ ] Π A B R i † Π A B = ∑ i r i Π A B [ O A ⊗ O B , ρ ] ( I − Π A B ) R i † Π A B + r i ∗ Π A B R i ( I − Π A B ) [ O A ⊗ O B , ρ ] Π A B = Π A B [ O A ⊗ O B , ρ ] × 0 + 0 × [ O A ⊗ O B , ρ ] Π A B = 0. \begin{aligned}
R([O_A \otimes O_B,\rho]) &= \Pi_{AB}R([O_A \otimes O_B,\rho])\Pi_{AB}\\
&= \sum_i \Pi_{AB}R_i\Pi_{AB}[O_A \otimes O_B,\rho](I-\Pi_{AB})R_i^\dagger\Pi_{AB}\\
&\quad+ \Pi_{AB}R_i(I-\Pi_{AB})[O_A \otimes O_B,\rho]\Pi_{AB}R_i^\dagger\Pi_{AB}\\
&= \sum_i r_i\Pi_{AB}[O_A \otimes O_B,\rho](I-\Pi_{AB})R_i^\dagger\Pi_{AB}\\
&\quad+ r_i^*\Pi_{AB}R_i(I-\Pi_{AB})[O_A \otimes O_B,\rho]\Pi_{AB}\\
&= \Pi_{AB}[O_A \otimes O_B,\rho] \times0 + 0 \times[O_A \otimes O_B,\rho]\Pi_{AB}\\
&=0.
\end{aligned} R ([ O A ⊗ O B , ρ ]) = Π A B R ([ O A ⊗ O B , ρ ]) Π A B = i ∑ Π A B R i Π A B [ O A ⊗ O B , ρ ] ( I − Π A B ) R i † Π A B + Π A B R i ( I − Π A B ) [ O A ⊗ O B , ρ ] Π A B R i † Π A B = i ∑ r i Π A B [ O A ⊗ O B , ρ ] ( I − Π A B ) R i † Π A B + r i ∗ Π A B R i ( I − Π A B ) [ O A ⊗ O B , ρ ] Π A B = Π A B [ O A ⊗ O B , ρ ] × 0 + 0 × [ O A ⊗ O B , ρ ] Π A B = 0. □
Constructing an imperfect local QEC code
Since perfect correction eliminates the desired interaction, the protocol considered in the main text relies instead on imperfect error correction. In this subsection, we provide an explicit derivation of this local QEC code and analyze how the microscopic interaction and decoherence processes are mapped onto the encoded degrees of freedom. We show this for the model that combines Newtonian gravitational interaction and the GRW-type of noise. Other models are considered in App. C3. Keeping only the leading non-trivial contributions, the dynamics can be expanded as
d d t ρ A B = − i ℏ [ G m A m B ∣ d + x A − x B ∣ , ρ A B ] + L G R W , A ( ρ A B ) + L G R W , B ( ρ A B ) = − 2 i G m A m B ℏ d 3 [ x A x B , ρ A B ] + ⋯ + λ G R W n ( ∫ d 3 x 1 d 3 x 2 ( 1 − ( x 1 − x 2 ) 2 4 r C 2 ) ∣ x 1 ⟩ A ⟨ x 1 ∣ ρ ∣ x 2 ⟩ A ⟨ x 2 ∣ − ρ A B ) + L G R W , B ( ρ A B ) = − 2 i G m A m B ℏ d 3 [ x A x B , ρ A B ] + λ G R W n 2 r C 2 D [ x A ] ( ρ A B ) + λ G R W n 2 r C 2 D [ x B ] ( ρ A B ) + ⋯ (51) \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\rho^{AB} &= -\frac{i}{\hbar}\left[\frac{Gm_A m_B}{\lvert d+x_A-x_B\rvert},\rho^{AB}\right]+\mathcal{L}_{\mathrm{GRW},A}(\rho^{AB})+\mathcal{L}_{\mathrm{GRW},B}(\rho^{AB})\\
&=-\frac{2iGm_A m_B}{\hbar d^3}[x_Ax_B,\rho^{AB}]+\cdots\\
&\quad+\lambda_{\mathrm{GRW}}n\left(\int\mathrm{d}^3x_1\mathrm{d}^3x_2\left(1-\frac{(x_1-x_2)^2}{4r_C^2}\right)\lvert x_1\rangle_A\langle x_1\lvert\rho\lvert x_2\rangle_A\langle x_2\lvert-\rho^{AB}\right)+\mathcal{L}_{\mathrm{GRW},B}(\rho^{AB})\\
&=-\frac{2iGm_A m_B}{\hbar d^3}[x_Ax_B,\rho^{AB}]+\frac{\lambda_{\mathrm{GRW}}n}{2r_C^2}\mathcal{D}[x_A](\rho^{AB})+\frac{\lambda_{\mathrm{GRW}}n}{2r_C^2}\mathcal{D}[x_B](\rho^{AB})+\cdots \tag*{(51)}
\end{aligned} d t d ρ A B = − ℏ i [ ∣ d + x A − x B ∣ G m A m B , ρ A B ] + L GRW , A ( ρ A B ) + L GRW , B ( ρ A B ) = − ℏ d 3 2 i G m A m B [ x A x B , ρ A B ] + ⋯ + λ GRW n ( ∫ d 3 x 1 d 3 x 2 ( 1 − 4 r C 2 ( x 1 − x 2 ) 2 ) ∣ x 1 ⟩ A ⟨ x 1 ∣ ρ ∣ x 2 ⟩ A ⟨ x 2 ∣ − ρ A B ) + L GRW , B ( ρ A B ) = − ℏ d 3 2 i G m A m B [ x A x B , ρ A B ] + 2 r C 2 λ GRW n D [ x A ] ( ρ A B ) + 2 r C 2 λ GRW n D [ x B ] ( ρ A B ) + ⋯ (51) Thus, to leading order, the dynamics contains a nonlocal interaction term proportional to x A x B x_Ax_B x A x B x A x_A x A x B x_B x B
L N G ( ρ A B ) = i ℏ [ G m A m B ∣ d + x A − x B ∣ , ρ A B ] . \mathcal{L}_{\mathrm{NG}}(\rho^{AB})=\frac{i}{\hbar}\left[\frac{Gm_A m_B}{\lvert d+x_A-x_B\rvert},\rho^{AB}\right]. L NG ( ρ A B ) = ℏ i [ ∣ d + x A − x B ∣ G m A m B , ρ A B ] . Our goal is to preserve the potential entanglement generation while suppressing the collapse-induced decoherence as much as possible. However, the analysis below shows a fundamental contradiction between these two requirements for this class of error models.
We now consider a code space C = span { ∣ i ˉ ⟩ A B } \mathcal{C}=\operatorname{span}\{\lvert\bar{i}\rangle_{AB}\} C = span {∣ i ˉ ⟩ A B } x A x B x_Ax_B x A x B
A B ⟨ i ˉ ∣ x A x B ∣ j ˉ ⟩ A B ∝̸ δ i j , (52) {}_{AB}\langle\bar{i}\lvert x_Ax_B\lvert\bar{j}\rangle_{AB}\not\propto\delta_{ij}, \tag*{(52)} A B ⟨ i ˉ ∣ x A x B ∣ j ˉ ⟩ A B ∝ δ ij , (52) i.e. x A x B x_Ax_B x A x B C \mathcal{C} C (3) in the main text, the collapse errors x A x_A x A x B x_B x B
A B ⟨ i ˉ ∣ E k † E l ∣ j ˉ ⟩ A B ∝ δ i j , E i ∈ { 1 , x A , x B } . (53) {}_{AB}\langle\bar{i}\lvert E_k^\dagger E_l\lvert\bar{j}\rangle_{AB}\propto\delta_{ij},\qquad E_i\in\{1,x_A,x_B\}. \tag*{(53)} A B ⟨ i ˉ ∣ E k † E l ∣ j ˉ ⟩ A B ∝ δ ij , E i ∈ { 1 , x A , x B } . (53) Therefore, these two requirements are incompatible in general. In particular, if A B ⟨ 0 ˉ ∣ x A x B ∣ 1 ˉ ⟩ A B ≠ 0 {}_{AB}\langle\bar{0}\lvert x_Ax_B\lvert\bar{1}\rangle_{AB}\ne0 A B ⟨ 0 ˉ ∣ x A x B ∣ 1 ˉ ⟩ A B = 0 x A x B x_Ax_B x A x B ∣ 0 ˉ ⟩ A B \lvert\bar{0}\rangle_{AB} ∣ 0 ˉ ⟩ A B ∣ 1 ˉ ⟩ A B \lvert\bar{1}\rangle_{AB} ∣ 1 ˉ ⟩ A B ∣ 1 ˉ ⟩ A B \lvert\bar{1}\rangle_{AB} ∣ 1 ˉ ⟩ A B x B x_B x B ∣ 0 ˉ ⟩ A B \lvert\bar{0}\rangle_{AB} ∣ 0 ˉ ⟩ A B x A x_A x A ( A B ⟨ 0 ˉ ∣ x A ) ( x B ∣ 1 ˉ ⟩ A B ) ≠ 0 ({}_{AB}\langle\bar{0}\lvert x_A)(x_B\lvert\bar{1}\rangle_{AB})\ne0 ( A B ⟨ 0 ˉ ∣ x A ) ( x B ∣ 1 ˉ ⟩ A B ) = 0 [53 ] . More broadly, whenever one combines a Newtonian interaction with localization in the position basis (from collapse or from environmental decoherence), the local error operators cannot be cleanly distinguished from the nonlocal interaction, and an analogous contradiction persists.
We therefore focus on imperfect QEC: we retain a non-trivial nonlocal Hamiltonian capable of generating entanglement, while protecting part of the encoded information against localization mechanisms.
We restrict to local QEC codes of the form C = C A ⊗ C B \mathcal{C}=\mathcal{C}_A\otimes\mathcal{C}_B C = C A ⊗ C B local recovery maps R = R A ⊗ R B R=R_A\otimes R_B R = R A ⊗ R B
In the following, we consider a local QEC code defined in the phononic Hilbert space of each center-of-mass motional mode, namely
C α = span { ∣ 0 ˉ ⟩ α = ∣ 0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣ 2 N + 1 ⟩ α } . \mathcal{C}_\alpha=\operatorname{span}\{\lvert\bar{0}\rangle_\alpha=\lvert0\rangle_\alpha,\lvert\bar{1}\rangle_\alpha=\lvert2N+1\rangle_\alpha\}. C α = span {∣ 0 ˉ ⟩ α = ∣ 0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣ 2 N + 1 ⟩ α } . We denote the Fock basis by kets without overlines, e.g. ∣ i ⟩ α \lvert i\rangle_\alpha ∣ i ⟩ α N N N Π = ∑ i = 0 , 1 ∣ i ˉ ⟩ ⟨ i ˉ ∣ \Pi=\sum_{i=0,1}\lvert\bar{i}\rangle\langle\bar{i}\lvert Π = ∑ i = 0 , 1 ∣ i ˉ ⟩ ⟨ i ˉ ∣ σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ \sigma^+=\lvert\bar{1}\rangle\langle\bar{0}\lvert σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ σ − = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ \sigma^-=\lvert\bar{0}\rangle\langle\bar{1}\lvert σ − = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ σ x = σ + + σ − \sigma^x=\sigma^++\sigma^- σ x = σ + + σ − σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ \sigma^z=\lvert\bar{0}\rangle\langle\bar{0}\lvert-\lvert\bar{1}\rangle\langle\bar{1}\lvert σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣
The discussion above kept only leading-order terms. In a full-order treatment of the Newtonian potential and collapse models, higher powers of the displacement operators, i.e. x α i x_\alpha^i x α i x α = x 0 ( a α + a α † ) x_\alpha=x_0(a_\alpha+a_\alpha^\dagger) x α = x 0 ( a α + a α † ) x 0 x_0 x 0 α ∈ { A , B } \alpha\in\{A,B\} α ∈ { A , B } a A , a B a_A,a_B a A , a B x α i x_\alpha^i x α i a α a_\alpha a α a α † a_\alpha^\dagger a α † i , i − 2 , i − 4 , … i,i-2,i-4,\ldots i , i − 2 , i − 4 , …
α ⟨ 0 ‾ ∣ x α i ∣ 1 ‾ ⟩ α ≠ 0 \alpha\langle\overline{0}|x_{\alpha}^{i}|\overline{1}\rangle_{\alpha}\ne0 α ⟨ 0 ∣ x α i ∣ 1 ⟩ α = 0 i ≥ 2 N + 1 i\ge2N+1 i ≥ 2 N + 1 i i i
α ⟨ 0 ‾ ∣ x α i ∣ 0 ‾ ⟩ α = α ⟨ 1 ‾ ∣ x α i ∣ 1 ‾ ⟩ α = 0 \alpha\langle\overline{0}|x_{\alpha}^{i}|\overline{0}\rangle_{\alpha}=\alpha\langle\overline{1}|x_{\alpha}^{i}|\overline{1}\rangle_{\alpha}=0 α ⟨ 0 ∣ x α i ∣ 0 ⟩ α = α ⟨ 1 ∣ x α i ∣ 1 ⟩ α = 0 i i i α ⟨ 0 ‾ ∣ x α i ∣ 0 ‾ ⟩ α ≠ α ⟨ 1 ‾ ∣ x α i ∣ 1 ‾ ⟩ α \alpha\langle\overline{0}|x_{\alpha}^{i}|\overline{0}\rangle_{\alpha}\ne\alpha\langle\overline{1}|x_{\alpha}^{i}|\overline{1}\rangle_{\alpha} α ⟨ 0 ∣ x α i ∣ 0 ⟩ α = α ⟨ 1 ∣ x α i ∣ 1 ⟩ α
According to the QEC conditions in Eq. (3) in the main text and the standard discussion in [53 ] , these matrix elements imply the following logical error structure (we omit subscripts for simplicity):
σ x \sigma^{x} σ x ⟨ 0 ‾ ∣ x i x j ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|x^{i}x^{j}|\overline{1}\rangle=0 ⟨ 0 ∣ x i x j ∣ 1 ⟩ = 0 x i ∣ 0 ‾ ⟩ x^{i}|\overline{0}\rangle x i ∣ 0 ⟩ x j ∣ 1 ‾ ⟩ x^{j}|\overline{1}\rangle x j ∣ 1 ⟩ ∣ 0 ‾ ⟩ |\overline{0}\rangle ∣ 0 ⟩ ∣ 1 ‾ ⟩ |\overline{1}\rangle ∣ 1 ⟩ i + j ≤ 2 N i+j\le2N i + j ≤ 2 N ⟨ 0 ‾ ∣ x i x j ∣ 1 ‾ ⟩ ≠ 0 \langle\overline{0}|x^{i}x^{j}|\overline{1}\rangle\ne0 ⟨ 0 ∣ x i x j ∣ 1 ⟩ = 0 x i ∣ 0 ‾ ⟩ x^{i}|\overline{0}\rangle x i ∣ 0 ⟩ x j ∣ 1 ‾ ⟩ x^{j}|\overline{1}\rangle x j ∣ 1 ⟩ ⟨ 0 ‾ ∣ x N x N + 1 ∣ 1 ‾ ⟩ ≠ 0 \langle\overline{0}|x^{N}x^{N+1}|\overline{1}\rangle\ne0 ⟨ 0 ∣ x N x N + 1 ∣ 1 ⟩ = 0 x N ∣ 0 ‾ ⟩ x^{N}|\overline{0}\rangle x N ∣ 0 ⟩ x N + 1 ∣ 1 ‾ ⟩ x^{N+1}|\overline{1}\rangle x N + 1 ∣ 1 ⟩ x N x^{N} x N ∣ 0 ‾ ⟩ ⟨ ψ ∣ |\overline{0}\rangle\langle\psi| ∣ 0 ⟩ ⟨ ψ ∣ ∣ ψ ⟩ ∝ x N ∣ 0 ‾ ⟩ |\psi\rangle\propto x^{N}|\overline{0}\rangle ∣ ψ ⟩ ∝ x N ∣ 0 ⟩ x N + 1 x^{N+1} x N + 1 ∣ 1 ‾ ⟩ |\overline{1}\rangle ∣ 1 ⟩ ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ x N + 1 ∣ 1 ‾ ⟩ x^{N+1}|\overline{1}\rangle x N + 1 ∣ 1 ⟩ ∣ 0 ‾ ⟩ |\overline{0}\rangle ∣ 0 ⟩ σ − \sigma^{-} σ − ⟨ 0 ‾ ∣ x N + 1 x N ∣ 1 ‾ ⟩ ≠ 0 \langle\overline{0}|x^{N+1}x^{N}|\overline{1}\rangle\ne0 ⟨ 0 ∣ x N + 1 x N ∣ 1 ⟩ = 0 σ + \sigma^{+} σ + σ x \sigma^{x} σ x
σ z \sigma^{z} σ z ⟨ 0 ‾ ∣ x i ∣ 0 ‾ ⟩ ≠ ⟨ 1 ‾ ∣ x i ∣ 1 ‾ ⟩ \langle\overline{0}|x^{i}|\overline{0}\rangle\ne\langle\overline{1}|x^{i}|\overline{1}\rangle ⟨ 0 ∣ x i ∣ 0 ⟩ = ⟨ 1 ∣ x i ∣ 1 ⟩ ( ⟨ 0 ‾ ∣ + ⟨ 1 ‾ ∣ ) x i ( ∣ 0 ‾ ⟩ − ∣ 1 ‾ ⟩ ) ≠ 0 (\langle\overline{0}|+\langle\overline{1}|)x^{i}(|\overline{0}\rangle-|\overline{1}\rangle)\ne0 (⟨ 0 ∣ + ⟨ 1 ∣ ) x i ( ∣ 0 ⟩ − ∣ 1 ⟩) = 0 σ z \sigma^{z} σ z
nonlocal coupling: α ⟨ 0 ‾ ∣ x α i ∣ 1 ‾ ⟩ α ≠ 0 \alpha\langle\overline{0}|x_{\alpha}^{i}|\overline{1}\rangle_{\alpha}\ne0 α ⟨ 0 ∣ x α i ∣ 1 ⟩ α = 0 i ≥ 2 N + 1 i\ge2N+1 i ≥ 2 N + 1 ∣ 0 ‾ ⟩ A ∣ 0 ‾ ⟩ B |\overline{0}\rangle_{A}|\overline{0}\rangle_{B} ∣ 0 ⟩ A ∣ 0 ⟩ B ∣ 1 ‾ ⟩ A ∣ 1 ‾ ⟩ B |\overline{1}\rangle_{A}|\overline{1}\rangle_{B} ∣ 1 ⟩ A ∣ 1 ⟩ B x A i x B j x_{A}^{i}x_{B}^{j} x A i x B j i , j ≥ 2 N + 1 i,j\ge2N+1 i , j ≥ 2 N + 1 i , j i,j i , j
More precisely, these errors can be characterized using Krylov spaces. For error operators x , x 2 , x 3 , … , x N x,x^{2},x^{3},\ldots,x^{N} x , x 2 , x 3 , … , x N ∣ i ‾ x k ⟩ ∝ ( 1 − ∣ i ‾ ⟩ ⟨ i ‾ ∣ − ∑ j = 1 k − 1 ∣ i ‾ x j ⟩ ⟨ i ‾ x j ∣ ) x k ∣ i ‾ ⟩ |\overline{i}x_{k}\rangle\propto(1-|\overline{i}\rangle\langle\overline{i}|-\sum_{j=1}^{k-1}|\overline{i}x_{j}\rangle\langle\overline{i}x_{j}|)x^{k}|\overline{i}\rangle ∣ i x k ⟩ ∝ ( 1 − ∣ i ⟩ ⟨ i ∣ − ∑ j = 1 k − 1 ∣ i x j ⟩ ⟨ i x j ∣ ) x k ∣ i ⟩ k = 1 , 2 , … , N k=1,2,\ldots,N k = 1 , 2 , … , N { ∣ i ‾ ⟩ , ∣ i ‾ x ⟩ , … , ∣ i ‾ x N ⟩ } i = 0 , 1 \{|\overline{i}\rangle,|\overline{i}x\rangle,\ldots,|\overline{i}x_{N}\rangle\}_{i=0,1} { ∣ i ⟩ , ∣ i x ⟩ , … , ∣ i x N ⟩ } i = 0 , 1 ⟨ 0 ‾ ∣ x k ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|x^{k}|\overline{1}\rangle=0 ⟨ 0 ∣ x k ∣ 1 ⟩ = 0 k = 1 , 2 , 3 , … , 2 N k=1,2,3,\ldots,2N k = 1 , 2 , 3 , … , 2 N ⟨ 0 ‾ ∣ x ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|x|\overline{1}\rangle=0 ⟨ 0 ∣ x ∣ 1 ⟩ = 0 ∣ 0 ‾ ⟩ ⊥ ∣ 1 ‾ x ⟩ |\overline{0}\rangle\perp|\overline{1}x\rangle ∣ 0 ⟩ ⊥ ∣ 1 x ⟩ ∣ 1 ‾ ⟩ ⊥ ∣ 0 ‾ x ⟩ |\overline{1}\rangle\perp|\overline{0}x\rangle ∣ 1 ⟩ ⊥ ∣ 0 x ⟩ ⟨ 0 ‾ ∣ x 2 ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|x^{2}|\overline{1}\rangle=0 ⟨ 0 ∣ x 2 ∣ 1 ⟩ = 0 ⟨ 0 ‾ ∣ 1 ‾ x 2 ⟩ = ⟨ 0 ‾ x ∣ 1 ‾ x ⟩ = ⟨ 0 ‾ x 2 ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|\overline{1}x_{2}\rangle=\langle\overline{0}x|\overline{1}x\rangle=\langle\overline{0}x_{2}|\overline{1}\rangle=0 ⟨ 0 ∣ 1 x 2 ⟩ = ⟨ 0 x ∣ 1 x ⟩ = ⟨ 0 x 2 ∣ 1 ⟩ = 0 ⟨ 0 ‾ ∣ x 3 ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|x^{3}|\overline{1}\rangle=0 ⟨ 0 ∣ x 3 ∣ 1 ⟩ = 0 ⟨ 0 ‾ ∣ 1 ‾ x 3 ⟩ = ⟨ 0 ‾ x ∣ 1 ‾ x 2 ⟩ = ⟨ 0 ‾ x 2 ∣ 1 ‾ x ⟩ = ⟨ 0 ‾ x 3 ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|\overline{1}x_{3}\rangle=\langle\overline{0}x|\overline{1}x_{2}\rangle=\langle\overline{0}x_{2}|\overline{1}x\rangle=\langle\overline{0}x_{3}|\overline{1}\rangle=0 ⟨ 0 ∣ 1 x 3 ⟩ = ⟨ 0 x ∣ 1 x 2 ⟩ = ⟨ 0 x 2 ∣ 1 x ⟩ = ⟨ 0 x 3 ∣ 1 ⟩ = 0 ⟨ 0 ‾ ∣ x 2 N ∣ 1 ‾ ⟩ = 0 \langle\overline{0}|x^{2N}|\overline{1}\rangle=0 ⟨ 0 ∣ x 2 N ∣ 1 ⟩ = 0 ⟨ 0 ‾ x N ∣ 1 ‾ x N ⟩ = 0 \langle\overline{0}x_{N}|\overline{1}x_{N}\rangle=0 ⟨ 0 x N ∣ 1 x N ⟩ = 0 ∣ 0 ‾ x k ⟩ = ∣ k ⟩ |\overline{0}x_{k}\rangle=|k\rangle ∣ 0 x k ⟩ = ∣ k ⟩
In this setting, we can correct these errors with the following local QEC channel:
R ( ρ ) = Π ρ Π + ∑ k = 1 N ( ∣ 0 ‾ ⟩ ⟨ 0 ‾ x k ∣ + ∣ 1 ‾ ⟩ ⟨ 1 ‾ x k ∣ ) ρ ( ∣ 0 ‾ x k ⟩ ⟨ 0 ‾ ∣ + ∣ 1 ‾ x k ⟩ ⟨ 1 ‾ ∣ ) . (54) R(\rho)=\Pi\rho\Pi+\sum_{k=1}^{N}\left(|\overline{0}\rangle\langle\overline{0}x_{k}|+|\overline{1}\rangle\langle\overline{1}x_{k}|\right)\rho\left(|\overline{0}x_{k}\rangle\langle\overline{0}|+|\overline{1}x_{k}\rangle\langle\overline{1}|\right). \tag*{(54)} R ( ρ ) = Π ρ Π + k = 1 ∑ N ( ∣ 0 ⟩ ⟨ 0 x k ∣ + ∣ 1 ⟩ ⟨ 1 x k ∣ ) ρ ( ∣ 0 x k ⟩ ⟨ 0 ∣ + ∣ 1 x k ⟩ ⟨ 1 ∣ ) . (54) In general, this map is not trace preserving. However, for pedagogical purposes, we use it as a recovery operation in this appendix only. Restricting the discussion to the code subspace makes the action of the recovery channel on the code states particularly transparent, without the additional structure required to define a trace-preserving map on the full Hilbert space. And this recovery map can of course be completed to be trace-preserving. In the rigorous treatment of App. C3, we instead introduce a recovery map that is trace-preserving in the whole Hilbert space, and all the results of the main text rest on that construction. Note, nevertheless, that if ρ \rho ρ (54) , since Tr ( R ( D i ) ) = 0 \operatorname{Tr}(R(\mathcal{D}^{i}) )=0 Tr ( R ( D i )) = 0 i ≤ N i\le N i ≤ N R R R σ x \sigma^{x} σ x x N x^{N} x N ρ \rho ρ C \mathcal{C} C
R ∘ D [ x i ] ( ρ ) ∝ D [ σ z ] ( ρ ) , for i = 1 , 2 , … , N . (55) R\circ\mathcal{D}[x^{i}](\rho)\propto\mathcal{D}[\sigma^{z}](\rho),\qquad\text{for }i=1,2,\ldots,N. \tag*{(55)} R ∘ D [ x i ] ( ρ ) ∝ D [ σ z ] ( ρ ) , for i = 1 , 2 , … , N . (55) For higher-order errors, e.g. an x N + 1 x^{N+1} x N + 1 σ x \sigma^{x} σ x ⟨ 0 ‾ ∣ x A N x B N + 1 ∣ 1 ‾ ⟩ ≠ 0 \langle\overline{0}|x_{A}^{N}x_{B}^{N+1}|\overline{1}\rangle\ne0 ⟨ 0 ∣ x A N x B N + 1 ∣ 1 ⟩ = 0 R ∘ D [ x N + 1 ] ( ρ ) R\circ\mathcal{D}[x^{N+1}](\rho) R ∘ D [ x N + 1 ] ( ρ ) σ x \sigma^{x} σ x
R ∘ L N G R\circ\mathcal{L}_{\mathrm{NG}} R ∘ L NG σ A x σ B x \sigma_{A}^{x}\sigma_{B}^{x} σ A x σ B x x A 2 N + 1 x B 2 N + 1 x_{A}^{2N+1}x_{B}^{2N+1} x A 2 N + 1 x B 2 N + 1
Error-corrected decoherence terms: for example, R ∘ L G W R\circ\mathcal{L}_{\mathrm{GW}} R ∘ L GW σ z \sigma^{z} σ z σ x \sigma^{x} σ x x α i x_{\alpha}^{i} x α i σ z \sigma^{z} σ z x α N + 1 x_{\alpha}^{N+1} x α N + 1 σ x \sigma^{x} σ x
Note that we have so far ignored the contribution from the free Hamiltonian H 0 H_{0} H 0
Π A Π B H 0 Π A Π B = ℏ ω ( 2 N + 2 − 2 N + 1 2 σ A z − 2 N + 1 2 σ B z ) . (56) \Pi_{A}\Pi_{B}H_{0}\Pi_{A}\Pi_{B}=\hbar\omega\left(2N+2-\frac{2N+1}{2}\sigma_{A}^{z}-\frac{2N+1}{2}\sigma_{B}^{z}\right). \tag*{(56)} Π A Π B H 0 Π A Π B = ℏ ω ( 2 N + 2 − 2 2 N + 1 σ A z − 2 2 N + 1 σ B z ) . (56) In general, this encoding will also introduce σ A x σ B z \sigma_A^x\sigma_B^z σ A x σ B z σ A z σ B x \sigma_A^z\sigma_B^x σ A z σ B x σ A z σ B z \sigma_A^z\sigma_B^z σ A z σ B z x A i x B j x_A^i x_B^j x A i x B j σ A x σ B x \sigma_A^x\sigma_B^x σ A x σ B x
Effective logical error channels
The encoded logical dynamics derived above can be expressed in terms of a small set of effective logical interactions and logical error channels. This subsection introduces a convenient parametrization of these processes that will be used in the subsequent separability analysis and in the calculation of effective error rates.
Throughout this section we write the effective corrected decoherence generator as
L e f f = R ∘ L d e c o h , (57) \mathcal{L}_{\mathrm{eff}}=\mathcal{R}\circ\mathcal{L}_{\mathrm{decoh}}, \tag*{(57)} L eff = R ∘ L decoh , (57) and we restrict attention to its action on operators supported on the code subspace.
Consider first the effective dynamics on a single encoded qubit (either party α = A \alpha=A α = A B B B L e f f \mathcal{L}_{\mathrm{eff}} L eff { ∣ i ˉ ⟩ α ⟨ j ˉ ∣ α } i , j ∈ 0 , 1 \{|\bar{i}\rangle_\alpha\langle\bar{j}|_\alpha\}_{i,j\in0,1} { ∣ i ˉ ⟩ α ⟨ j ˉ ∣ α } i , j ∈ 0 , 1
∣ i ˉ j ˉ ⟩ ⟩ α ≡ ∣ i ˉ ⟩ α ⟨ j ˉ ∣ α , (58) |\bar{i}\bar{j}\rangle\rangle_\alpha\equiv|\bar{i}\rangle_\alpha\langle\bar{j}|_\alpha, \tag*{(58)} ∣ i ˉ j ˉ ⟩ ⟩ α ≡ ∣ i ˉ ⟩ α ⟨ j ˉ ∣ α , (58) we will encounter generators of the form
Let us first study single-qubit errors. Without loss of generality, for the calculations in the following we will encounter effective Lindbladians in the double-ket notation as follows:
L e f f ( ∣ 0 ˉ 0 ˉ ⟩ ⟩ α ∣ 0 ˉ 1 ˉ ⟩ ⟩ α ∣ 1 ˉ 0 ˉ ⟩ ⟩ α ∣ 1 ˉ 1 ˉ ⟩ ⟩ α ) = ( − a 0 0 b 0 − c d 0 0 d − c 0 a 0 0 − b ) ( ∣ 0 ˉ 0 ˉ ⟩ ⟩ α ∣ 0 ˉ 1 ˉ ⟩ ⟩ α ∣ 1 ˉ 0 ˉ ⟩ ⟩ α ∣ 1 ˉ 1 ˉ ⟩ ⟩ α ) , (59) \mathcal{L}_{\mathrm{eff}}
\begin{pmatrix}
|\bar{0}\bar{0}\rangle\rangle_\alpha\\
|\bar{0}\bar{1}\rangle\rangle_\alpha\\
|\bar{1}\bar{0}\rangle\rangle_\alpha\\
|\bar{1}\bar{1}\rangle\rangle_\alpha
\end{pmatrix}
=
\begin{pmatrix}
-a & 0 & 0 & b\\
0 & -c & d & 0\\
0 & d & -c & 0\\
a & 0 & 0 & -b
\end{pmatrix}
\begin{pmatrix}
|\bar{0}\bar{0}\rangle\rangle_\alpha\\
|\bar{0}\bar{1}\rangle\rangle_\alpha\\
|\bar{1}\bar{0}\rangle\rangle_\alpha\\
|\bar{1}\bar{1}\rangle\rangle_\alpha
\end{pmatrix}, \tag*{(59)} L eff ∣ 0 ˉ 0 ˉ ⟩ ⟩ α ∣ 0 ˉ 1 ˉ ⟩ ⟩ α ∣ 1 ˉ 0 ˉ ⟩ ⟩ α ∣ 1 ˉ 1 ˉ ⟩ ⟩ α = − a 0 0 a 0 − c d 0 0 d − c 0 b 0 0 − b ∣ 0 ˉ 0 ˉ ⟩ ⟩ α ∣ 0 ˉ 1 ˉ ⟩ ⟩ α ∣ 1 ˉ 0 ˉ ⟩ ⟩ α ∣ 1 ˉ 1 ˉ ⟩ ⟩ α , (59) where a a a b b b c c c d d d R α R_\alpha R α
This class of generators can be rewritten in the standard Lindblad form on the logical qubit as
L e f f ( ρ ) = Γ e f f + D [ σ α + ] ( ρ ) + Γ e f f − D [ σ α − ] ( ρ ) + Γ e f f x D [ σ α x ] ( ρ ) + Γ e f f z D [ σ α z ] ( ρ ) , \mathcal{L}_{\mathrm{eff}}(\rho)=\Gamma_{\mathrm{eff}}^+D[\sigma_\alpha^+](\rho)+\Gamma_{\mathrm{eff}}^-D[\sigma_\alpha^-](\rho)+\Gamma_{\mathrm{eff}}^xD[\sigma_\alpha^x](\rho)+\Gamma_{\mathrm{eff}}^zD[\sigma_\alpha^z](\rho), L eff ( ρ ) = Γ eff + D [ σ α + ] ( ρ ) + Γ eff − D [ σ α − ] ( ρ ) + Γ eff x D [ σ α x ] ( ρ ) + Γ eff z D [ σ α z ] ( ρ ) , where Γ e f f x = d \Gamma_{\mathrm{eff}}^x=d Γ eff x = d Γ e f f + = a − d \Gamma_{\mathrm{eff}}^+=a-d Γ eff + = a − d Γ e f f − = b − d \Gamma_{\mathrm{eff}}^-=b-d Γ eff − = b − d Γ e f f z = 2 c − a − b 4 \Gamma_{\mathrm{eff}}^z=\frac{2c-a-b}{4} Γ eff z = 4 2 c − a − b
In addition to local terms, some microscopic models generate symmetric two-body decoherence contributions that, after the local recovery, take the structured form, see also App. C 3:
L e f f ( ρ ) = + Γ e f f z z ( D [ σ A z − σ B z ] − D [ σ A z ] − D [ σ B z ] ) ( ρ ) + Γ e f f x x ( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ] ) ( ρ ) + Γ e f f x z ( D [ σ A x − σ B z ] − D [ σ A x ] − D [ σ B z ] − D [ σ A x − σ B z ] + D [ σ A x ] + D [ σ B z ] ) ( ρ ) , (60) \begin{aligned}
\mathcal{L}_{\mathrm{eff}}(\rho)={}&+\Gamma_{\mathrm{eff}}^{zz}\left(D[\sigma_A^z-\sigma_B^z]-D[\sigma_A^z]-D[\sigma_B^z]\right)(\rho)+\Gamma_{\mathrm{eff}}^{xx}\left(D[\sigma_A^x-\sigma_B^x]-D[\sigma_A^x]-D[\sigma_B^x]\right)(\rho)\\
&+\Gamma_{\mathrm{eff}}^{xz}\left(D[\sigma_A^x-\sigma_B^z]-D[\sigma_A^x]-D[\sigma_B^z]-D[\sigma_A^x-\sigma_B^z]+D[\sigma_A^x]+D[\sigma_B^z]\right)(\rho), \tag*{(60)}
\end{aligned} L eff ( ρ ) = + Γ eff z z ( D [ σ A z − σ B z ] − D [ σ A z ] − D [ σ B z ] ) ( ρ ) + Γ eff xx ( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ] ) ( ρ ) + Γ eff x z ( D [ σ A x − σ B z ] − D [ σ A x ] − D [ σ B z ] − D [ σ A x − σ B z ] + D [ σ A x ] + D [ σ B z ] ) ( ρ ) , (60) This structure reflects the symmetry of the underlying decoherence Lindbladian and isolates the genuinely two-party contributions into differences of local logical operators.
A particularly useful identity is
( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ] ) ( ρ ) = σ A x ⊗ σ B x ρ + ρ σ A x ⊗ σ B x − σ A x ρ σ B x − σ B x ρ σ A x . (61) \left(D[\sigma_A^x-\sigma_B^x]-D[\sigma_A^x]-D[\sigma_B^x]\right)(\rho)=\sigma_A^x\otimes\sigma_B^x\rho+\rho\sigma_A^x\otimes\sigma_B^x-\sigma_A^x\rho\sigma_B^x-\sigma_B^x\rho\sigma_A^x. \tag*{(61)} ( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ] ) ( ρ ) = σ A x ⊗ σ B x ρ + ρ σ A x ⊗ σ B x − σ A x ρ σ B x − σ B x ρ σ A x . (61) which makes it explicit that this term correlates the two systems even though it is generated by a single Lindblad operator of the form σ A x − σ B x \sigma_A^x-\sigma_B^x σ A x − σ B x
Finally, it is convenient to extract the coefficients Γ e f f z z \Gamma_{\mathrm{eff}}^{zz} Γ eff z z Γ e f f x z \Gamma_{\mathrm{eff}}^{xz} Γ eff x z Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx
4 Γ e f f z z = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ L e f f ( ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B + A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ L e f f ( ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ L e f f ( ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B − A ⟨ 1 ˉ ∣ B ⟨ 0 ˉ ∣ L e f f ( ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B , 2 Γ e f f x z = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ L e f f ( ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ L e f f ( ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B , Γ e f f x x = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ L e f f ( ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B . (62) \begin{aligned}
4\Gamma_{\mathrm{eff}}^{zz}={}&{}_A\langle\bar{0}|{}_B\langle\bar{0}|\mathcal{L}_{\mathrm{eff}}(|\bar{0}\rangle_A|\bar{0}\rangle_B)_A\langle\bar{0}|_B\langle\bar{0}|{}_A\langle\bar{0}|_B\\
&+{}_A\langle\bar{1}|{}_B\langle\bar{1}|\mathcal{L}_{\mathrm{eff}}(|\bar{1}\rangle_A|\bar{1}\rangle_B)_A\langle\bar{0}|_B\langle\bar{0}|{}_A\langle\bar{0}|_B\\
&-{}_A\langle\bar{0}|{}_B\langle\bar{1}|\mathcal{L}_{\mathrm{eff}}(|\bar{0}\rangle_A|\bar{1}\rangle_B)_A\langle\bar{0}|_B\langle\bar{0}|{}_A\langle\bar{0}|_B\\
&-{}_A\langle\bar{1}|{}_B\langle\bar{0}|\mathcal{L}_{\mathrm{eff}}(|\bar{1}\rangle_A|\bar{0}\rangle_B)_A\langle\bar{0}|_B\langle\bar{0}|{}_A\langle\bar{0}|_B,\\
2\Gamma_{\mathrm{eff}}^{xz}={}&{}_A\langle\bar{0}|{}_B\langle\bar{0}|\mathcal{L}_{\mathrm{eff}}(|\bar{0}\rangle_A|\bar{1}\rangle_B)_A\langle\bar{0}|_B\langle\bar{0}|{}_A\langle\bar{0}|_B-{}_A\langle\bar{0}|{}_B\langle\bar{1}|\mathcal{L}_{\mathrm{eff}}(|\bar{1}\rangle_A|\bar{1}\rangle_B)_A\langle\bar{0}|_B\langle\bar{0}|{}_A\langle\bar{0}|_B,\\
\Gamma_{\mathrm{eff}}^{xx}={}&{}_A\langle\bar{0}|{}_B\langle\bar{0}|\mathcal{L}_{\mathrm{eff}}(|\bar{1}\rangle_A|\bar{1}\rangle_B)_A\langle\bar{0}|_B\langle\bar{0}|{}_A\langle\bar{0}|_B. \tag*{(62)}
\end{aligned} 4 Γ eff z z = 2 Γ eff x z = Γ eff xx = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ L eff ( ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B + A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ L eff ( ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ L eff ( ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B − A ⟨ 1 ˉ ∣ B ⟨ 0 ˉ ∣ L eff ( ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B , A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ L eff ( ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ L eff ( ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B , A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ L eff ( ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ) A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ A ⟨ 0 ˉ ∣ B . (62) These relations will be used later to calculate effective logical error rates in a model-independent way and to connect them to the separability criteria discussed in App. B 5.
Separability of the effective Lindbladian
The separability criterion stated in Theorem 1 of the main text is derived here in detail. Using the effective logical interaction and error channels introduced above, we determine when the resulting Lindbladian is non-separable or entanglement-generating and when it admits an equivalent separable realization.
In general, in this section let us deal with the Lindbladian of the following form
L t o t = L e f f x x + L e f f z z + L e f f x z + L e f f ± \mathcal{L}_{\mathrm{tot}}=\mathcal{L}_{\mathrm{eff}}^{xx}+\mathcal{L}_{\mathrm{eff}}^{zz}+\mathcal{L}_{\mathrm{eff}}^{xz}+\mathcal{L}_{\mathrm{eff}}^{\pm} L tot = L eff xx + L eff z z + L eff x z + L eff ± L e f f x x ( ρ ) = − i λ e f f x x [ σ A x ⊗ σ B x , ρ ] + Γ e f f x ( σ A x ρ σ A x − ρ ) + Γ e f f x ( σ B x ρ σ B x − ρ ) + Γ e f f x x ( σ A x σ B x ρ + ρ σ A x σ B x − σ A x ρ σ B x − σ B x ρ σ A x ) , \mathcal{L}_{\mathrm{eff}}^{xx}(\rho)=-i\lambda_{\mathrm{eff}}^{xx}[\sigma_{A}^{x}\otimes\sigma_{B}^{x},\rho]+\Gamma_{\mathrm{eff}}^{x}(\sigma_{A}^{x}\rho\sigma_{A}^{x}-\rho)+\Gamma_{\mathrm{eff}}^{x}(\sigma_{B}^{x}\rho\sigma_{B}^{x}-\rho)+\Gamma_{\mathrm{eff}}^{xx}(\sigma_{A}^{x}\sigma_{B}^{x}\rho+\rho\sigma_{A}^{x}\sigma_{B}^{x}-\sigma_{A}^{x}\rho\sigma_{B}^{x}-\sigma_{B}^{x}\rho\sigma_{A}^{x}), L eff xx ( ρ ) = − i λ eff xx [ σ A x ⊗ σ B x , ρ ] + Γ eff x ( σ A x ρ σ A x − ρ ) + Γ eff x ( σ B x ρ σ B x − ρ ) + Γ eff xx ( σ A x σ B x ρ + ρ σ A x σ B x − σ A x ρ σ B x − σ B x ρ σ A x ) , L e f f z z ( ρ ) = − i λ e f f z z [ σ A z ⊗ σ B z , ρ ] + Γ e f f z ( σ A z ρ σ A z − ρ ) + Γ e f f z ( σ B z ρ σ B z − ρ ) + Γ e f f z z ( σ A z σ B z ρ + ρ σ A z σ B z − σ A z ρ σ B z − σ B z ρ σ A z ) , \mathcal{L}_{\mathrm{eff}}^{zz}(\rho)=-i\lambda_{\mathrm{eff}}^{zz}[\sigma_{A}^{z}\otimes\sigma_{B}^{z},\rho]+\Gamma_{\mathrm{eff}}^{z}(\sigma_{A}^{z}\rho\sigma_{A}^{z}-\rho)+\Gamma_{\mathrm{eff}}^{z}(\sigma_{B}^{z}\rho\sigma_{B}^{z}-\rho)+\Gamma_{\mathrm{eff}}^{zz}(\sigma_{A}^{z}\sigma_{B}^{z}\rho+\rho\sigma_{A}^{z}\sigma_{B}^{z}-\sigma_{A}^{z}\rho\sigma_{B}^{z}-\sigma_{B}^{z}\rho\sigma_{A}^{z}), L eff z z ( ρ ) = − i λ eff z z [ σ A z ⊗ σ B z , ρ ] + Γ eff z ( σ A z ρ σ A z − ρ ) + Γ eff z ( σ B z ρ σ B z − ρ ) + Γ eff z z ( σ A z σ B z ρ + ρ σ A z σ B z − σ A z ρ σ B z − σ B z ρ σ A z ) , L e f f x z ( ρ ) = − i λ e f f x z [ σ A x ⊗ σ B z − σ A z ⊗ σ B x , ρ ] + Γ e f f x z ( σ A x σ B z ρ + ρ σ A z σ B x − σ A x ρ σ B z − σ B z ρ σ A x − σ A z σ B x ρ − ρ σ A x σ B z + σ A x ρ σ B z + σ B z ρ σ A x ) , \begin{aligned}
\mathcal{L}_{\mathrm{eff}}^{xz}(\rho)={}&-i\lambda_{\mathrm{eff}}^{xz}[\sigma_{A}^{x}\otimes\sigma_{B}^{z}-\sigma_{A}^{z}\otimes\sigma_{B}^{x},\rho]\\
&+\Gamma_{\mathrm{eff}}^{xz}(\sigma_{A}^{x}\sigma_{B}^{z}\rho+\rho\sigma_{A}^{z}\sigma_{B}^{x}-\sigma_{A}^{x}\rho\sigma_{B}^{z}-\sigma_{B}^{z}\rho\sigma_{A}^{x}\\
&\qquad-\sigma_{A}^{z}\sigma_{B}^{x}\rho-\rho\sigma_{A}^{x}\sigma_{B}^{z}+\sigma_{A}^{x}\rho\sigma_{B}^{z}+\sigma_{B}^{z}\rho\sigma_{A}^{x}),
\end{aligned} L eff x z ( ρ ) = − i λ eff x z [ σ A x ⊗ σ B z − σ A z ⊗ σ B x , ρ ] + Γ eff x z ( σ A x σ B z ρ + ρ σ A z σ B x − σ A x ρ σ B z − σ B z ρ σ A x − σ A z σ B x ρ − ρ σ A x σ B z + σ A x ρ σ B z + σ B z ρ σ A x ) , L e f f ± ( ρ ) = ∑ α = A , B ( Γ e f f + D [ σ α + ] ( ρ ) + Γ e f f − D [ σ α − ] ( ρ ) ) . (63) \mathcal{L}_{\mathrm{eff}}^{\pm}(\rho)=\sum_{\alpha=A,B}\left(\Gamma_{\mathrm{eff}}^{+}\mathcal{D}[\sigma_{\alpha}^{+}](\rho)+\Gamma_{\mathrm{eff}}^{-}\mathcal{D}[\sigma_{\alpha}^{-}](\rho)\right). \tag*{(63)} L eff ± ( ρ ) = α = A , B ∑ ( Γ eff + D [ σ α + ] ( ρ ) + Γ eff − D [ σ α − ] ( ρ ) ) . (63) which is the most general form of the effective Lindbladian in Eq. (8) and App. C, up to local Hamiltonian terms, which does not affect the separability classification.
Separability classification
Let us first prove a lemma that will be used for the proof of the theorem below. This lemma considers a bipartite system A , B A,B A , B O A , O B O_A,O_B O A , O B
Lemma 3. Consider the following Lindbladian defined on a bipartite system:
L ( ρ ) = − i λ [ O A ⊗ O B , ρ ] + Γ A D [ O A ] ( ρ ) + Γ B D [ O B ] ( ρ ) + Γ A B ( O A O B ρ + ρ O A O B − O A ρ O B − O B ρ O A ) , (64) \mathcal{L}(\rho)=-i\lambda[O_A\otimes O_B,\rho]+\Gamma_A\mathcal{D}[O_A](\rho)+\Gamma_B\mathcal{D}[O_B](\rho)+\Gamma_{AB}(O_AO_B\rho+\rho O_AO_B-O_A\rho O_B-O_B\rho O_A), \tag*{(64)} L ( ρ ) = − iλ [ O A ⊗ O B , ρ ] + Γ A D [ O A ] ( ρ ) + Γ B D [ O B ] ( ρ ) + Γ A B ( O A O B ρ + ρ O A O B − O A ρ O B − O B ρ O A ) , (64) where O A , O B O_A,O_B O A , O B A , B A,B A , B λ \lambda λ Γ A \Gamma_A Γ A Γ B \Gamma_B Γ B Γ A B \Gamma_{AB} Γ A B
When λ 2 + Γ A B 2 > Γ A Γ B \lambda^{2}+\Gamma_{AB}^{2}>\Gamma_A\Gamma_B λ 2 + Γ A B 2 > Γ A Γ B L \mathcal{L} L
When λ 2 + Γ A B 2 ≤ Γ A Γ B \lambda^{2}+\Gamma_{AB}^{2}\le\Gamma_A\Gamma_B λ 2 + Γ A B 2 ≤ Γ A Γ B L \mathcal{L} L λ 2 ≤ ( Γ A − ∣ Γ A B ∣ ) ( Γ B − ∣ Γ A B ∣ ) \lambda^{2}\le(\Gamma_A-\lvert\Gamma_{AB}\rvert)(\Gamma_B-\lvert\Gamma_{AB}\rvert) λ 2 ≤ ( Γ A − ∣ Γ A B ∣) ( Γ B − ∣ Γ A B ∣)
Note that complete positivity of L \mathcal{L} L Γ A , Γ B ≥ 0 \Gamma_A,\Gamma_B\ge0 Γ A , Γ B ≥ 0 ∣ Γ A B ∣ ≤ Γ A Γ B \lvert\Gamma_{AB}\rvert\le\sqrt{\Gamma_A\Gamma_B} ∣ Γ A B ∣ ≤ Γ A Γ B
Proof. Let us first study the non-separable side, i.e. when λ 2 + Γ A B 2 > Γ A Γ B \lambda^{2}+\Gamma_{AB}^{2}>\Gamma_A\Gamma_B λ 2 + Γ A B 2 > Γ A Γ B ∣ ψ ⟩ A ∣ ϕ ⟩ B \lvert\psi\rangle_A\lvert\phi\rangle_B ∣ ψ ⟩ A ∣ ϕ ⟩ B L \mathcal{L} L O A → O A + a I A , O B → O B + b I B O_A\to O_A+aI_A,O_B\to O_B+bI_B O A → O A + a I A , O B → O B + b I B a , b a,b a , b L \mathcal{L} L ⟨ ψ ∣ O A ∣ ψ ⟩ A = ⟨ ϕ ∣ O B ∣ ϕ ⟩ B = 0 \langle\psi\lvert O_A\lvert\psi\rangle_A=\langle\phi\lvert O_B\lvert\phi\rangle_B=0 ⟨ ψ ∣ O A ∣ ψ ⟩ A = ⟨ ϕ ∣ O B ∣ ϕ ⟩ B = 0 O A ∣ ψ ⟩ A = ∣ ψ O ⟩ A , O B ∣ ϕ ⟩ B = ∣ ϕ O ⟩ B O_A\lvert\psi\rangle_A=\lvert\psi_O\rangle_A,O_B\lvert\phi\rangle_B=\lvert\phi_O\rangle_B O A ∣ ψ ⟩ A = ∣ ψ O ⟩ A , O B ∣ ϕ ⟩ B = ∣ ϕ O ⟩ B O A , O B O_A,O_B O A , O B span { ∣ ψ ⟩ A ∣ ϕ ⟩ B , ∣ ψ ⟩ A ∣ ϕ O ⟩ B , ∣ ψ O ⟩ A ∣ ϕ ⟩ B , ∣ ψ O ⟩ A ∣ ϕ O ⟩ B } \operatorname{span}\{\lvert\psi\rangle_A\lvert\phi\rangle_B,\lvert\psi\rangle_A\lvert\phi_O\rangle_B,\lvert\psi_O\rangle_A\lvert\phi\rangle_B,\lvert\psi_O\rangle_A\lvert\phi_O\rangle_B\} span {∣ ψ ⟩ A ∣ ϕ ⟩ B , ∣ ψ ⟩ A ∣ ϕ O ⟩ B , ∣ ψ O ⟩ A ∣ ϕ ⟩ B , ∣ ψ O ⟩ A ∣ ϕ O ⟩ B }
L ( ∣ ψ ⟩ A ∣ ψ ⟩ A ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣ ) = ( − Γ A − Γ B − Γ B ⟨ ϕ ∣ O B 2 ∣ ϕ ⟩ B 2 − Γ A ⟨ ψ ∣ O A 2 ∣ ψ ⟩ A 2 Γ A B + i λ − Γ B ⟨ ϕ ∣ O B 2 ∣ ϕ ⟩ B 2 Γ B − Γ A B 0 − Γ A ⟨ ψ ∣ O A 2 ∣ ψ ⟩ A 2 − Γ A B Γ A 0 Γ A B − i λ 0 0 0 ) + ⋯ (65) \begin{aligned}
\mathcal{L}(\lvert\psi\rangle_A\lvert\psi\rangle_A\otimes\lvert\phi\rangle_B\langle\phi\rvert)={}&
\begin{pmatrix}
-\Gamma_A-\Gamma_B & -\Gamma_B\frac{\langle\phi\lvert O_B^2\lvert\phi\rangle_B}{2} & -\Gamma_A\frac{\langle\psi\lvert O_A^2\lvert\psi\rangle_A}{2} & \Gamma_{AB}+i\lambda\\
-\Gamma_B\frac{\langle\phi\lvert O_B^2\lvert\phi\rangle_B}{2} & \Gamma_B & -\Gamma_{AB} & 0\\
-\Gamma_A\frac{\langle\psi\lvert O_A^2\lvert\psi\rangle_A}{2} & -\Gamma_{AB} & \Gamma_A & 0\\
\Gamma_{AB}-i\lambda& 0 & 0 & 0
\end{pmatrix}
+\cdots \tag*{(65)}
\end{aligned} L (∣ ψ ⟩ A ∣ ψ ⟩ A ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣) = − Γ A − Γ B − Γ B 2 ⟨ ϕ ∣ O B 2 ∣ ϕ ⟩ B − Γ A 2 ⟨ ψ ∣ O A 2 ∣ ψ ⟩ A Γ A B − iλ − Γ B 2 ⟨ ϕ ∣ O B 2 ∣ ϕ ⟩ B Γ B − Γ A B 0 − Γ A 2 ⟨ ψ ∣ O A 2 ∣ ψ ⟩ A − Γ A B Γ A 0 Γ A B + iλ 0 0 0 + ⋯ (65) Note that in the subspace Π = ∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ O ⟩ B ⟨ ϕ ∣ + ∣ ψ O ⟩ A ⟨ ψ O ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣ \Pi=\lvert\psi\rangle_A\langle\psi\rvert\otimes\lvert\phi_O\rangle_B\langle\phi\rvert+\lvert\psi_O\rangle_A\langle\psi_O\rvert\otimes\lvert\phi\rangle_B\langle\phi\rvert Π = ∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ O ⟩ B ⟨ ϕ ∣ + ∣ ψ O ⟩ A ⟨ ψ O ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣
det Π × L T B ( ∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣ ) × Π = det ( Γ B Γ A B − i λ Γ A B + i λ Γ A ) = Γ A Γ B − Γ A B 2 − λ 2 (66) \det\Pi\times\mathcal{L}_{T_B}(\lvert\psi\rangle_A\langle\psi\rvert\otimes\lvert\phi\rangle_B\langle\phi\rvert)\times\Pi
=\det\begin{pmatrix}\Gamma_B&\Gamma_{AB}-i\lambda\\\Gamma_{AB}+i\lambda&\Gamma_A\end{pmatrix}
=\Gamma_A\Gamma_B-\Gamma_{AB}^{2}-\lambda^{2} \tag*{(66)} det Π × L T B (∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣) × Π = det ( Γ B Γ A B + iλ Γ A B − iλ Γ A ) = Γ A Γ B − Γ A B 2 − λ 2 (66) where T B T_B T B B B B L \mathcal{L} L λ 2 + Γ A B 2 > Γ A Γ B \lambda^2+\Gamma_{AB}^2>\Gamma_A\Gamma_B λ 2 + Γ A B 2 > Γ A Γ B exp L t \exp\mathcal{L}t exp L t t > 0 t>0 t > 0
On the other hand, let us explicitly construct a separable construction of the evolution channel. Let us consider the following Kraus operators:
K 1 , 2 = ( I A ± c O A ) ( 2 I A + 2 ∣ c ∣ 2 O A 2 ) − 1 / 2 ⊗ ( I B ± d O B ) ( 2 I B + 2 ∣ d ∣ 2 O B 2 ) − 1 / 2 , K 3 , 4 = ( I A ± e O A ) ( 2 I A + 2 ∣ e ∣ 2 O A 2 ) − 1 / 2 ⊗ ( I B ± f O B ) ( 2 I B + 2 ∣ f ∣ 2 O B 2 ) − 1 / 2 . (67) \begin{aligned}
K_{1,2} &=(I_A\pm cO_A)(2I_A+2|c|^2O_A^2)^{-1/2}\otimes(I_B\pm dO_B)(2I_B+2|d|^2O_B^2)^{-1/2},\\
K_{3,4} &=(I_A\pm eO_A)(2I_A+2|e|^2O_A^2)^{-1/2}\otimes(I_B\pm fO_B)(2I_B+2|f|^2O_B^2)^{-1/2}. \tag*{(67)}
\end{aligned} K 1 , 2 K 3 , 4 = ( I A ± c O A ) ( 2 I A + 2∣ c ∣ 2 O A 2 ) − 1/2 ⊗ ( I B ± d O B ) ( 2 I B + 2∣ d ∣ 2 O B 2 ) − 1/2 , = ( I A ± e O A ) ( 2 I A + 2∣ e ∣ 2 O A 2 ) − 1/2 ⊗ ( I B ± f O B ) ( 2 I B + 2∣ f ∣ 2 O B 2 ) − 1/2 . (67) These Kraus operators are trace-preserving as ∑ i = 1 4 K i † K i = I A B \sum_{i=1}^{4}K_i^\dagger K_i=I_{AB} ∑ i = 1 4 K i † K i = I A B ∣ c ∣ 2 = ∣ e ∣ 2 |c|^2=|e|^2 ∣ c ∣ 2 = ∣ e ∣ 2 ∣ d ∣ 2 = ∣ f ∣ 2 |d|^2=|f|^2 ∣ d ∣ 2 = ∣ f ∣ 2 Re c Re d + Re e Re f = 0 \operatorname{Re}c\operatorname{Re}d+\operatorname{Re}e\operatorname{Re}f=0 Re c Re d + Re e Re f = 0 c , d c,d c , d O ( d t ) O(\sqrt{dt}) O ( d t ) d t dt d t
E ( ρ ) = ∑ i = 1 4 K i ρ K i † = ρ + ∣ c ∣ 2 D [ O A ] ( ρ ) + ∣ d ∣ 2 D [ O B ] ( ρ ) + Re c d + e f 2 ( O A O B ρ + ρ O A O B − O A ρ O B − O B ρ O A ) + i Im c d + e f 2 [ O A O B , ρ ] + i Im c d ∗ + e f ∗ 2 ( O A ρ O B − O B ρ O A ) + O ( d t 2 ) . (68) \begin{aligned}
\mathcal{E}(\rho)&=\sum_{i=1}^{4}K_i\rho K_i^\dagger\\
&=\rho+|c|^2\mathcal{D}[O_A](\rho)+|d|^2\mathcal{D}[O_B](\rho)+\operatorname{Re}\frac{cd+ef}{2}(O_AO_B\rho+\rho O_AO_B-O_A\rho O_B-O_B\rho O_A)\\
&\quad+i\operatorname{Im}\frac{cd+ef}{2}[O_AO_B,\rho]+i\operatorname{Im}\frac{cd^*+ef^*}{2}(O_A\rho O_B-O_B\rho O_A)+O(dt^2). \tag*{(68)}
\end{aligned} E ( ρ ) = i = 1 ∑ 4 K i ρ K i † = ρ + ∣ c ∣ 2 D [ O A ] ( ρ ) + ∣ d ∣ 2 D [ O B ] ( ρ ) + Re 2 c d + e f ( O A O B ρ + ρ O A O B − O A ρ O B − O B ρ O A ) + i Im 2 c d + e f [ O A O B , ρ ] + i Im 2 c d ∗ + e f ∗ ( O A ρ O B − O B ρ O A ) + O ( d t 2 ) . (68) Note that the i O A ρ O B − i O B ρ O A iO_A\rho O_B-iO_B\rho O_A i O A ρ O B − i O B ρ O A
∣ c ∣ 2 = Γ A d t , ∣ d ∣ 2 = Γ B d t , Re c d + e f 2 = Γ A B d t , Im c d + e f 2 = − λ d t , Im c d ∗ + e f ∗ 2 = 0. (69) |c|^2=\Gamma_A dt,\qquad|d|^2=\Gamma_B dt,\qquad\operatorname{Re}\frac{cd+ef}{2}=\Gamma_{AB}dt,\qquad\operatorname{Im}\frac{cd+ef}{2}=-\lambda dt,\qquad\operatorname{Im}\frac{cd^*+ef^*}{2}=0. \tag*{(69)} ∣ c ∣ 2 = Γ A d t , ∣ d ∣ 2 = Γ B d t , Re 2 c d + e f = Γ A B d t , Im 2 c d + e f = − λ d t , Im 2 c d ∗ + e f ∗ = 0. (69) More specifically, these equations have solution so long as ( λ 2 + Γ A B 2 ) d t 2 = ∣ c d + e f ∣ 2 / 4 ≤ ∣ c ∣ 2 ∣ d ∣ 2 = Γ A Γ B d t 2 (\lambda^2+\Gamma_{AB}^2)dt^2=|cd+ef|^2/4\leq|c|^2|d|^2=\Gamma_A\Gamma_Bdt^2 ( λ 2 + Γ A B 2 ) d t 2 = ∣ c d + e f ∣ 2 /4 ≤ ∣ c ∣ 2 ∣ d ∣ 2 = Γ A Γ B d t 2
c = Γ A d t e i α + β + θ 2 , d = Γ B d t e i α + β − θ 2 , e = Γ A d t e i α − β − θ 2 , f = Γ B d t e i α − β + θ 2 , c=\sqrt{\Gamma_A dt}e^{i\frac{\alpha+\beta+\theta}{2}},\qquad d=\sqrt{\Gamma_Bdt}e^{i\frac{\alpha+\beta-\theta}{2}},\qquad e=\sqrt{\Gamma_A dt}e^{i\frac{\alpha-\beta-\theta}{2}},\qquad f=\sqrt{\Gamma_Bdt}e^{i\frac{\alpha-\beta+\theta}{2}}, c = Γ A d t e i 2 α + β + θ , d = Γ B d t e i 2 α + β − θ , e = Γ A d t e i 2 α − β − θ , f = Γ B d t e i 2 α − β + θ , where α = arg Γ A B − i λ \alpha=\arg\Gamma_{AB}-i\lambda α = arg Γ A B − iλ β = arccos Γ A B 2 + λ 2 Γ A Γ B \beta=\arccos\sqrt{\frac{\Gamma_{AB}^2+\lambda^2}{\Gamma_A\Gamma_B}} β = arccos Γ A Γ B Γ A B 2 + λ 2 θ = arccos − Γ A B Γ A Γ B \theta=\arccos-\frac{\Gamma_{AB}}{\sqrt{\Gamma_A\Gamma_B}} θ = arccos − Γ A Γ B Γ A B
{#eq:B26}
and we take α = 0 \alpha=0 α = 0 Γ A B = λ = 0 \Gamma_{AB}=\lambda=0 Γ A B = λ = 0 E ( ρ , d t ) = ρ + L ( ρ ) d t + O ( d t 2 ) \mathcal{E}(\rho,dt)=\rho+\mathcal{L}(\rho)dt+O(dt^2) E ( ρ , d t ) = ρ + L ( ρ ) d t + O ( d t 2 ) exp L t \exp\mathcal{L}t exp L t exp L t \exp\mathcal{L}t exp L t
exp L t ( ρ ) = lim N → ∞ E N ( ρ , t / N ) (70) \exp\mathcal{L}t(\rho)=\lim_{N\to\infty}\mathcal{E}^N(\rho,t/N) \tag*{(70)} exp L t ( ρ ) = N → ∞ lim E N ( ρ , t / N ) (70) can be realized by composition of separable channels.
On the other hand, let us show this Lindbladian can also be realized in a LOCC fashion with a smaller parameter regime. Following the KTM construction in App. A 1, we may take O A O_A O A O B O_B O B
L L O C C ( ρ ) = − i [ g O A ⊗ h O B , ρ ] + D [ g O A − i h O B ] ( ρ ) − i [ m O A ⊗ n O B , ρ ] + D [ n O B − i m O A ] ( ρ ) = − i ( g h + m n ) [ O A ⊗ O B , ρ ] + ( g 2 + m 2 ) D [ O A ] ( ρ ) + ( h 2 + n 2 ) D [ O B ] ( ρ ) , (71) \begin{aligned}
\mathcal{L}_{\mathrm{LOCC}}(\rho)&=-i[gO_A\otimes hO_B,\rho]+\mathcal{D}[gO_A-ihO_B](\rho)\\
&\quad-i[mO_A\otimes nO_B,\rho]+\mathcal{D}[nO_B-imO_A](\rho)\\
&=-i(gh+mn)[O_A\otimes O_B,\rho]+(g^2+m^2)\mathcal{D}[O_A](\rho)+(h^2+n^2)\mathcal{D}[O_B](\rho), \tag*{(71)}
\end{aligned} L LOCC ( ρ ) = − i [ g O A ⊗ h O B , ρ ] + D [ g O A − ih O B ] ( ρ ) − i [ m O A ⊗ n O B , ρ ] + D [ n O B − im O A ] ( ρ ) = − i ( g h + mn ) [ O A ⊗ O B , ρ ] + ( g 2 + m 2 ) D [ O A ] ( ρ ) + ( h 2 + n 2 ) D [ O B ] ( ρ ) , (71) where g , h , m , n > 0 g,h,m,n>0 g , h , m , n > 0 g h = m n gh=mn g h = mn Γ A B \Gamma_{AB} Γ A B
E η e i η O A e − i η O B ρ e − i η O A e i η O B = ρ + E η i η [ ( O A − O B ) ρ − ρ ( O A − O B ) ] + E η η 2 [ ( O A − O B ) ρ ( O A − O B ) − ( O A − O B ) 2 2 ρ − ρ ( O A − O B ) 2 2 ] + ⋯ = ρ + E η η 2 D [ O A − O B ] ( ρ ) + O ( η 4 ) , (72) \begin{aligned}
\mathbb{E}_{\eta}e^{i\eta O_A}e^{-i\eta O_B}\rho e^{-i\eta O_A}e^{i\eta O_B}
&=\rho+\mathbb{E}_{\eta}i\eta[(O_A-O_B)\rho-\rho(O_A-O_B)]\\
&\quad+\mathbb{E}_{\eta}\eta^2\left[(O_A-O_B)\rho(O_A-O_B)-\frac{(O_A-O_B)^2}{2}\rho-\rho\frac{(O_A-O_B)^2}{2}\right]+\cdots\\
&=\rho+\mathbb{E}_{\eta}\eta^2\mathcal{D}[O_A-O_B](\rho)+O(\eta^4), \tag*{(72)}
\end{aligned} E η e i η O A e − i η O B ρ e − i η O A e i η O B = ρ + E η i η [( O A − O B ) ρ − ρ ( O A − O B )] + E η η 2 [ ( O A − O B ) ρ ( O A − O B ) − 2 ( O A − O B ) 2 ρ − ρ 2 ( O A − O B ) 2 ] + ⋯ = ρ + E η η 2 D [ O A − O B ] ( ρ ) + O ( η 4 ) , (72) assuming the random variable satisfied E η η = E η η 3 = 0 \mathbb{E}_{\eta}\eta=\mathbb{E}_{\eta}\eta^3=0 E η η = E η η 3 = 0
E η e i η O A e i η O B ρ e − i η O A e − i η O B = ρ + E η η 2 D [ O A + O B ] ( ρ ) + O ( η 4 ) , (73) \mathbb{E}_{\eta}e^{i\eta O_A}e^{i\eta O_B}\rho e^{-i\eta O_A}e^{-i\eta O_B}=\rho+\mathbb{E}_{\eta}\eta^2\mathcal{D}[O_A+O_B](\rho)+O(\eta^4), \tag*{(73)} E η e i η O A e i η O B ρ e − i η O A e − i η O B = ρ + E η η 2 D [ O A + O B ] ( ρ ) + O ( η 4 ) , (73) Combining Eq. (71) –(B30), and assuming E η 2 = p d t \mathbb{E}\eta^{2}=p\,dt E η 2 = p d t
− i ( g h + m n ) [ O A ⊗ O B , ρ ] + ( g 2 + m 2 + p ) D [ O A ] ( ρ ) + ( h 2 + n 2 + p ) D [ O B ] ( ρ ) ± p ( O A O B ρ + ρ O A O B − O A ρ O B − O B ρ O A ) . (74) -i(gh+mn)[O_{A}\otimes O_{B},\rho]+(g^{2}+m^{2}+p)D[O_{A}](\rho)+(h^{2}+n^{2}+p)D[O_{B}](\rho)\pm p(O_{A}O_{B}\rho+\rho O_{A}O_{B}-O_{A}\rho O_{B}-O_{B}\rho O_{A}). \tag*{(74)} − i ( g h + mn ) [ O A ⊗ O B , ρ ] + ( g 2 + m 2 + p ) D [ O A ] ( ρ ) + ( h 2 + n 2 + p ) D [ O B ] ( ρ ) ± p ( O A O B ρ + ρ O A O B − O A ρ O B − O B ρ O A ) . (74) Therefore, L \mathcal{L} L
λ 2 = ( g h + m n ) 2 ≤ ( g 2 + m 2 ) ( h 2 + n 2 ) = ( Γ A − ∣ Γ A B ∣ ) ( Γ B − ∣ Γ A B ∣ ) , (75) \lambda^{2}=(gh+mn)^{2}\leq(g^{2}+m^{2})(h^{2}+n^{2})=(\Gamma_{A}-\lvert\Gamma_{AB}\rvert)(\Gamma_{B}-\lvert\Gamma_{AB}\rvert), \tag*{(75)} λ 2 = ( g h + mn ) 2 ≤ ( g 2 + m 2 ) ( h 2 + n 2 ) = ( Γ A − ∣ Γ A B ∣) ( Γ B − ∣ Γ A B ∣) , (75) in the case Γ A , Γ B ≥ ∣ Γ A B ∣ \Gamma_{A},\Gamma_{B}\geq\lvert\Gamma_{AB}\rvert Γ A , Γ B ≥ ∣ Γ A B ∣ g = Γ A − ∣ Γ A B ∣ cos ξ g=\sqrt{\Gamma_{A}-\lvert\Gamma_{AB}\rvert}\cos\xi g = Γ A − ∣ Γ A B ∣ cos ξ m = Γ A − ∣ Γ A B ∣ sin ξ m=\sqrt{\Gamma_{A}-\lvert\Gamma_{AB}\rvert}\sin\xi m = Γ A − ∣ Γ A B ∣ sin ξ h = λ / 2 g h=\lambda/2g h = λ /2 g n = λ / 2 m n=\lambda/2m n = λ /2 m sin 2 ξ = ∣ λ ∣ / ( Γ A − ∣ Γ A B ∣ ) ( Γ B − ∣ Γ A B ∣ ) \sin2\xi=\lvert\lambda\rvert/\sqrt{(\Gamma_{A}-\lvert\Gamma_{AB}\rvert)(\Gamma_{B}-\lvert\Gamma_{AB}\rvert)} sin 2 ξ = ∣ λ ∣ / ( Γ A − ∣ Γ A B ∣) ( Γ B − ∣ Γ A B ∣)
Note that there is a finite gap, where L \mathcal{L} L
Γ A Γ B − ( Γ A + Γ B ) ∣ Γ A B ∣ + Γ A B 2 < λ 2 ≤ Γ A Γ B − Γ A B 2 , (76) \Gamma_{A}\Gamma_{B}-(\Gamma_{A}+\Gamma_{B})\lvert\Gamma_{AB}\rvert+\Gamma_{AB}^{2}<\lambda^{2}\leq\Gamma_{A}\Gamma_{B}-\Gamma_{AB}^{2}, \tag*{(76)} Γ A Γ B − ( Γ A + Γ B ) ∣ Γ A B ∣ + Γ A B 2 < λ 2 ≤ Γ A Γ B − Γ A B 2 , (76) since complete positivity require ∣ Γ A B ∣ ≤ Γ A Γ B ≤ ( Γ A + Γ B ) / 2 \lvert\Gamma_{AB}\rvert\leq\sqrt{\Gamma_{A}\Gamma_{B}}\leq(\Gamma_{A}+\Gamma_{B})/2 ∣ Γ A B ∣ ≤ Γ A Γ B ≤ ( Γ A + Γ B ) /2
Let us now consider the separability classification of L t o t \mathcal{L}_{\mathrm{tot}} L tot
L e f f x + L e f f ± = L e f f x ′ + L e f f ⊥ , L e f f x ′ ( ρ ) = − i λ e f f x x [ σ A x ⊗ σ B x , ρ ] + Γ ˉ e f f x ( σ A x ρ σ A x − ρ ) + Γ ˉ e f f x ( σ B x ρ σ B x − ρ ) + Γ e f f x x ( σ A x σ B x ρ + ρ σ A x σ B x − σ A x ρ σ B x − σ B x ρ σ A x ) , L e f f ⊥ ( ρ ) = D [ L A ⊥ ] ( ρ ) + D [ L B ⊥ ] ( ρ ) , (77) \begin{aligned}
\mathcal{L}_{\mathrm{eff}}^{x}+\mathcal{L}_{\mathrm{eff}}^{\pm}&=\mathcal{L}_{\mathrm{eff}}^{x'}+\mathcal{L}_{\mathrm{eff}}^{\perp},\\
\mathcal{L}_{\mathrm{eff}}^{x'}(\rho)&=-i\lambda_{\mathrm{eff}}^{xx}[\sigma_{A}^{x}\otimes\sigma_{B}^{x},\rho]+\bar{\Gamma}_{\mathrm{eff}}^{x}(\sigma_{A}^{x}\rho\sigma_{A}^{x}-\rho)+\bar{\Gamma}_{\mathrm{eff}}^{x}(\sigma_{B}^{x}\rho\sigma_{B}^{x}-\rho)\\
&\quad+\Gamma_{\mathrm{eff}}^{xx}(\sigma_{A}^{x}\sigma_{B}^{x}\rho+\rho\sigma_{A}^{x}\sigma_{B}^{x}-\sigma_{A}^{x}\rho\sigma_{B}^{x}-\sigma_{B}^{x}\rho\sigma_{A}^{x}),\\
\mathcal{L}_{\mathrm{eff}}^{\perp}(\rho)&=D[L_{A}^{\perp}](\rho)+D[L_{B}^{\perp}](\rho), \tag*{(77)}
\end{aligned} L eff x + L eff ± L eff x ′ ( ρ ) L eff ⊥ ( ρ ) = L eff x ′ + L eff ⊥ , = − i λ eff xx [ σ A x ⊗ σ B x , ρ ] + Γ ˉ eff x ( σ A x ρ σ A x − ρ ) + Γ ˉ eff x ( σ B x ρ σ B x − ρ ) + Γ eff xx ( σ A x σ B x ρ + ρ σ A x σ B x − σ A x ρ σ B x − σ B x ρ σ A x ) , = D [ L A ⊥ ] ( ρ ) + D [ L B ⊥ ] ( ρ ) , (77) where Γ ˉ e f f x = Γ e f f x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − \bar{\Gamma}_{\mathrm{eff}}^{x}=\Gamma_{\mathrm{eff}}^{x}+\frac{\Gamma_{\mathrm{eff}}^{+}\Gamma_{\mathrm{eff}}^{-}}{\Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}} Γ ˉ eff x = Γ eff x + Γ eff + + Γ eff − Γ eff + Γ eff − L α ⊥ = i Γ e f f + σ α + − i Γ e f f − σ α − Γ e f f + + Γ e f f − \quad L_{\alpha}^{\perp}=\frac{i\Gamma_{\mathrm{eff}}^{+}\sigma_{\alpha}^{+}-i\Gamma_{\mathrm{eff}}^{-}\sigma_{\alpha}^{-}}{\sqrt{\Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}}} L α ⊥ = Γ eff + + Γ eff − i Γ eff + σ α + − i Γ eff − σ α −
The numerical results in App. C3 shows Γ e f f + < 0 \Gamma_{\mathrm{eff}}^{+}<0 Γ eff + < 0 Γ e f f + + Γ e f f − > 0 \Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}>0 Γ eff + + Γ eff − > 0 Γ ˉ e f f x > 0 \bar{\Gamma}_{\mathrm{eff}}^{x}>0 Γ ˉ eff x > 0 [54 , 55 ] . In the case of Γ e f f + + Γ e f f − = 0 \Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}=0 Γ eff + + Γ eff − = 0 Γ e f f + = Γ e f f − = 0 \Gamma_{\mathrm{eff}}^{+}=\Gamma_{\mathrm{eff}}^{-}=0 Γ eff + = Γ eff − = 0 Γ ˉ e f f x = Γ e f f x \bar{\Gamma}_{\mathrm{eff}}^{x}=\Gamma_{\mathrm{eff}}^{x} Γ ˉ eff x = Γ eff x L α ⊥ = 0 L_{\alpha}^{\perp}=0 L α ⊥ = 0 L e f f ⊥ \mathcal{L}_{\mathrm{eff}}^{\perp} L eff ⊥
Let us restate Theorem 1 as follows:
Theorem 1. Given the Lindbladian L t o t \mathcal{L}_{\mathrm{tot}} L tot
η = ∥ ( Γ e f f x x − i λ e f f x x Γ e f f x Γ e f f x z − i λ e f f x z Γ e f f z Γ e f f z Γ e f f z x − i λ e f f z x Γ e f f z Γ e f f x Γ e f f z z − i λ e f f z z Γ e f f z ) ∥ 2 . (78) \eta=\left\Vert\begin{pmatrix}
\dfrac{\Gamma_{\mathrm{eff}}^{xx}-i\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^{x}} & \dfrac{\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{z}\Gamma_{\mathrm{eff}}^{z}}}\\
\dfrac{\Gamma_{\mathrm{eff}}^{zx}-i\lambda_{\mathrm{eff}}^{zx}}{\sqrt{\Gamma_{\mathrm{eff}}^{z}\Gamma_{\mathrm{eff}}^{x}}} & \dfrac{\Gamma_{\mathrm{eff}}^{zz}-i\lambda_{\mathrm{eff}}^{zz}}{\Gamma_{\mathrm{eff}}^{z}}
\end{pmatrix}\right\Vert_{2}. \tag*{(78)} η = Γ eff x Γ eff xx − i λ eff xx Γ eff z Γ eff x Γ eff z x − i λ eff z x Γ eff z Γ eff z Γ eff x z − i λ eff x z Γ eff z Γ eff z z − i λ eff z z 2 . (78) The separability classification is determined as follows:
Therefore, η > 1 \eta>1 η > 1
Consider the simplified situation of Γ e f f x z = λ e f f x z = λ e f f z z = Γ e f f z z = 0 \Gamma_{\mathrm{eff}}^{xz}=\lambda_{\mathrm{eff}}^{xz}=\lambda_{\mathrm{eff}}^{zz}=\Gamma_{\mathrm{eff}}^{zz}=0 Γ eff x z = λ eff x z = λ eff z z = Γ eff z z = 0 η = ∣ Γ e f f x x − i λ e f f x x ∣ / Γ ˉ e f f x \eta=\lvert\Gamma_{\mathrm{eff}}^{xx}-i\lambda_{\mathrm{eff}}^{xx}\rvert/\bar{\Gamma}_{\mathrm{eff}}^{x} η = ∣ Γ eff xx − i λ eff xx ∣ / Γ ˉ eff x η > 1 \eta>1 η > 1 ∣ Γ e f f x z − i λ e f f x z ∣ / Γ e f f z Γ e f f x , ∣ Γ e f f z z − i λ e f f z z ∣ / Γ e f f z ≪ ∣ Γ e f f x x − i λ e f f x x ∣ / Γ ˉ e f f x \lvert\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}\rvert/\sqrt{\Gamma_{\mathrm{eff}}^{z}\Gamma_{\mathrm{eff}}^{x}},\lvert\Gamma_{\mathrm{eff}}^{zz}-i\lambda_{\mathrm{eff}}^{zz}\rvert/\Gamma_{\mathrm{eff}}^{z}\ll\lvert\Gamma_{\mathrm{eff}}^{xx}-i\lambda_{\mathrm{eff}}^{xx}\rvert/\bar{\Gamma}_{\mathrm{eff}}^{x} ∣ Γ eff x z − i λ eff x z ∣ / Γ eff z Γ eff x , ∣ Γ eff z z − i λ eff z z ∣ / Γ eff z ≪ ∣ Γ eff xx − i λ eff xx ∣ / Γ ˉ eff x η \eta η
η = ∣ a ∣ + ∣ b ∣ 2 ∣ a ∣ − Re c ∗ b 2 ∣ a ∣ a + O ( max ( ∣ b ∣ ∣ a ∣ , ∣ c ∣ ∣ a ∣ ) 4 ) , \eta=\lvert a\rvert+\frac{\lvert b\rvert^{2}}{\lvert a\rvert}-\operatorname{Re}\frac{c^{*}b^{2}}{\lvert a\rvert a}+\mathcal{O}\left(\max\left(\frac{\lvert b\rvert}{\lvert a\rvert},\frac{\lvert c\rvert}{\lvert a\rvert}\right)^{4}\right), η = ∣ a ∣ + ∣ a ∣ ∣ b ∣ 2 − Re ∣ a ∣ a c ∗ b 2 + O ( max ( ∣ a ∣ ∣ b ∣ , ∣ a ∣ ∣ c ∣ ) 4 ) , where a = Γ e f f x x − i λ e f f x x Γ ˉ e f f x a=\frac{\Gamma_{\mathrm{eff}}^{xx}-i\lambda_{\mathrm{eff}}^{xx}}{\bar{\Gamma}_{\mathrm{eff}}^{x}} a = Γ ˉ eff x Γ eff xx − i λ eff xx b = Γ e f f x z − i λ e f f x z Γ e f f z Γ e f f x b=\frac{\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{z}\Gamma_{\mathrm{eff}}^{x}}} b = Γ eff z Γ eff x Γ eff x z − i λ eff x z c = Γ e f f z z − i λ e f f z z Γ e f f z c=\frac{\Gamma_{\mathrm{eff}}^{zz}-i\lambda_{\mathrm{eff}}^{zz}}{\Gamma_{\mathrm{eff}}^{z}} c = Γ eff z Γ eff z z − i λ eff z z
Therefore, it is well approximated by
η ≈ ( Γ e f f x x ) 2 + ( λ e f f x x ) 2 Γ ˉ e f f . (79) \eta\approx\frac{\sqrt{\left(\Gamma_{\mathrm{eff}}^{xx}\right)^2+\left(\lambda_{\mathrm{eff}}^{xx}\right)^2}}{\bar{\Gamma}_{\mathrm{eff}}}. \tag*{(79)} η ≈ Γ ˉ eff ( Γ eff xx ) 2 + ( λ eff xx ) 2 . (79) On the other hand, in general we have
η ≥ η s e = ( Γ e f f x x ) 2 + ( λ e f f x x ) 2 Γ e f f x x . (80) \eta\ge\eta_{\mathrm{se}}=\frac{\sqrt{\left(\Gamma_{\mathrm{eff}}^{xx}\right)^2+\left(\lambda_{\mathrm{eff}}^{xx}\right)^2}}{\Gamma_{\mathrm{eff}}^{xx}}. \tag*{(80)} η ≥ η se = Γ eff xx ( Γ eff xx ) 2 + ( λ eff xx ) 2 . (80) Hence ( Γ e f f x x ) 2 + ( λ e f f x x ) 2 > Γ ˉ e f f \sqrt{\left(\Gamma_{\mathrm{eff}}^{xx}\right)^2+\left(\lambda_{\mathrm{eff}}^{xx}\right)^2}>\bar{\Gamma}_{\mathrm{eff}} ( Γ eff xx ) 2 + ( λ eff xx ) 2 > Γ ˉ eff (86) . Therefore, it is sufficient to consider the rough scaling analysis as in the main text, to account for the non-separability of the effective dynamics. In other words, it is sufficient to claim non-separability of the error-corrected dynamics given λ e f f x x ≫ λ ˉ e f f x x \lambda_{\mathrm{eff}}^{xx}\gg\bar{\lambda}_{\mathrm{eff}}^{xx} λ eff xx ≫ λ ˉ eff xx
Proof. In the following proof, for simplicity let us omit the eff subscripts. Let us first consider the non-separable side. The non-separability of the Lindbladian L t o t \mathcal{L}_{\mathrm{tot}} L tot ∣ Φ ⟩ = ( ∣ 00 ⟩ + ∣ 11 ⟩ ) / 2 |\Phi\rangle=(|00\rangle+|11\rangle)/\sqrt{2} ∣Φ ⟩ = ( ∣00 ⟩ + ∣11 ⟩) / 2 ∣ Ψ ⟩ A 1 A 2 ; B 1 B 2 = ∣ Φ ⟩ A 1 A 2 ⊗ ∣ Φ ⟩ B 1 B 2 |\Psi\rangle_{A_1A_2;B_1B_2}=|\Phi\rangle_{A_1A_2}\otimes|\Phi\rangle_{B_1B_2} ∣Ψ ⟩ A 1 A 2 ; B 1 B 2 = ∣Φ ⟩ A 1 A 2 ⊗ ∣Φ ⟩ B 1 B 2 Lemma 3 , let us consider the six-dimensional subspace span { σ A 1 x ∣ Ψ ⟩ , σ A 1 y ∣ Ψ ⟩ , σ A 1 z ∣ Ψ ⟩ , σ B 1 x ∣ Ψ ⟩ , σ B 1 y ∣ Ψ ⟩ , σ B 1 z ∣ Ψ ⟩ } \operatorname{span}\{\sigma_{A_1}^{x}|\Psi\rangle,\sigma_{A_1}^{y}|\Psi\rangle,\sigma_{A_1}^{z}|\Psi\rangle,\sigma_{B_1}^{x}|\Psi\rangle,\sigma_{B_1}^{y}|\Psi\rangle,\sigma_{B_1}^{z}|\Psi\rangle\} span { σ A 1 x ∣Ψ ⟩ , σ A 1 y ∣Ψ ⟩ , σ A 1 z ∣Ψ ⟩ , σ B 1 x ∣Ψ ⟩ , σ B 1 y ∣Ψ ⟩ , σ B 1 z ∣Ψ ⟩} Π 6 \Pi_6 Π 6 1
G = Π 6 L t o t , A 1 B 1 ( ∣ Ψ ⟩ A 1 A 2 ; B 1 B 2 ⟨ Ψ ∣ ) T B 1 B 2 Π 6 = ( Γ x + Γ + + Γ − 4 − i Γ + − Γ − 4 Γ x x − i λ x x Γ x z − i λ x z i Γ + − Γ − 4 Γ + + Γ − 4 Γ z − Γ x z + i λ x z Γ x x + i λ x x − Γ x z − i λ x z Γ x + Γ + + Γ − 4 i Γ + − Γ − 4 Γ x z + i λ x z Γ z z + i λ z z − i Γ + − Γ − 4 Γ z ) . (81) G=\Pi_6\mathcal{L}_{\mathrm{tot},A_1B_1}\left(|\Psi\rangle_{A_1A_2;B_1B_2}\langle\Psi| \right)^{T_{B_1B_2}}\Pi_6=
\begin{pmatrix}
\Gamma^x+\frac{\Gamma^++\Gamma^-}{4} & -i\frac{\Gamma^+-\Gamma^-}{4} & \Gamma^{xx}-i\lambda^{xx} & \Gamma^{xz}-i\lambda^{xz}\\
i\frac{\Gamma^+-\Gamma^-}{4} & \frac{\Gamma^++\Gamma^-}{4} & \Gamma^z & -\Gamma^{xz}+i\lambda^{xz}\\
\Gamma^{xx}+i\lambda^{xx} & -\Gamma^{xz}-i\lambda^{xz} & \Gamma^x+\frac{\Gamma^++\Gamma^-}{4} & i\frac{\Gamma^+-\Gamma^-}{4}\\
\Gamma^{xz}+i\lambda^{xz} & \Gamma^{zz}+i\lambda^{zz} & -i\frac{\Gamma^+-\Gamma^-}{4} & \Gamma^z
\end{pmatrix}. \tag*{(81)} G = Π 6 L tot , A 1 B 1 ( ∣Ψ ⟩ A 1 A 2 ; B 1 B 2 ⟨ Ψ∣ ) T B 1 B 2 Π 6 = Γ x + 4 Γ + + Γ − i 4 Γ + − Γ − Γ xx + i λ xx Γ x z + i λ x z − i 4 Γ + − Γ − 4 Γ + + Γ − − Γ x z − i λ x z Γ z z + i λ z z Γ xx − i λ xx Γ z Γ x + 4 Γ + + Γ − − i 4 Γ + − Γ − Γ x z − i λ x z − Γ x z + i λ x z i 4 Γ + − Γ − Γ z . (81) Note that complete positivity requires Γ + + Γ − ≥ 0 \Gamma^++\Gamma^-\ge0 Γ + + Γ − ≥ 0 G G G 4 ⊕ 2 4\oplus2 4 ⊕ 2
G ≅ ( Γ x + Γ + + Γ − 4 Γ x z − i λ x z Γ x x − i λ x x − i Γ + − Γ − 4 Γ x z + i λ x z Γ z Γ z z + i λ z z Γ x + Γ + − Γ − 4 Γ x x + i λ x x − Γ x z − i λ x z Γ x + Γ + + Γ − 4 i Γ + − Γ − 4 i Γ + − Γ − 4 − i Γ + − Γ − 4 Γ + + Γ − 4 ) . (82) G\cong
\begin{pmatrix}
\Gamma^x+\frac{\Gamma^++\Gamma^-}{4} & \Gamma^{xz}-i\lambda^{xz} & \Gamma^{xx}-i\lambda^{xx} & -i\frac{\Gamma^+-\Gamma^-}{4}\\
\Gamma^{xz}+i\lambda^{xz} & \Gamma^z & \Gamma^{zz}+i\lambda^{zz} & \Gamma^{x}+\frac{\Gamma^+-\Gamma^-}{4}\\
\Gamma^{xx}+i\lambda^{xx} & -\Gamma^{xz}-i\lambda^{xz} & \Gamma^x+\frac{\Gamma^++\Gamma^-}{4} & i\frac{\Gamma^+-\Gamma^-}{4}\\
i\frac{\Gamma^+-\Gamma^-}{4} & & -i\frac{\Gamma^+-\Gamma^-}{4} & \frac{\Gamma^++\Gamma^-}{4}
\end{pmatrix}. \tag*{(82)} G ≅ Γ x + 4 Γ + + Γ − Γ x z + i λ x z Γ xx + i λ xx i 4 Γ + − Γ − Γ x z − i λ x z Γ z − Γ x z − i λ x z Γ xx − i λ xx Γ z z + i λ z z Γ x + 4 Γ + + Γ − − i 4 Γ + − Γ − − i 4 Γ + − Γ − Γ x + 4 Γ + − Γ − i 4 Γ + − Γ − 4 Γ + + Γ − . (82) The positivity of G G G
( Γ x + Γ + + Γ − 4 Γ x z − i λ x z Γ x x − i λ x x Γ x z + i λ x z Γ z Γ z z + i λ z z Γ x x + i λ x x − Γ x z − i λ x z Γ x + Γ + + Γ − 4 ) − ( − i Γ + − Γ − 4 i Γ + − Γ − 4 ) 4 I 2 Γ + + Γ − ( − i Γ + − Γ − 4 i Γ + − Γ − 4 ) = ( Γ ˉ x Γ x z − i λ x z Γ x x − i λ x x Γ x z + i λ x z Γ z Γ z z + i λ z z Γ x x + i λ x x − Γ x z − i λ x z Γ ˉ x ) . (83) \begin{aligned}
\begin{pmatrix}
\Gamma^x+\frac{\Gamma^++\Gamma^-}{4} & \Gamma^{xz}-i\lambda^{xz} & \Gamma^{xx}-i\lambda^{xx}\\
\Gamma^{xz}+i\lambda^{xz} & \Gamma^z & \Gamma^{zz}+i\lambda^{zz}\\
\Gamma^{xx}+i\lambda^{xx} & -\Gamma^{xz}-i\lambda^{xz} & \Gamma^x+\frac{\Gamma^++\Gamma^-}{4}
\end{pmatrix}
-
\begin{pmatrix}
-i\frac{\Gamma^+-\Gamma^-}{4}\\
i\frac{\Gamma^+-\Gamma^-}{4}
\end{pmatrix}
\frac{4I_2}{\Gamma^++\Gamma^-}
\begin{pmatrix}
-i\frac{\Gamma^+-\Gamma^-}{4} & i\frac{\Gamma^+-\Gamma^-}{4}
\end{pmatrix} \\
&=
\begin{pmatrix}
\bar{\Gamma}^x & \Gamma^{xz}-i\lambda^{xz} & \Gamma^{xx}-i\lambda^{xx}\\
\Gamma^{xz}+i\lambda^{xz} & \Gamma^z & \Gamma^{zz}+i\lambda^{zz}\\
\Gamma^{xx}+i\lambda^{xx} & -\Gamma^{xz}-i\lambda^{xz} & \bar{\Gamma}^x
\end{pmatrix}. \tag*{(83)}
\end{aligned} Γ x + 4 Γ + + Γ − Γ x z + i λ x z Γ xx + i λ xx Γ x z − i λ x z Γ z − Γ x z − i λ x z Γ xx − i λ xx Γ z z + i λ z z Γ x + 4 Γ + + Γ − − ( − i 4 Γ + − Γ − i 4 Γ + − Γ − ) Γ + + Γ − 4 I 2 ( − i 4 Γ + − Γ − i 4 Γ + − Γ − ) = Γ ˉ x Γ x z + i λ x z Γ xx + i λ xx Γ x z − i λ x z Γ z − Γ x z − i λ x z Γ xx − i λ xx Γ z z + i λ z z Γ ˉ x . (83) Note that this last 4-by-4 matrix can be rewritten as a 2 ⊕ 2 2\oplus2 2 ⊕ 2
\begin{pmatrix}
\bar{\Gamma}^x & \Gamma^z & \Gamma^{xx}+i\lambda^{xx} & \Gamma^{xz}+i\lambda^{xz}\\
\Gamma^x-\!i\lambda^{xx} & -\Gamma^{xz}+i\lambda^{xz} & \bar{\Gamma}^x & \Gamma^z\\
\Gamma^{xz}-i\lambda^{xz} & \Gamma^{zz}-i\lambda^{zz} & & \Gamma^z
\end{pmatrix}. \tag*{(84)}\end{pmatrix}
Therefore, the separability of this dynamics is controlled by the rewritten L x ′ \mathcal{L}^{x^\prime} L x ′ L ⊥ \mathcal{L}^{\perp} L ⊥ Γ ˉ x , Γ z ≠ 0 \bar{\Gamma}^{x},\Gamma^{z}\ne0 Γ ˉ x , Γ z = 0 Lemma 3 . Then its positivity is also equivalent to the positivity of the following matrix, given Γ ˉ x , Γ z > 0 \bar{\Gamma}^{x},\Gamma^{z}>0 Γ ˉ x , Γ z > 0
( Γ ˉ x Γ z ) − ( Γ x x + i λ x x Γ x z + i λ x z − Γ x z − i λ x z Γ z z + i λ z z ) ( 1 Γ ˉ x 0 0 1 Γ z ) ( Γ x x − i λ x x − Γ x z + i λ x z Γ x z − i λ x z Γ z z − i λ z z ) = ( Γ ˉ x − ( Γ x x ) 2 + ( λ x x ) 2 Γ ˉ x − ( Γ x z ) 2 + ( λ x z ) 2 Γ z ( Γ x x + i λ x x ) ( Γ x z − i λ x z ) Γ ˉ x − ( Γ x z + i λ x z ) ( Γ z z − i λ z z ) Γ z ( Γ x x − i λ x x ) ( Γ x z + i λ x z ) Γ ˉ x − ( Γ x z − i λ x z ) ( Γ z z + i λ z z ) Γ z Γ z − ( Γ x z ) 2 + ( λ x z ) 2 Γ ˉ x − ( Γ z z ) 2 + ( λ z z ) 2 Γ z ) ≥ 0 (85) \begin{aligned}
\begin{pmatrix}\bar{\Gamma}^{x}\\ \Gamma^{z}\end{pmatrix}
-
\begin{pmatrix}
\Gamma^{xx}+i\lambda^{xx} & \Gamma^{xz}+i\lambda^{xz}\\
-\Gamma^{xz}-i\lambda^{xz} & \Gamma^{zz}+i\lambda^{zz}
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\bar{\Gamma}^{x}} & 0\\
0 & \frac{1}{\Gamma^{z}}
\end{pmatrix}
\begin{pmatrix}
\Gamma^{xx}-i\lambda^{xx} & -\Gamma^{xz}+i\lambda^{xz}\\
\Gamma^{xz}-i\lambda^{xz} & \Gamma^{zz}-i\lambda^{zz}
\end{pmatrix} \\
&=
\begin{pmatrix}
\bar{\Gamma}^{x}-\frac{(\Gamma^{xx})^{2}+(\lambda^{xx})^{2}}{\bar{\Gamma}^{x}}-\frac{(\Gamma^{xz})^{2}+(\lambda^{xz})^{2}}{\Gamma^{z}}
&
\frac{(\Gamma^{xx}+i\lambda^{xx})(\Gamma^{xz}-i\lambda^{xz})}{\bar{\Gamma}^{x}}-\frac{(\Gamma^{xz}+i\lambda^{xz})(\Gamma^{zz}-i\lambda^{zz})}{\Gamma^{z}}\\
\frac{(\Gamma^{xx}-i\lambda^{xx})(\Gamma^{xz}+i\lambda^{xz})}{\bar{\Gamma}^{x}}-\frac{(\Gamma^{xz}-i\lambda^{xz})(\Gamma^{zz}+i\lambda^{zz})}{\Gamma^{z}}
&
\Gamma^{z}-\frac{(\Gamma^{xz})^{2}+(\lambda^{xz})^{2}}{\bar{\Gamma}^{x}}-\frac{(\Gamma^{zz})^{2}+(\lambda^{zz})^{2}}{\Gamma^{z}}
\end{pmatrix}\ge0 \tag*{(85)}
\end{aligned} ( Γ ˉ x Γ z ) − ( Γ xx + i λ xx − Γ x z − i λ x z Γ x z + i λ x z Γ z z + i λ z z ) ( Γ ˉ x 1 0 0 Γ z 1 ) ( Γ xx − i λ xx Γ x z − i λ x z − Γ x z + i λ x z Γ z z − i λ z z ) = ( Γ ˉ x − Γ ˉ x ( Γ xx ) 2 + ( λ xx ) 2 − Γ z ( Γ x z ) 2 + ( λ x z ) 2 Γ ˉ x ( Γ xx − i λ xx ) ( Γ x z + i λ x z ) − Γ z ( Γ x z − i λ x z ) ( Γ z z + i λ z z ) Γ ˉ x ( Γ xx + i λ xx ) ( Γ x z − i λ x z ) − Γ z ( Γ x z + i λ x z ) ( Γ z z − i λ z z ) Γ z − Γ ˉ x ( Γ x z ) 2 + ( λ x z ) 2 − Γ z ( Γ z z ) 2 + ( λ z z ) 2 ) ≥ 0 (85) In other words, we have the following separability necessary condition:
( Γ x x ) 2 + ( λ x x ) 2 ( Γ ˉ x ) 2 + ( Γ x z ) 2 + ( λ x z ) 2 Γ ˉ x Γ z ≤ 1 , ( Γ x z ) 2 + ( λ x z ) 2 Γ ˉ x Γ z + ( Γ z z ) 2 + ( λ z z ) 2 ( Γ z ) 2 ≤ 1 , \frac{(\Gamma^{xx})^{2}+(\lambda^{xx})^{2}}{(\bar{\Gamma}^{x})^{2}}+\frac{(\Gamma^{xz})^{2}+(\lambda^{xz})^{2}}{\bar{\Gamma}^{x}\Gamma^{z}}\le1,\qquad
\frac{(\Gamma^{xz})^{2}+(\lambda^{xz})^{2}}{\bar{\Gamma}^{x}\Gamma^{z}}+\frac{(\Gamma^{zz})^{2}+(\lambda^{zz})^{2}}{(\Gamma^{z})^{2}}\le1, ( Γ ˉ x ) 2 ( Γ xx ) 2 + ( λ xx ) 2 + Γ ˉ x Γ z ( Γ x z ) 2 + ( λ x z ) 2 ≤ 1 , Γ ˉ x Γ z ( Γ x z ) 2 + ( λ x z ) 2 + ( Γ z ) 2 ( Γ z z ) 2 + ( λ z z ) 2 ≤ 1 , ∣ ( Γ x x + i λ x x ) ( Γ x z − i λ x z ) Γ ˉ x − ( Γ x z + i λ x z ) ( Γ z z − i λ z z ) Γ z ∣ 2 ≤ ( Γ ˉ x − ( Γ x x ) 2 + ( λ x x ) 2 Γ ˉ x − ( Γ x z ) 2 + ( λ x z ) 2 Γ z ) ( Γ z − ( Γ x z ) 2 + ( λ x z ) 2 Γ ˉ x − ( Γ z z ) 2 + ( λ z z ) 2 Γ z ) . (86) \left|\frac{(\Gamma^{xx}+i\lambda^{xx})(\Gamma^{xz}-i\lambda^{xz})}{\bar{\Gamma}^{x}}-\frac{(\Gamma^{xz}+i\lambda^{xz})(\Gamma^{zz}-i\lambda^{zz})}{\Gamma^{z}}\right|^{2}
\le
\left(\bar{\Gamma}^{x}-\frac{(\Gamma^{xx})^{2}+(\lambda^{xx})^{2}}{\bar{\Gamma}^{x}}-\frac{(\Gamma^{xz})^{2}+(\lambda^{xz})^{2}}{\Gamma^{z}}\right)
\left(\Gamma^{z}-\frac{(\Gamma^{xz})^{2}+(\lambda^{xz})^{2}}{\bar{\Gamma}^{x}}-\frac{(\Gamma^{zz})^{2}+(\lambda^{zz})^{2}}{\Gamma^{z}}\right). \tag*{(86)} Γ ˉ x ( Γ xx + i λ xx ) ( Γ x z − i λ x z ) − Γ z ( Γ x z + i λ x z ) ( Γ z z − i λ z z ) 2 ≤ ( Γ ˉ x − Γ ˉ x ( Γ xx ) 2 + ( λ xx ) 2 − Γ z ( Γ x z ) 2 + ( λ x z ) 2 ) ( Γ z − Γ ˉ x ( Γ x z ) 2 + ( λ x z ) 2 − Γ z ( Γ z z ) 2 + ( λ z z ) 2 ) . (86) Equivalently, Eq. (85) can also be rewritten as
0 ≤ I − ( 1 Γ ˉ x 1 Γ z ) ( Γ x x + i λ x x Γ x z + i λ x z − Γ x z − i λ x z Γ z z + i λ z z ) ( 1 Γ ˉ x 1 Γ z ) ( Γ x x − i λ x x − Γ x z + i λ x z Γ x z − i λ x z Γ z z − i λ z z ) ( 1 Γ ˉ x 1 Γ z ) (87) 0\le I-
\begin{pmatrix}
\frac{1}{\sqrt{\bar{\Gamma}^{x}}}\\
\frac{1}{\sqrt{\Gamma^{z}}}
\end{pmatrix}
\begin{pmatrix}
\Gamma^{xx}+i\lambda^{xx} & \Gamma^{xz}+i\lambda^{xz}\\
-\Gamma^{xz}-i\lambda^{xz} & \Gamma^{zz}+i\lambda^{zz}
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{\bar{\Gamma}^{x}}} & \frac{1}{\sqrt{\Gamma^{z}}}
\end{pmatrix}
\begin{pmatrix}
\Gamma^{xx}-i\lambda^{xx} & -\Gamma^{xz}+i\lambda^{xz}\\
\Gamma^{xz}-i\lambda^{xz} & \Gamma^{zz}-i\lambda^{zz}
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{\bar{\Gamma}^{x}}}\\
\frac{1}{\sqrt{\Gamma^{z}}}
\end{pmatrix} \tag*{(87)} 0 ≤ I − ( Γ ˉ x 1 Γ z 1 ) ( Γ xx + i λ xx − Γ x z − i λ x z Γ x z + i λ x z Γ z z + i λ z z ) ( Γ ˉ x 1 Γ z 1 ) ( Γ xx − i λ xx Γ x z − i λ x z − Γ x z + i λ x z Γ z z − i λ z z ) ( Γ ˉ x 1 Γ z 1 ) (87) Therefore, we can define the 2-norm, i.e. the largest singular value of J † J \sqrt{J^{\dagger}J} J † J
η = ∥ J ∥ 2 ≤ 1 , where J = ( Γ x x − i λ x x Γ ˉ x Γ x z − i λ x z Γ z Γ ˉ x Γ x z − i λ x z Γ z Γ ˉ x Γ z z − i λ z z Γ z ) . (88) \eta=\lVert J\rVert_{2}\le1,\qquad\text{where}\qquad
J=
\begin{pmatrix}
\frac{\Gamma^{xx}-i\lambda^{xx}}{\bar{\Gamma}^{x}} &
\frac{\Gamma^{xz}-i\lambda^{xz}}{\sqrt{\Gamma^{z}\bar{\Gamma}^{x}}}\\
\frac{\Gamma^{xz}-i\lambda^{xz}}{\sqrt{\Gamma^{z}\bar{\Gamma}^{x}}} &
\frac{\Gamma^{zz}-i\lambda^{zz}}{\Gamma^{z}}
\end{pmatrix}. \tag*{(88)} η = ∥ J ∥ 2 ≤ 1 , where J = ( Γ ˉ x Γ xx − i λ xx Γ z Γ ˉ x Γ x z − i λ x z Γ z Γ ˉ x Γ x z − i λ x z Γ z Γ z z − i λ z z ) . (88) This is because Eq. (87) is equivalent to I − J † J ≥ 0 I-J^{\dagger}J\ge0 I − J † J ≥ 0 η > 1 \eta>1 η > 1 L t o t \mathcal{L}_{\mathrm{tot}} L tot L e f f \mathcal{L}_{\mathrm{eff}} L eff
Then let us consider the separable side. Note that the Choi state above is a 4 × 4 4\times4 4 × 4 exp L t o t t \exp\mathcal{L}_{\mathrm{tot}}t exp L tot t L ′ = L x ′ + L z + L x z \mathcal{L}^{\prime}=\mathcal{L}^{x^\prime}+\mathcal{L}^{z}+\mathcal{L}^{xz} L ′ = L x ′ + L z + L x z L ⊥ \mathcal{L}^{\perp} L ⊥ L ⊥ \mathcal{L}^{\perp} L ⊥ Lemma 3 , it’s sufficient to construct L ′ \mathcal{L}^{\prime} L ′ L ′ \mathcal{L}^{\prime} L ′
L ′ = ∑ j L j O A , j , O B , j ( ρ ) , \mathcal{L}^{\prime}=\sum_{j}\mathcal{L}^{O_{A,j},O_{B,j}}_{j}(\rho), L ′ = j ∑ L j O A , j , O B , j ( ρ ) , L j O A , j , O B , j = − i λ j [ O A , j ⊗ O B , j , ρ ] + Γ A , j D [ O A , j ] ( ρ ) + Γ B , j D [ O B , j ] ( ρ ) + Γ A B , j ( O A , j O B , j ρ + ρ O A , j O B , j − O A , j ρ O B , j − O B , j ρ O A , j ) , \mathcal{L}^{O_{A,j},O_{B,j}}_{j}
=-i\lambda_{j}[O_{A,j}\otimes O_{B,j},\rho]
+\Gamma_{A,j}\mathcal{D}[O_{A,j}](\rho)
+\Gamma_{B,j}\mathcal{D}[O_{B,j}](\rho)
+\Gamma_{AB,j}(O_{A,j}O_{B,j}\rho+\rho O_{A,j}O_{B,j}-O_{A,j}\rho O_{B,j}-O_{B,j}\rho O_{A,j}), L j O A , j , O B , j = − i λ j [ O A , j ⊗ O B , j , ρ ] + Γ A , j D [ O A , j ] ( ρ ) + Γ B , j D [ O B , j ] ( ρ ) + Γ A B , j ( O A , j O B , j ρ + ρ O A , j O B , j − O A , j ρ O B , j − O B , j ρ O A , j ) , where O A , j ≡ a j ⋅ σ A T = a j 1 σ A x + a j 2 σ A z , O B , j ≡ b j ⋅ σ B T = b j 1 σ B x + b j 2 σ B z , (89) \text{where}\quad O_{A,j}\equiv\mathbf{a}_{j}\cdot\boldsymbol{\sigma}_{A}^{T}=a_{j}^{1}\sigma_{A}^{x}+a_{j}^{2}\sigma_{A}^{z},\qquad
O_{B,j}\equiv\mathbf{b}_{j}\cdot\boldsymbol{\sigma}_{B}^{T}=b_{j}^{1}\sigma_{B}^{x}+b_{j}^{2}\sigma_{B}^{z}, \tag*{(89)} where O A , j ≡ a j ⋅ σ A T = a j 1 σ A x + a j 2 σ A z , O B , j ≡ b j ⋅ σ B T = b j 1 σ B x + b j 2 σ B z , (89) where we denote a j = ( a j 1 , a j 2 ) \mathbf{a}_{j}=(a_{j}^{1},a_{j}^{2}) a j = ( a j 1 , a j 2 ) b j = ( b j 1 , b j 2 ) \mathbf{b}_{j}=(b_{j}^{1},b_{j}^{2}) b j = ( b j 1 , b j 2 ) σ α = ( σ α x , σ α z ) \boldsymbol{\sigma}_{\alpha}=(\sigma_{\alpha}^{x},\sigma_{\alpha}^{z}) σ α = ( σ α x , σ α z ) a j , b j \mathbf{a}_{j},\mathbf{b}_{j} a j , b j
∑ j ( Γ A B , j − i λ j ) a j T b j = ( Γ x x − i λ x x Γ x z − i λ x z − Γ x z + i λ x z Γ z z − i λ z z ) , ∑ j Γ A , j a j T a j = ∑ j Γ B , j b j T b j = ( Γ ˉ x Γ z ) . (90) \sum_{j}(\Gamma_{AB,j}-i\lambda_{j})\mathbf{a}_{j}^{T}\mathbf{b}_{j}
=
\begin{pmatrix}
\Gamma^{xx}-i\lambda^{xx} & \Gamma^{xz}-i\lambda^{xz}\\
-\Gamma^{xz}+i\lambda^{xz} & \Gamma^{zz}-i\lambda^{zz}
\end{pmatrix},
\qquad
\sum_{j}\Gamma_{A,j}\mathbf{a}_{j}^{T}\mathbf{a}_{j}
=
\sum_{j}\Gamma_{B,j}\mathbf{b}_{j}^{T}\mathbf{b}_{j}
=
\begin{pmatrix}
\bar{\Gamma}^{x}\\
\Gamma^{z}
\end{pmatrix}. \tag*{(90)} j ∑ ( Γ A B , j − i λ j ) a j T b j = ( Γ xx − i λ xx − Γ x z + i λ x z Γ x z − i λ x z Γ z z − i λ z z ) , j ∑ Γ A , j a j T a j = j ∑ Γ B , j b j T b j = ( Γ ˉ x Γ z ) . (90) For simplicity, let us denote Q = diag ( Γ ˉ x , Γ z ) Q=\operatorname{diag}(\bar{\Gamma}^{x},\Gamma^{z}) Q = diag ( Γ ˉ x , Γ z ) Γ ˉ x , Γ z > 0 \bar{\Gamma}^{x},\Gamma^{z}>0 Γ ˉ x , Γ z > 0 c j = Γ A , j a j Q − 1 / 2 \mathbf{c}_{j}=\sqrt{\Gamma_{A,j}}\mathbf{a}_{j}Q^{-1/2} c j = Γ A , j a j Q − 1/2 d j = Γ ˉ B , j b j Q − 1 / 2 d_j=\sqrt{\bar{\Gamma}_{B,j}}b_jQ^{-1/2} d j = Γ ˉ B , j b j Q − 1/2
∑ j Γ A B , j − i λ j Γ A , j Γ B , j d j T c j = Q − 1 / 2 ( Γ x x − i λ x x − Γ x z + i λ x z Γ x z − i λ x z Γ z z − i λ z z ) Q − 1 / 2 = J , ∑ j c j T c j = ∑ j d j T d j = ( 1 1 ) (91) \sum_j\frac{\Gamma_{AB,j}-i\lambda_j}{\sqrt{\Gamma_{A,j}\Gamma_{B,j}}}d_j^Tc_j=Q^{-1/2}\begin{pmatrix}\Gamma^{xx}-i\lambda^{xx}&-\Gamma^{xz}+i\lambda^{xz}\\ \Gamma^{xz}-i\lambda^{xz}&\Gamma^{zz}-i\lambda^{zz}\end{pmatrix}Q^{-1/2}=J,\qquad\sum_jc_j^Tc_j=\sum_jd_j^Td_j=\begin{pmatrix}1\\1\end{pmatrix} \tag*{(91)} j ∑ Γ A , j Γ B , j Γ A B , j − i λ j d j T c j = Q − 1/2 ( Γ xx − i λ xx Γ x z − i λ x z − Γ x z + i λ x z Γ z z − i λ z z ) Q − 1/2 = J , j ∑ c j T c j = j ∑ d j T d j = ( 1 1 ) (91) Assuming there exists a solution of c , d c,d c , d e , f e,f e , f
∣ e J f † ∣ = ∣ ∑ j Γ A B , j − i λ j Γ A , j Γ B , j ( e ⋅ d j T ) ( c j ⋅ f † ) ∣ ≤ ∑ j ∣ ( e ⋅ d j T ) ∣ ∣ ( c j ⋅ f † ) ∣ ≤ ∑ j , k ∣ e ⋅ d j T ∣ 2 ∣ c k ⋅ f † ∣ 2 = ∑ j e ⋅ ( d j T d j ) ⋅ e † ∑ k f ⋅ ( c k T c k ) f † = ∣ e ∣ ∣ f ∣ (92) \begin{aligned}
|eJf^\dagger|&=\left|\sum_j\frac{\Gamma_{AB,j}-i\lambda_j}{\sqrt{\Gamma_{A,j}\Gamma_{B,j}}}(e\cdot d_j^T)(c_j\cdot f^\dagger)\right|\\
&\leq\sum_j|(e\cdot d_j^T)||(c_j\cdot f^\dagger)|\\
&\leq\sqrt{\sum_{j,k}|e\cdot d_j^T|^2|c_k\cdot f^\dagger|^2}\\
&=\sqrt{\sum_j e\cdot(d_j^Td_j)\cdot e^\dagger}\sqrt{\sum_k f\cdot(c_k^Tc_k)f^\dagger}\\
&=|e||f| \tag*{(92)}
\end{aligned} ∣ e J f † ∣ = j ∑ Γ A , j Γ B , j Γ A B , j − i λ j ( e ⋅ d j T ) ( c j ⋅ f † ) ≤ j ∑ ∣ ( e ⋅ d j T ) ∣∣ ( c j ⋅ f † ) ∣ ≤ j , k ∑ ∣ e ⋅ d j T ∣ 2 ∣ c k ⋅ f † ∣ 2 = j ∑ e ⋅ ( d j T d j ) ⋅ e † k ∑ f ⋅ ( c k T c k ) f † = ∣ e ∣∣ f ∣ (92) For the first inequality we have used Γ A B , j 2 + λ j 2 ≤ Γ A , j Γ B , j \Gamma_{AB,j}^2+\lambda_j^2\leq\Gamma_{A,j}\Gamma_{B,j} Γ A B , j 2 + λ j 2 ≤ Γ A , j Γ B , j Lemma 3 . This implies the separability condition ∥ J ∥ 2 ≤ 1 \lVert J\rVert_2\leq1 ∥ J ∥ 2 ≤ 1
Then let us consider the sufficiency of this condition. Let us assume the number of j j j O 1 = ( d 1 T , d 2 T ) , O 2 = ( c 1 T , c 2 T ) O_1=(d_1^T,d_2^T),O_2=(c_1^T,c_2^T) O 1 = ( d 1 T , d 2 T ) , O 2 = ( c 1 T , c 2 T )
\begin{pmatrix}1\\1\end{pmatrix}=\sum_jc_j^Tc_j=O_2O_2^T=\sum_jd_j^Td_j=O_1O_1^T \tag*{(93)}\end{pmatrix}
Therefore we can take O 1 , O 2 O_1,O_2 O 1 , O 2 (91) can be rewritten as
J = O 1 ( z 1 z 2 ) O 2 T , where z j = Γ A B , j − i λ j Γ A , j Γ B , j . (94) J=O_1\begin{pmatrix}z_1\\z_2\end{pmatrix}O_2^T,\qquad\text{where }z_j=\frac{\Gamma_{AB,j}-i\lambda_j}{\sqrt{\Gamma_{A,j}\Gamma_{B,j}}}. \tag*{(94)} J = O 1 ( z 1 z 2 ) O 2 T , where z j = Γ A , j Γ B , j Γ A B , j − i λ j . (94) For the convenience of discussion, let us first consider the boundary case that both singular value of J J J J J J J J J
∣ Γ x x − i λ x x ∣ Γ ˉ x = ∣ Γ z z − i λ z z ∣ Γ ˉ z , ∣ Γ x x − i λ x x ∣ 2 ( Γ ˉ x ) 2 + ∣ Γ x z − i λ x z ∣ 2 Γ ˉ x Γ ˉ z = 1 , \frac{|\Gamma^{xx}-i\lambda^{xx}|}{\bar{\Gamma}^{x}}=\frac{|\Gamma^{zz}-i\lambda^{zz}|}{\bar{\Gamma}^{z}},\qquad\frac{|\Gamma^{xx}-i\lambda^{xx}|^2}{(\bar{\Gamma}^{x})^2}+\frac{|\Gamma^{xz}-i\lambda^{xz}|^2}{\bar{\Gamma}^{x}\bar{\Gamma}^{z}}=1, Γ ˉ x ∣ Γ xx − i λ xx ∣ = Γ ˉ z ∣ Γ z z − i λ z z ∣ , ( Γ ˉ x ) 2 ∣ Γ xx − i λ xx ∣ 2 + Γ ˉ x Γ ˉ z ∣ Γ x z − i λ x z ∣ 2 = 1 , arg ( Γ x x − i λ x x ) + arg ( Γ z z − i λ z z ) − 2 arg ( Γ x z − i λ x z ) ≡ 0 ( m o d 2 π ) . (95) \arg(\Gamma^{xx}-i\lambda^{xx})+\arg(\Gamma^{zz}-i\lambda^{zz})-2\arg(\Gamma^{xz}-i\lambda^{xz})\equiv0\pmod{2\pi}. \tag*{(95)} arg ( Γ xx − i λ xx ) + arg ( Γ z z − i λ z z ) − 2 arg ( Γ x z − i λ x z ) ≡ 0 ( mod 2 π ) . (95) Let us denote
g = ∣ Γ x x − i λ x x ∣ Γ ˉ x = 1 − h 2 , θ = − arg ( Γ x z − i λ x z ) , γ = θ + arg ( Γ x x − i λ x x ) , J = e − i θ ( g e i γ − h h g e − i γ ) . (96) g=\frac{|\Gamma^{xx}-i\lambda^{xx}|}{\bar{\Gamma}^{x}}=\sqrt{1-h^2},\qquad\theta=-\arg(\Gamma^{xz}-i\lambda^{xz}),\qquad\gamma=\theta+\arg(\Gamma^{xx}-i\lambda^{xx}),\qquad J=e^{-i\theta}\begin{pmatrix}ge^{i\gamma}&-h\\h&ge^{-i\gamma}\end{pmatrix}. \tag*{(96)} g = Γ ˉ x ∣ Γ xx − i λ xx ∣ = 1 − h 2 , θ = − arg ( Γ x z − i λ x z ) , γ = θ + arg ( Γ xx − i λ xx ) , J = e − i θ ( g e iγ h − h g e − iγ ) . (96) This equation can be solved in two-fold ways. Firstly, let us solve
J T J = O 2 ( z 1 2 z 2 2 ) O 2 T = e − 2 i θ − i π / 2 ( i h 2 + i g 2 e 2 i γ 2 g h sin γ 2 g h sin γ i h 2 + i g 2 e − 2 i γ ) = e − 2 i θ − i π / 2 ( m e i β n n − m e − i β ) , J^TJ=O_2\begin{pmatrix}z_1^2\\z_2^2\end{pmatrix}O_2^T=e^{-2i\theta-i\pi/2}\begin{pmatrix}ih^2+ig^2e^{2i\gamma}&2gh\sin\gamma\\2gh\sin\gamma&ih^2+ig^2e^{-2i\gamma}\end{pmatrix}=e^{-2i\theta-i\pi/2}\begin{pmatrix}me^{i\beta}&n\\n&-me^{-i\beta}\end{pmatrix}, J T J = O 2 ( z 1 2 z 2 2 ) O 2 T = e − 2 i θ − iπ /2 ( i h 2 + i g 2 e 2 iγ 2 g h sin γ 2 g h sin γ i h 2 + i g 2 e − 2 iγ ) = e − 2 i θ − iπ /2 ( m e i β n n − m e − i β ) , where m = ( h 2 + g 2 cos 2 γ ) 2 + g 4 sin 2 2 γ , β = arg ( i h 2 + i g 2 e 2 i γ ) , n = 2 g h sin γ (97) \text{where }m=\sqrt{(h^2+g^2\cos2\gamma)^2+g^4\sin^22\gamma},\qquad\beta=\arg(ih^2+ig^2e^{2i\gamma}),\qquad n=2gh\sin\gamma \tag*{(97)} where m = ( h 2 + g 2 cos 2 γ ) 2 + g 4 sin 2 2 γ , β = arg ( i h 2 + i g 2 e 2 iγ ) , n = 2 g h sin γ (97) This equation is readily solved as follows:
c 1 = ( x , s 1 − x 2 ) , c 2 = ( s 1 − x 2 , − x ) , x = 1 + m cos β m 2 cos 2 β + n 2 2 , s = n ∣ n ∣ , c_1=(x,s\sqrt{1-x^2}),\qquad c_2=(s\sqrt{1-x^2},-x),\qquad x=\sqrt{\frac{1+\sqrt{\frac{m\cos\beta}{\sqrt{m^2\cos^2\beta+n^2}}}}{2}},\qquad s=\frac{n}{|n|}, c 1 = ( x , s 1 − x 2 ) , c 2 = ( s 1 − x 2 , − x ) , x = 2 1 + m 2 c o s 2 β + n 2 m c o s β , s = ∣ n ∣ n , z 1 2 = e − 2 i θ − i π / 2 ( m 2 cos 2 β + n 2 + i m sin β ) , z 2 2 = e − 2 i θ − i π / 2 ( − m 2 cos 2 β + n 2 + i m sin β ) (98) z_1^2=e^{-2i\theta-i\pi/2}\left(\sqrt{m^2\cos^2\beta+n^2}+im\sin\beta\right),\qquad z_2^2=e^{-2i\theta-i\pi/2}\left(-\sqrt{m^2\cos^2\beta+n^2}+im\sin\beta\right) \tag*{(98)} z 1 2 = e − 2 i θ − iπ /2 ( m 2 cos 2 β + n 2 + im sin β ) , z 2 2 = e − 2 i θ − iπ /2 ( − m 2 cos 2 β + n 2 + im sin β ) (98) Note that here ∣ z 1 ∣ = ∣ z 2 ∣ = 1 |z_1| = |z_2| = 1 ∣ z 1 ∣ = ∣ z 2 ∣ = 1 L ρ A , j , ρ B , j \mathcal{L}^{\rho_{A,j},\rho_{B,j}} L ρ A , j , ρ B , j
J J T = O 1 ( z 1 2 0 0 z 2 2 ) O 1 T = e − 2 i θ − i π / 2 ( i h 2 + i g 2 e 2 i γ − 2 g h sin γ − 2 g h sin γ i h 2 + i g 2 e − 2 i γ ) = e − 2 i θ − i π / 2 ( m e i β − n − n − m e − i β ) , (99) JJ^{\mathsf{T}} = O_1 \begin{pmatrix} z_1^2 & 0 \\ 0 & z_2^2 \end{pmatrix} O_1^{\mathsf{T}} = e^{-2i\theta-i\pi/2} \begin{pmatrix} ih^2 + ig^2e^{2i\gamma} & -2gh\sin\gamma\\ -2gh\sin\gamma& ih^2 + ig^2e^{-2i\gamma} \end{pmatrix} = e^{-2i\theta-i\pi/2} \begin{pmatrix} me^{i\beta} & -n \\ -n & -me^{-i\beta} \end{pmatrix}, \tag*{(99)} J J T = O 1 ( z 1 2 0 0 z 2 2 ) O 1 T = e − 2 i θ − iπ /2 ( i h 2 + i g 2 e 2 iγ − 2 g h sin γ − 2 g h sin γ i h 2 + i g 2 e − 2 iγ ) = e − 2 i θ − iπ /2 ( m e i β − n − n − m e − i β ) , (99) where only the sign of off-diagonal element is altered. Therefore it is solved by
d 1 = ( x , − s 1 − x 2 ) , d 2 = ( − s 1 − x 2 , − x ) (100) d_1 = \left(x,-s\sqrt{1-x^2}\right), \qquad d_2 = \left(-s\sqrt{1-x^2},-x\right) \tag*{(100)} d 1 = ( x , − s 1 − x 2 ) , d 2 = ( − s 1 − x 2 , − x ) (100) More specifically, this L ′ \mathcal{L}' L ′
L ′ = L 1 O A , 1 , O B , 1 + L 2 O A , 2 , O B , 2 , \mathcal{L}'=\mathcal{L}^{O_{A,1},O_{B,1}}_1+\mathcal{L}^{O_{A,2},O_{B,2}}_2, L ′ = L 1 O A , 1 , O B , 1 + L 2 O A , 2 , O B , 2 , O A , 1 = x Γ ‾ σ A x + s 1 − x 2 Γ σ A z Γ A , 1 , O A , 2 = s 1 − x 2 Γ ‾ σ A x − x Γ σ A z Γ A , 2 , O_{A,1}=\frac{x\sqrt{\overline{\Gamma}}\sigma_A^x+s\sqrt{1-x^2}\sqrt{\Gamma}\sigma_A^z}{\sqrt{\Gamma_{A,1}}}, \qquad O_{A,2}=\frac{s\sqrt{1-x^2}\sqrt{\overline{\Gamma}}\sigma_A^x-x\sqrt{\Gamma}\sigma_A^z}{\sqrt{\Gamma_{A,2}}}, O A , 1 = Γ A , 1 x Γ σ A x + s 1 − x 2 Γ σ A z , O A , 2 = Γ A , 2 s 1 − x 2 Γ σ A x − x Γ σ A z , O B , 1 = x Γ ‾ σ B x − s 1 − x 2 Γ σ B z Γ B , 1 , O B , 2 = − s 1 − x 2 Γ ‾ σ B x − x Γ σ B z Γ B , 2 , O_{B,1}=\frac{x\sqrt{\overline{\Gamma}}\sigma_B^x-s\sqrt{1-x^2}\sqrt{\Gamma}\sigma_B^z}{\sqrt{\Gamma_{B,1}}}, \qquad O_{B,2}=\frac{-s\sqrt{1-x^2}\sqrt{\overline{\Gamma}}\sigma_B^x-x\sqrt{\Gamma}\sigma_B^z}{\sqrt{\Gamma_{B,2}}}, O B , 1 = Γ B , 1 x Γ σ B x − s 1 − x 2 Γ σ B z , O B , 2 = Γ B , 2 − s 1 − x 2 Γ σ B x − x Γ σ B z , Γ A B , 1 − i λ 1 = Γ A , 1 Γ B , 1 z 1 , Γ A B , 2 − i λ 2 = Γ A , 2 Γ B , 2 z 2 (101) \Gamma_{AB,1}-i\lambda_1=\sqrt{\Gamma_{A,1}\Gamma_{B,1}}z_1,\qquad\Gamma_{AB,2}-i\lambda_2=\sqrt{\Gamma_{A,2}\Gamma_{B,2}}z_2 \tag*{(101)} Γ A B , 1 − i λ 1 = Γ A , 1 Γ B , 1 z 1 , Γ A B , 2 − i λ 2 = Γ A , 2 Γ B , 2 z 2 (101) and Γ α , j > 0 \Gamma_{\alpha,j}>0 Γ α , j > 0
In general, J J J ∥ J ∥ 2 ≤ 1 \lVert J\rVert_2\leq1 ∥ J ∥ 2 ≤ 1 J J J
J = U ( s 1 0 0 s 2 ) V † , (102) J=U\begin{pmatrix}s_1&0\\0&s_2\end{pmatrix}V^{\dagger}, \tag*{(102)} J = U ( s 1 0 0 s 2 ) V † , (102) where U U U V V V 1 ≥ s 1 , s 2 ≥ 0 1\geq s_1,s_2\geq0 1 ≥ s 1 , s 2 ≥ 0 J J J
J = 1 2 U ( s 1 + i 1 − s 1 2 0 0 s 2 + i 1 − s 2 2 ) V † + 1 2 U ( s 1 − i 1 − s 1 2 0 0 s 2 − i 1 − s 2 2 ) V † . (103) J=\frac{1}{2}U\begin{pmatrix}s_1+i\sqrt{1-s_1^2}&0\\0&s_2+i\sqrt{1-s_2^2}\end{pmatrix}V^{\dagger}+\frac{1}{2}U\begin{pmatrix}s_1-i\sqrt{1-s_1^2}&0\\0&s_2-i\sqrt{1-s_2^2}\end{pmatrix}V^{\dagger}. \tag*{(103)} J = 2 1 U ( s 1 + i 1 − s 1 2 0 0 s 2 + i 1 − s 2 2 ) V † + 2 1 U ( s 1 − i 1 − s 1 2 0 0 s 2 − i 1 − s 2 2 ) V † . (103) Therefore, similar to the discussions above, using product of the unitary and its transpose, we can write down a decomposition of the first unitary as z 3 c 3 T d 3 + z 4 c 4 T d 4 z_3c_3^{\mathsf{T}}d_3+z_4c_4^{\mathsf{T}}d_4 z 3 c 3 T d 3 + z 4 c 4 T d 4 z 5 d 5 T c 5 + z 6 d 6 T c 6 z_5d_5^{\mathsf{T}}c_5+z_6d_6^{\mathsf{T}}c_6 z 5 d 5 T c 5 + z 6 d 6 T c 6
z 3 c 3 T d 3 2 + z 4 c 4 T d 4 2 + z 5 c 5 T d 5 2 + z 6 c 6 T d 6 2 = J , ∑ j = 3 6 c j T c j 2 = ∑ j = 3 6 d j T d j 2 = ( 1 0 0 1 ) . (104) z_3\frac{c_3^{\mathsf{T}}d_3}{2}+z_4\frac{c_4^{\mathsf{T}}d_4}{2}+z_5\frac{c_5^{\mathsf{T}}d_5}{2}+z_6\frac{c_6^{\mathsf{T}}d_6}{2}=J,\qquad\sum_{j=3}^{6}\frac{c_j^{\mathsf{T}}c_j}{2}=\sum_{j=3}^{6}\frac{d_j^{\mathsf{T}}d_j}{2}=\begin{pmatrix}1&0\\0&1\end{pmatrix}. \tag*{(104)} z 3 2 c 3 T d 3 + z 4 2 c 4 T d 4 + z 5 2 c 5 T d 5 + z 6 2 c 6 T d 6 = J , j = 3 ∑ 6 2 c j T c j = j = 3 ∑ 6 2 d j T d j = ( 1 0 0 1 ) . (104) Therefore, L ′ \mathcal{L}' L ′ L t o t = L ′ + L ⊥ \mathcal{L}_{\mathrm{tot}}=\mathcal{L}'+\mathcal{L}^{\perp} L tot = L ′ + L ⊥ L ⊥ \mathcal{L}^{\perp} L ⊥
In summary, we have shown that η = ∥ J ∥ 2 \eta=\lVert J\rVert_2 η = ∥ J ∥ 2 L t o t \mathcal{L}_{\mathrm{tot}} L tot L t o t \mathcal{L}_{\mathrm{tot}} L tot η ≤ 1 \eta\leq1 η ≤ 1 Lemma 3 ; L t o t \mathcal{L}_{\mathrm{tot}} L tot η > 1 \eta>1 η > 1
Entangling condition
On the other hand, non-separability does not necessarily leads to entanglement. In general, the evolution channels could be SEPP\ \backslash \ [56 ] .
Proposition 4. Given the Lindbladian in Eq. (B19), it is entangling/not separability-preserving at t = 0 + t=0+ t = 0 + ∣ ψ ⟩ A ⊗ ∣ ϕ ⟩ B |\psi\rangle_A\otimes|\phi\rangle_B ∣ ψ ⟩ A ⊗ ∣ ϕ ⟩ B
inf ∣ ψ ⟩ A , ∣ ϕ ⟩ B ( ( ⟨ h C h † ⟩ ) ( ⟨ g C g † ⟩ ) − ∣ g ∗ ( − D + i H ) h † ∣ 2 ) < 0. (105) \underset{|\psi\rangle_A,|\phi\rangle_B}{\inf}\left((\langle hC h^{\dagger}\rangle)(\langle gC g^{\dagger}\rangle)-\left|g^*(-D+iH)h^{\dagger}\right|^2\right)<0. \tag*{(105)} ∣ ψ ⟩ A , ∣ ϕ ⟩ B inf ( (⟨ h C h † ⟩) (⟨ g C g † ⟩) − g ∗ ( − D + i H ) h † 2 ) < 0. (105) The definition of these symbols are presented below. As a simple sufficient condition, if we take ∣ ψ ⟩ A ⊗ ∣ ϕ ⟩ B = ∣ 00 ⟩ A B |\psi\rangle_A \otimes|\phi\rangle_B = |00\rangle_{AB} ∣ ψ ⟩ A ⊗ ∣ ϕ ⟩ B = ∣00 ⟩ A B ∣ 11 ⟩ A B |11\rangle_{AB} ∣11 ⟩ A B
( λ x x ) 2 + ( Γ x x ) 2 > min ( Γ x + Γ + , Γ x + Γ − ) 2 (106) (\lambda^{xx})^2 + (\Gamma^{xx})^2 > \min(\Gamma^x+\Gamma^+,\Gamma^x+\Gamma^-)^2 \tag*{(106)} ( λ xx ) 2 + ( Γ xx ) 2 > min ( Γ x + Γ + , Γ x + Γ − ) 2 (106) In the situation that only λ e f f z z , Γ e f f z z , Γ e f f z ≠ 0 \lambda_{\mathrm{eff}}^{zz},\Gamma_{\mathrm{eff}}^{zz},\Gamma_{\mathrm{eff}}^z \ne0 λ eff z z , Γ eff z z , Γ eff z = 0 ( λ e f f z z ) 2 + ( Γ e f f z z ) 2 > ( Γ e f f z ) 2 (\lambda_{\mathrm{eff}}^{zz})^2 + (\Gamma_{\mathrm{eff}}^{zz})^2 > (\Gamma_{\mathrm{eff}}^z)^2 ( λ eff z z ) 2 + ( Γ eff z z ) 2 > ( Γ eff z ) 2
Proof. In general, if the channel exp L t o t t \exp\mathcal{L}_{\mathrm{tot}}t exp L tot t t = 0 + t=0^+ t = 0 + L t o t \mathcal{L}_{\mathrm{tot}} L tot ∣ ψ ⟩ A ∣ ϕ ⟩ B |\psi\rangle_A|\phi\rangle_B ∣ ψ ⟩ A ∣ ϕ ⟩ B ∣ ψ ⊥ ⟩ A |\psi_\perp\rangle_A ∣ ψ ⊥ ⟩ A ∣ ϕ ⊥ ⟩ B |\phi_\perp\rangle_B ∣ ϕ ⊥ ⟩ B O ( d t ) O(dt) O ( d t ) span { ∣ ψ ⊥ ⟩ A ∣ ϕ ⊥ ⟩ B , ∣ ψ ⟩ A ∣ ϕ ⟩ B } \operatorname{span}\{|\psi_\perp\rangle_A|\phi_\perp\rangle_B,|\psi\rangle_A|\phi\rangle_B\} span { ∣ ψ ⊥ ⟩ A ∣ ϕ ⊥ ⟩ B , ∣ ψ ⟩ A ∣ ϕ ⟩ B } Π ⊥ \Pi_\perp Π ⊥ [54 , 55 ]
L t o t ( ρ ) = − i [ ∑ i j H i j σ A i ⊗ σ B j , ρ ] + ∑ i j ( C i j ( σ A i ρ σ A j − 1 2 { σ A j σ A i , ρ } ) + D i j ( σ A i ρ σ B j − 1 2 { σ A i σ B j , ρ } ) + D i j ∗ ( σ B i ρ σ A j − 1 2 { σ B i σ A j , ρ } ) + E i j ( σ B i ρ σ B j − 1 2 { σ B j σ B i , ρ } ) ) , \begin{aligned}
\mathcal{L}_{\mathrm{tot}}(\rho)={}&-i\left[\sum_{ij}H_{ij}\sigma_A^i\otimes\sigma_B^j,\rho\right]\\
&+\sum_{ij}\left(C_{ij}\left(\sigma_A^i\rho\sigma_A^j-\frac{1}{2}\{\sigma_A^j\sigma_A^i,\rho\}\right)+D_{ij}\left(\sigma_A^i\rho\sigma_B^j-\frac{1}{2}\{\sigma_A^i\sigma_B^j,\rho\}\right)\right.\\
&\left.+D_{ij}^*\left(\sigma_B^i\rho\sigma_A^j-\frac{1}{2}\{\sigma_B^i\sigma_A^j,\rho\}\right)+E_{ij}\left(\sigma_B^i\rho\sigma_B^j-\frac{1}{2}\{\sigma_B^j\sigma_B^i,\rho\}\right)\right),
\end{aligned} L tot ( ρ ) = − i [ ij ∑ H ij σ A i ⊗ σ B j , ρ ] + ij ∑ ( C ij ( σ A i ρ σ A j − 2 1 { σ A j σ A i , ρ } ) + D ij ( σ A i ρ σ B j − 2 1 { σ A i σ B j , ρ } ) + D ij ∗ ( σ B i ρ σ A j − 2 1 { σ B i σ A j , ρ } ) + E ij ( σ B i ρ σ B j − 2 1 { σ B j σ B i , ρ } ) ) , where
H = ( λ x x λ x z − λ x z λ z z ) , C = E = ( Γ + + Γ − i Γ + − Γ − 4 − i Γ + − Γ − 4 Γ + + Γ − 4 Γ z ) , D = ( − Γ x x − Γ x z Γ x z − Γ z z ) , (107) H=\begin{pmatrix}\lambda^{xx}&\lambda^{xz}\\-\lambda^{xz}&\lambda^{zz}\end{pmatrix},\quad
C=E=\begin{pmatrix}\Gamma^+ + \Gamma^- & i\frac{\Gamma^+ - \Gamma^-}{4}\\-i\frac{\Gamma^+ - \Gamma^-}{4}&\frac{\Gamma^+ + \Gamma^-}{4}\\&&\Gamma^z\end{pmatrix},\quad
D=\begin{pmatrix}-\Gamma^{xx}&-\Gamma^{xz}\\\Gamma^{xz}&-\Gamma^{zz}\end{pmatrix}, \tag*{(107)} H = ( λ xx − λ x z λ x z λ z z ) , C = E = Γ + + Γ − − i 4 Γ + − Γ − i 4 Γ + − Γ − 4 Γ + + Γ − Γ z , D = ( − Γ xx Γ x z − Γ x z − Γ z z ) , (107) where we have taken σ 1 = σ x , σ 2 = σ y , σ 3 = σ z \sigma^1=\sigma^x,\sigma^2=\sigma^y,\sigma^3=\sigma^z σ 1 = σ x , σ 2 = σ y , σ 3 = σ z A ⟨ ψ ⊥ ∣ σ A i ∣ ψ ⟩ A = g i {}_A\langle\psi_\perp|\sigma_A^i|\psi\rangle_A=g^i A ⟨ ψ ⊥ ∣ σ A i ∣ ψ ⟩ A = g i B ⟨ ϕ ⊥ ∣ σ B i ∣ ϕ ⟩ B = h i {}_B\langle\phi_\perp|\sigma_B^i|\phi\rangle_B=h^i B ⟨ ϕ ⊥ ∣ σ B i ∣ ϕ ⟩ B = h i
Π ⊥ L t o t ( ∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣ ) T B Π ⊥ = ( h E h † g ∗ ( − Re D + i H ) h † g ( − Re D − i H ) h T g C g † ) = ( h C h † g ∗ ( − D + i H ) h † g ( − D − i H ) h T g C g † ) . (108) \Pi_\perp\mathcal{L}_{\mathrm{tot}}(|\psi\rangle_A\langle\psi|\otimes|\phi\rangle_B\langle\phi|)^{T_B}\Pi_\perp=
\begin{pmatrix}
hEh^\dagger&g^*(-\operatorname{Re}D+iH)h^\dagger\\
g(-\operatorname{Re}D-iH)h^T&gCg^\dagger
\end{pmatrix}
=
\begin{pmatrix}
hCh^\dagger&g^*(-D+iH)h^\dagger\\
g(-D-iH)h^T&gCg^\dagger
\end{pmatrix}. \tag*{(108)} Π ⊥ L tot ( ∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣ ) T B Π ⊥ = ( h E h † g ( − Re D − i H ) h T g ∗ ( − Re D + i H ) h † g C g † ) = ( h C h † g ( − D − i H ) h T g ∗ ( − D + i H ) h † g C g † ) . (108) Since C = E ≥ 0 C=E\ge0 C = E ≥ 0
( h C h † ) ( g C g † ) ≥ ∣ g ∗ ( − D + i H ) h † ∣ 2 (109) (hCh^\dagger)(gCg^\dagger)\ge\left|g^*(-D+iH)h^\dagger\right|^2 \tag*{(109)} ( h C h † ) ( g C g † ) ≥ g ∗ ( − D + i H ) h † 2 (109) and therefore L t o t \mathcal{L}_{\mathrm{tot}} L tot t = 0 + t=0^+ t = 0 +
inf ∣ ψ ⟩ A , ∣ ϕ ⟩ B ( h C h † ) ( g C g † ) − ∣ g ∗ ( − D + i H ) h † ∣ 2 < 0. (110) \inf_{|\psi\rangle_A,|\phi\rangle_B}(hCh^\dagger)(gCg^\dagger)-\left|g^*(-D+iH)h^\dagger\right|^2<0. \tag*{(110)} ∣ ψ ⟩ A , ∣ ϕ ⟩ B inf ( h C h † ) ( g C g † ) − g ∗ ( − D + i H ) h † 2 < 0. (110) On the other hand, note that this criterion is only at the leading order. If Π ⊥ L t o t ( ∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣ ) T B Π ⊥ \Pi_\perp\mathcal{L}_{\mathrm{tot}}(|\psi\rangle_A\langle\psi|\otimes|\phi\rangle_B\langle\phi|)^{T_B}\Pi_\perp Π ⊥ L tot ( ∣ ψ ⟩ A ⟨ ψ ∣ ⊗ ∣ ϕ ⟩ B ⟨ ϕ ∣ ) T B Π ⊥
As an example, let us take ∣ ψ ⟩ A = ∣ 0 ⟩ , ∣ ϕ ⟩ B = ∣ 0 ⟩ B |\psi\rangle_A=|0\rangle,|\phi\rangle_B=|0\rangle_B ∣ ψ ⟩ A = ∣0 ⟩ , ∣ ϕ ⟩ B = ∣0 ⟩ B g = h = ( 1 , i , 0 ) g=h=(1,i,0) g = h = ( 1 , i , 0 )
( h C h † ) ( g C g † ) − ∣ g ∗ ( − D + i H ) h † ∣ 2 = ( Γ x + Γ + ) 2 − ( λ x x ) 2 − ( Γ x x ) 2 . (111) (hCh^\dagger)(gCg^\dagger)-\left|g^*(-D+iH)h^\dagger\right|^2=(\Gamma^x+\Gamma^+)^2-(\lambda^{xx})^2-(\Gamma^{xx})^2. \tag*{(111)} ( h C h † ) ( g C g † ) − g ∗ ( − D + i H ) h † 2 = ( Γ x + Γ + ) 2 − ( λ xx ) 2 − ( Γ xx ) 2 . (111) Similarly, if we take ∣ ψ ⟩ A = ∣ 1 ⟩ , ∣ ϕ ⟩ B = ∣ 1 ⟩ B |\psi\rangle_A=|1\rangle,|\phi\rangle_B=|1\rangle_B ∣ ψ ⟩ A = ∣1 ⟩ , ∣ ϕ ⟩ B = ∣1 ⟩ B g = h = ( 1 , − i , 0 ) g=h=(1,-i,0) g = h = ( 1 , − i , 0 )
( h C h † ) ( g C g † ) − ∣ g ∗ ( − D + i H ) h † ∣ 2 = ( Γ x + Γ − ) 2 − ( λ x x ) 2 − ( Γ x x ) 2 . (112) (hCh^\dagger)(gCg^\dagger)-\left|g^*(-D+iH)h^\dagger\right|^2=(\Gamma^x+\Gamma^-)^2-(\lambda^{xx})^2-(\Gamma^{xx})^2. \tag*{(112)} ( h C h † ) ( g C g † ) − g ∗ ( − D + i H ) h † 2 = ( Γ x + Γ − ) 2 − ( λ xx ) 2 − ( Γ xx ) 2 . (112) Hence we conclude L t o t \mathcal{L}_{\mathrm{tot}} L tot
( λ x x ) 2 + ( Γ x x ) 2 > min ( Γ x + Γ + , Γ x + Γ − ) 2 (113) (\lambda^{xx})^2+(\Gamma^{xx})^2>\min(\Gamma^x+\Gamma^+,\Gamma^x+\Gamma^-)^2 \tag*{(113)} ( λ xx ) 2 + ( Γ xx ) 2 > min ( Γ x + Γ + , Γ x + Γ − ) 2 (113) In summary, for the separability classification of the Lindbladian dynamics L t o t \mathcal{L}_{\mathrm{tot}} L tot
η = ∥ ( Γ e f f x x − i λ e f f x x Γ e f f x Γ e f f x z − i λ e f f x z Γ e f f x Γ e f f z Γ e f f x z − i λ e f f x z Γ e f f x Γ e f f z Γ e f f z z − i λ e f f z z Γ e f f z ) ∥ 2 , (114) \eta=\left\Vert\begin{pmatrix}
\dfrac{\Gamma_{\mathrm{eff}}^{xx}-i\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^{x}} & \dfrac{\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{x}\Gamma_{\mathrm{eff}}^{z}}}\\
\dfrac{\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{x}\Gamma_{\mathrm{eff}}^{z}}} & \dfrac{\Gamma_{\mathrm{eff}}^{zz}-i\lambda_{\mathrm{eff}}^{zz}}{\Gamma_{\mathrm{eff}}^{z}}
\end{pmatrix}\right\Vert_{2}, \tag*{(114)} η = Γ eff x Γ eff xx − i λ eff xx Γ eff x Γ eff z Γ eff x z − i λ eff x z Γ eff x Γ eff z Γ eff x z − i λ eff x z Γ eff z Γ eff z z − i λ eff z z 2 , (114) such that L t o t \mathcal{L}_{\mathrm{tot}} L tot η ≤ 1 \eta\leq1 η ≤ 1 L t o t \mathcal{L}_{\mathrm{tot}} L tot η > 1 \eta>1 η > 1 L t o t \mathcal{L}_{\mathrm{tot}} L tot
( λ e f f x x ) 2 + ( Γ e f f x x ) 2 min ( Γ e f f x + Γ e f f + , Γ e f f x + Γ e f f − ) > 1. (115) \frac{\sqrt{\left(\lambda_{\mathrm{eff}}^{xx}\right)^{2}+\left(\Gamma_{\mathrm{eff}}^{xx}\right)^{2}}}{\min\left(\Gamma_{\mathrm{eff}}^{x}+\Gamma_{\mathrm{eff}}^{+},\Gamma_{\mathrm{eff}}^{x}+\Gamma_{\mathrm{eff}}^{-}\right)}>1. \tag*{(115)} min ( Γ eff x + Γ eff + , Γ eff x + Γ eff − ) ( λ eff xx ) 2 + ( Γ eff xx ) 2 > 1. (115) Beyond local QEC codes
In the previous sections we analyzed an imperfect local QEC protocol, where both the code space and the recovery operations are defined locally, ensuring that entanglement is not introduced by the QEC setting itself. More generally, one may consider separable recovery operations that also do not introduce entanglement. In this section, we investigate whether one can design a better imperfect QEC code equipped with a general separable recovery map.
In App. B 3 we considered a QEC code that encodes two logical qubits. We start from the leading-order master equation in Eq. (51) , which contains local x A x_A x A x B x_B x B
d d t ρ = L x A , x B ( ρ ) = − i λ [ x A ⊗ x B , ρ ] + Γ ( D [ x A ⊗ I B ] + D [ I A ⊗ x B ] ) ( ρ ) , (116) \frac{\mathrm{d}}{\mathrm{d}t}\rho=\mathcal{L}_{x_A,x_B}(\rho)=-i\lambda[x_A\otimes x_B,\rho]+\Gamma\left(\mathcal{D}[x_A\otimes I_B]+\mathcal{D}[I_A\otimes x_B]\right)(\rho), \tag*{(116)} d t d ρ = L x A , x B ( ρ ) = − iλ [ x A ⊗ x B , ρ ] + Γ ( D [ x A ⊗ I B ] + D [ I A ⊗ x B ] ) ( ρ ) , (116) If we choose a code with distance N = 0 N=0 N = 0 (3) in the main text reduce to
Π α x α Π α = x 0 σ α x , Π α x α 2 Π α = 2 x 0 2 Π α − x 0 2 σ α z , \Pi_\alpha x_\alpha\Pi_\alpha=x_0\sigma_\alpha^x,\qquad\Pi_\alpha x_\alpha^2\Pi_\alpha=2x_0^2\Pi_\alpha-x_0^2\sigma_\alpha^z, Π α x α Π α = x 0 σ α x , Π α x α 2 Π α = 2 x 0 2 Π α − x 0 2 σ α z , Π A ⊗ Π B × x A ⊗ x B × Π A ⊗ Π B = x 0 2 σ A x σ B x , (117) \Pi_A\otimes\Pi_B\times x_A\otimes x_B\times\Pi_A\otimes\Pi_B=x_0^2\sigma_A^x\sigma_B^x, \tag*{(117)} Π A ⊗ Π B × x A ⊗ x B × Π A ⊗ Π B = x 0 2 σ A x σ B x , (117) i.e. none of the error-correction conditions are satisfied. This failure is already enforced by the requirement that x A ⊗ x B x_A\otimes x_B x A ⊗ x B ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B |\bar{0}\rangle_A|\bar{0}\rangle_B ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B |\bar{0}\rangle_A|\bar{0}\rangle_B ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B |\bar{1}\rangle_A|\bar{1}\rangle_B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B
We now consider a more general QEC code in which the error-correction conditions can be satisfied except for the condition involving x A ⊗ x B x_A\otimes x_B x A ⊗ x B C = span { ∣ i ˉ ⟩ A B } i = 0 , 1 ⊂ H A ⊗ H B \mathcal{C}=\operatorname{span}\{|\bar{i}\rangle_{AB}\}_{i=0,1}\subset\mathcal{H}_A\otimes\mathcal{H}_B C = span { ∣ i ˉ ⟩ A B } i = 0 , 1 ⊂ H A ⊗ H B C \mathcal{C} C Π = ∑ i ∣ i ˉ ⟩ ⟨ i ˉ ∣ \Pi=\sum_i|\bar{i}\rangle\langle\bar{i}| Π = ∑ i ∣ i ˉ ⟩ ⟨ i ˉ ∣ σ x = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ \sigma^x=|\bar{0}\rangle\langle\bar{1}|+|\bar{1}\rangle\langle\bar{0}| σ x = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣
Assume that the following error conditions hold:
Π x A ⊗ I Π = a 1 Π , Π x A 2 ⊗ I Π = ( a 1 2 + a 2 2 ) Π , \Pi x_A\otimes I\Pi=a_1\Pi,\qquad\Pi x_A^2\otimes I\Pi=(a_1^2+a_2^2)\Pi, Π x A ⊗ I Π = a 1 Π , Π x A 2 ⊗ I Π = ( a 1 2 + a 2 2 ) Π , Π ⊗ x B Π = b 1 Π , Π ⊗ x B 2 Π = ( b 1 2 + b 2 2 ) Π , (118) \Pi\otimes x_B\Pi=b_1\Pi,\qquad\Pi\otimes x_B^2\Pi=(b_1^2+b_2^2)\Pi, \tag*{(118)} Π ⊗ x B Π = b 1 Π , Π ⊗ x B 2 Π = ( b 1 2 + b 2 2 ) Π , (118) where a 1 , a 2 , b 1 , b 2 a_1,a_2,b_1,b_2 a 1 , a 2 , b 1 , b 2 a 2 , b 2 ≠ 0 a_2,b_2\ne0 a 2 , b 2 = 0 a 2 = ⟨ x A 2 ⟩ − ⟨ x A ⟩ 2 = Δ x A a_2=\sqrt{\langle x_A^2\rangle-\langle x_A\rangle^2}=\Delta x_A a 2 = ⟨ x A 2 ⟩ − ⟨ x A ⟩ 2 = Δ x A b 2 = Δ x B b_2=\Delta x_B b 2 = Δ x B ∣ i ˉ x A ⟩ ∝ ( I − ∣ i ˉ ⟩ ⟨ i ˉ ∣ ) x A ∣ i ˉ ⟩ |\bar{i}_{x_A}\rangle\propto(I-|\bar{i}\rangle\langle\bar{i}|)x_A|\bar{i}\rangle ∣ i ˉ x A ⟩ ∝ ( I − ∣ i ˉ ⟩ ⟨ i ˉ ∣ ) x A ∣ i ˉ ⟩ ⟨ i ˉ x A ∣ j ˉ ⟩ = 0 \langle\bar{i}_{x_A}|\bar{j}\rangle=0 ⟨ i ˉ x A ∣ j ˉ ⟩ = 0 ⟨ i ˉ x A ∣ j ˉ x A ⟩ = δ i j \langle\bar{i}_{x_A}|\bar{j}_{x_A}\rangle=\delta_{ij} ⟨ i ˉ x A ∣ j ˉ x A ⟩ = δ ij x A ∣ i ˉ ⟩ = a 1 ∣ i ˉ ⟩ + a 2 ∣ i ˉ x A ⟩ x_A|\bar{i}\rangle=a_1|\bar{i}\rangle+a_2|\bar{i}_{x_A}\rangle x A ∣ i ˉ ⟩ = a 1 ∣ i ˉ ⟩ + a 2 ∣ i ˉ x A ⟩ C x A = { ∣ i ˉ x A ⟩ } \mathcal{C}_{x_A}=\{|\bar{i}_{x_A}\rangle\} C x A = { ∣ i ˉ x A ⟩} Figure 4 . This subspace is orthogonal to the code space C \mathcal{C} C { ∣ i ˉ x A ⟩ } \{|\bar{i}_{x_A}\rangle\} { ∣ i ˉ x A ⟩} x A x_A x A C x A \mathcal{C}_{x_A} C x A
Figure 4. Schematics of the change of bases ∣ i ˉ ⟩ |\bar{i}\rangle ∣ i ˉ ⟩ x A x_A x A x B x_B x B ∣ 0 ˉ ⟩ |\bar{0}\rangle ∣ 0 ˉ ⟩ ∣ 1 ˉ ⟩ |\bar{1}\rangle ∣ 1 ˉ ⟩ x A x_A x A x B x_B x B ∣ i ˉ ⟩ |\bar{i}\rangle ∣ i ˉ ⟩ ∣ i ˉ x A ⟩ |\bar{i}_{x_A}\rangle ∣ i ˉ x A ⟩ ∣ i ˉ x B ⟩ |\bar{i}_{x_B}\rangle ∣ i ˉ x B ⟩ ∣ i ˉ x B ′ ⟩ |\bar{i}^{\prime}_{x_B}\rangle ∣ i ˉ x B ′ ⟩ ∣ i ˉ x B ⟩ |\bar{i}_{x_B}\rangle ∣ i ˉ x B ⟩ ∣ 0 ˉ x A ⟩ |\bar{0}_{x_A}\rangle ∣ 0 ˉ x A ⟩ ∣ 1 ˉ x A ⟩ |\bar{1}_{x_A}\rangle ∣ 1 ˉ x A ⟩ ∣ 0 ˉ x B ′ ⟩ |\bar{0}^{\prime}_{x_B}\rangle ∣ 0 ˉ x B ′ ⟩ ∣ 1 ˉ x B ′ ⟩ |\bar{1}^{\prime}_{x_B}\rangle ∣ 1 ˉ x B ′ ⟩ ∣ 0 ˉ x B ′ ′ ⟩ |\bar{0}^{\prime\prime}_{x_B}\rangle ∣ 0 ˉ x B ′′ ⟩ ∣ 1 ˉ x B ′ ′ ⟩ |\bar{1}^{\prime\prime}_{x_B}\rangle ∣ 1 ˉ x B ′′ ⟩
Similarly, define ∣ i ˉ x B ⟩ ∝ ( I − ∣ i ˉ ⟩ ⟨ i ˉ ∣ ) x B ∣ i ˉ ⟩ |\bar{i}_{x_B}\rangle\propto(I-|\bar{i}\rangle\langle\bar{i}|)x_B|\bar{i}\rangle ∣ i ˉ x B ⟩ ∝ ( I − ∣ i ˉ ⟩ ⟨ i ˉ ∣ ) x B ∣ i ˉ ⟩ C x B = { ∣ i ˉ x B ⟩ } \mathcal{C}_{x_B}=\{|\bar{i}_{x_B}\rangle\} C x B = { ∣ i ˉ x B ⟩} ⟨ i ˉ ∣ j ˉ x B ⟩ = 0 \langle\bar{i}|\bar{j}_{x_B}\rangle=0 ⟨ i ˉ ∣ j ˉ x B ⟩ = 0 ⟨ i ˉ x B ∣ j ˉ x B ⟩ = δ i j \langle\bar{i}_{x_B}|\bar{j}_{x_B}\rangle=\delta_{ij} ⟨ i ˉ x B ∣ j ˉ x B ⟩ = δ ij x B ∣ i ˉ ⟩ = b 1 ∣ i ˉ ⟩ + b 2 ∣ i ˉ x B ⟩ x_B|\bar{i}\rangle=b_1|\bar{i}\rangle+b_2|\bar{i}_{x_B}\rangle x B ∣ i ˉ ⟩ = b 1 ∣ i ˉ ⟩ + b 2 ∣ i ˉ x B ⟩ Figure 4 . What remains undetermined is the relationship between C x A \mathcal{C}_{x_A} C x A C x B \mathcal{C}_{x_B} C x B Π x A = ∑ i ∣ i ˉ x A ⟩ ⟨ i ˉ x A ∣ \Pi_{x_A}=\sum_i|\bar{i}_{x_A}\rangle\langle\bar{i}_{x_A}| Π x A = ∑ i ∣ i ˉ x A ⟩ ⟨ i ˉ x A ∣ Π x B = ∑ i ∣ i ˉ x B ⟩ ⟨ i ˉ x B ∣ \Pi_{x_B}=\sum_i|\bar{i}_{x_B}\rangle\langle\bar{i}_{x_B}| Π x B = ∑ i ∣ i ˉ x B ⟩ ⟨ i ˉ x B ∣
To ensure that x A ⊗ x B x_A\otimes x_B x A ⊗ x B C x A \mathcal{C}_{x_A} C x A C x B \mathcal{C}_{x_B} C x B
Π x A ⊗ x B Π = ( a 1 b 1 + a 2 c 1 ) Π + a 2 c 2 σ x . (119) \Pi_{x_A}\otimes x_B\Pi=(a_1b_1+a_2c_1)\Pi+a_2c_2\sigma^x. \tag*{(119)} Π x A ⊗ x B Π = ( a 1 b 1 + a 2 c 1 ) Π + a 2 c 2 σ x . (119) where c 1 c_1 c 1 c 2 c_2 c 2 C \mathcal{C} C σ x \sigma^x σ x ∣ i ˉ x A ⟩ |\bar{i}_{x_A}\rangle ∣ i ˉ x A ⟩ ∣ j ˉ x B ⟩ |\bar{j}_{x_B}\rangle ∣ j ˉ x B ⟩ i ≠ j i\ne j i = j ∣ i ˉ x A ⟩ |\bar{i}_{x_A}\rangle ∣ i ˉ x A ⟩ ∣ j ˉ x B ⟩ |\bar{j}_{x_B}\rangle ∣ j ˉ x B ⟩ C x A \mathcal{C}_{x_A} C x A C x B \mathcal{C}_{x_B} C x B
Note that x A ⊗ I x_A\otimes I x A ⊗ I I ⊗ x B I\otimes x_B I ⊗ x B
0 = ⟨ i ˉ ∣ [ x A ⊗ I , I ⊗ x B ] ∣ j ˉ ⟩ = a 2 b 2 ( ⟨ i ˉ x A ∣ j ˉ x B ⟩ − ⟨ i ˉ x B ∣ j ˉ x A ⟩ ) . (120) 0=\langle\bar{i}|[x_A\otimes I,I\otimes x_B]|\bar{j}\rangle=a_2b_2\left(\langle\bar{i}_{x_A}|\bar{j}_{x_B}\rangle-\langle\bar{i}_{x_B}|\bar{j}_{x_A}\rangle\right). \tag*{(120)} 0 = ⟨ i ˉ ∣ [ x A ⊗ I , I ⊗ x B ] ∣ j ˉ ⟩ = a 2 b 2 ( ⟨ i ˉ x A ∣ j ˉ x B ⟩ − ⟨ i ˉ x B ∣ j ˉ x A ⟩ ) . (120) Define the component of C x B \mathcal{C}_{x_B} C x B C x A \mathcal{C}_{x_A} C x A ∣ i ˉ x B ′ ⟩ ∝ ( I − ∣ i ˉ ⟩ ⟨ i ˉ ∣ − Π x A ) x B ∣ i ˉ ⟩ |\bar{i}^{\prime}_{x_B}\rangle\propto(I-|\bar{i}\rangle\langle\bar{i}|-\Pi_{x_A})x_B|\bar{i}\rangle ∣ i ˉ x B ′ ⟩ ∝ ( I − ∣ i ˉ ⟩ ⟨ i ˉ ∣ − Π x A ) x B ∣ i ˉ ⟩ Figure 4 . One can then solve
x B ∣ i ˉ ⟩ = b 1 ∣ i ˉ ⟩ + b 2 ∣ i ˉ x B ⟩ = b 1 ∣ i ˉ ⟩ + c 1 ∣ i ˉ x A ⟩ + c 2 ∣ i ⊕ 1 ˉ x A ⟩ + c 3 ∣ i ˉ x B ′ ⟩ , (121) x_B|\bar{i}\rangle=b_1|\bar{i}\rangle+b_2|\bar{i}_{x_B}\rangle=b_1|\bar{i}\rangle+c_1|\bar{i}_{x_A}\rangle+c_2|\bar{i\oplus1}_{x_A}\rangle+c_3|\bar{i}^{\prime}_{x_B}\rangle, \tag*{(121)} x B ∣ i ˉ ⟩ = b 1 ∣ i ˉ ⟩ + b 2 ∣ i ˉ x B ⟩ = b 1 ∣ i ˉ ⟩ + c 1 ∣ i ˉ x A ⟩ + c 2 ∣ i ⊕ 1 ˉ x A ⟩ + c 3 ∣ i ˉ x B ′ ⟩ , (121) where c 3 = b 2 2 − c 1 2 − c 2 2 c_3=\sqrt{b_2^2-c_1^2-c_2^2} c 3 = b 2 2 − c 1 2 − c 2 2 c 1 , c 2 , c 3 ≥ 0 c_1,c_2,c_3\ge0 c 1 , c 2 , c 3 ≥ 0 ∣ 0 ˉ x B ′ ⟩ |\bar{0}^{\prime}_{x_B}\rangle ∣ 0 ˉ x B ′ ⟩ ∣ 1 ˉ x B ′ ⟩ |\bar{1}^{\prime}_{x_B}\rangle ∣ 1 ˉ x B ′ ⟩ ∣ 0 ˉ x B ⟩ |\bar{0}_{x_B}\rangle ∣ 0 ˉ x B ⟩ ∣ 1 ˉ x B ⟩ |\bar{1}_{x_B}\rangle ∣ 1 ˉ x B ⟩ ⟨ 0 ˉ x B ′ ∣ 1 ˉ x B ′ ⟩ = − z = − 2 c 1 c 2 / c 3 2 ≤ 0 \langle\bar{0}^{\prime}_{x_B}|\bar{1}^{\prime}_{x_B}\rangle=-z=-2c_1c_2/c_3^2\le0 ⟨ 0 ˉ x B ′ ∣ 1 ˉ x B ′ ⟩ = − z = − 2 c 1 c 2 / c 3 2 ≤ 0 ∣ i ˉ x B ′ ′ ⟩ |\bar{i}^{\prime\prime}_{x_B}\rangle ∣ i ˉ x B ′′ ⟩ Figure 4 :
∣ 0 ˉ x B ′ ⟩ = 1 + 1 − z 2 2 ∣ 0 ˉ x B ′ ′ ⟩ − 1 − 1 − z 2 2 ∣ 1 ˉ x B ′ ′ ⟩ , (122) |\bar{0}^{\prime}_{x_B}\rangle=\sqrt{\frac{1+\sqrt{1-z^2}}{2}}|\bar{0}^{\prime\prime}_{x_B}\rangle-\sqrt{\frac{1-\sqrt{1-z^2}}{2}}|\bar{1}^{\prime\prime}_{x_B}\rangle, \tag*{(122)} ∣ 0 ˉ x B ′ ⟩ = 2 1 + 1 − z 2 ∣ 0 ˉ x B ′′ ⟩ − 2 1 − 1 − z 2 ∣ 1 ˉ x B ′′ ⟩ , (122) ∣ 1 ˉ x B ′ ⟩ = − 1 − 1 − z 2 2 ∣ 0 ˉ x B ′ ′ ⟩ + 1 + 1 − z 2 2 ∣ 1 ˉ x B ′ ′ ⟩ . (123) |\bar{1}^{\prime}_{x_B}\rangle=-\sqrt{\frac{1-\sqrt{1-z^2}}{2}}|\bar{0}^{\prime\prime}_{x_B}\rangle+\sqrt{\frac{1+\sqrt{1-z^2}}{2}}|\bar{1}^{\prime\prime}_{x_B}\rangle. \tag*{(123)} ∣ 1 ˉ x B ′ ⟩ = − 2 1 − 1 − z 2 ∣ 0 ˉ x B ′′ ⟩ + 2 1 + 1 − z 2 ∣ 1 ˉ x B ′′ ⟩ . (123) Both C x A = { ∣ i ˉ x A ⟩ } \mathcal{C}_{x_A}=\{|\bar{i}_{x_A}\rangle\} C x A = { ∣ i ˉ x A ⟩} C x B = { ∣ i ˉ x B ⟩ } \mathcal{C}_{x_B}=\{|\bar{i}_{x_B}\rangle\} C x B = { ∣ i ˉ x B ⟩} x A x_A x A x B x_B x B ∣ i ˉ x A ⟩ |\bar{i}_{x_A}\rangle ∣ i ˉ x A ⟩ ∣ i ˉ x B ⟩ |\bar{i}_{x_B}\rangle ∣ i ˉ x B ⟩ ∣ i ˉ ⟩ |\bar{i}\rangle ∣ i ˉ ⟩ σ x \sigma^x σ x (119) prevents both errors from being corrected unambiguously within a single recovery procedure. Concretely: (i) Π x A x B ∣ 0 ˉ ⟩ = c 1 ∣ 0 ˉ x A ⟩ + c 2 ∣ 1 ˉ x A ⟩ \Pi_{x_A}x_B|\bar{0}\rangle=c_1|\bar{0}_{x_A}\rangle+c_2|\bar{1}_{x_A}\rangle Π x A x B ∣ 0 ˉ ⟩ = c 1 ∣ 0 ˉ x A ⟩ + c 2 ∣ 1 ˉ x A ⟩ Π x A x B ∣ 1 ˉ ⟩ = c 1 ∣ 1 ˉ x A ⟩ + c 2 ∣ 0 ˉ x A ⟩ \Pi_{x_A}x_B|\bar{1}\rangle=c_1|\bar{1}_{x_A}\rangle+c_2|\bar{0}_{x_A}\rangle Π x A x B ∣ 1 ˉ ⟩ = c 1 ∣ 1 ˉ x A ⟩ + c 2 ∣ 0 ˉ x A ⟩ x B x_B x B ∣ i ˉ x A ⟩ ↦ ∣ i ˉ ⟩ |\bar{i}_{x_A}\rangle\mapsto|\bar{i}\rangle ∣ i ˉ x A ⟩ ↦ ∣ i ˉ ⟩ c 2 c_2 c 2 σ x \sigma^x σ x ∣ 0 ˉ x A ⟩ |\bar{0}_{x_A}\rangle ∣ 0 ˉ x A ⟩ x A ∣ 0 ˉ ⟩ x_A|\bar{0}\rangle x A ∣ 0 ˉ ⟩ x B ∣ 1 ˉ ⟩ x_B|\bar{1}\rangle x B ∣ 1 ˉ ⟩ ∣ 0 ˉ x B ′ ⟩ |\bar{0}^{\prime}_{x_B}\rangle ∣ 0 ˉ x B ′ ⟩ ∣ 1 ˉ x B ′ ⟩ |\bar{1}^{\prime}_{x_B}\rangle ∣ 1 ˉ x B ′ ⟩ x B x_B x B x A x_A x A ⟨ i ˉ ∣ x A ⊗ x B ∣ j ˉ ⟩ ∝ δ i j \langle\bar{i}|x_A\otimes x_B|\bar{j}\rangle\propto\delta_{ij} ⟨ i ˉ ∣ x A ⊗ x B ∣ j ˉ ⟩ ∝ δ ij x A ⊗ x B x_A\otimes x_B x A ⊗ x B σ x \sigma^x σ x
Given the orthogonality structure above, we introduce the recovery map
R ( ρ ) = ∑ i = 0 , 1 , 2 R i ρ R i † , where R 0 = Π , R 1 = ∑ i = 0 , 1 ∣ i ˉ ⟩ ⟨ i ˉ x A ∣ , R 2 = ∑ i = 0 , 1 ∣ i ˉ ⟩ ⟨ i ˉ x B ′ ′ ∣ . (124) R(\rho)=\sum_{i=0,1,2}R_i\rho R_i^\dagger,\quad\text{where }R_0=\Pi,\ R_1=\sum_{i=0,1}|\bar{i}\rangle\langle\bar{i}_{x_A}|,\ R_2=\sum_{i=0,1}|\bar{i}\rangle\langle\bar{i}^{\prime\prime}_{x_B}|. \tag*{(124)} R ( ρ ) = i = 0 , 1 , 2 ∑ R i ρ R i † , where R 0 = Π , R 1 = i = 0 , 1 ∑ ∣ i ˉ ⟩ ⟨ i ˉ x A ∣ , R 2 = i = 0 , 1 ∑ ∣ i ˉ ⟩ ⟨ i ˉ x B ′′ ∣. (124) Note that this recovery map is actually not trace-preserving, also only for pedagogical purpose. In principle, one should take the recovery map supplemented with additional Kraus operators, such that it is trace-preserving. But Tr R ( L ( ρ ) ) = 0 \operatorname{Tr}R(\mathcal{L}(\rho))=0 Tr R ( L ( ρ )) = 0 R R R
R ( L x A , x B ( ρ ) ) = − i λ e f f [ σ x , ρ ] + Γ e f f ( σ x ρ σ x − ρ ) , (125) R(\mathcal{L}_{x_A,x_B}(\rho))=-i\lambda_{\mathrm{eff}}[\sigma^{x},\rho]+\Gamma_{\mathrm{eff}}(\sigma^{x}\rho\sigma^{x}-\rho), \tag*{(125)} R ( L x A , x B ( ρ )) = − i λ eff [ σ x , ρ ] + Γ eff ( σ x ρ σ x − ρ ) , (125) where λ e f f = λ a 2 c 2 \lambda_{\mathrm{eff}}=\lambda a_2c_2 λ eff = λ a 2 c 2 Γ e f f = Γ ( c 2 2 + c 3 2 ( 1 − 1 − z 2 ) / 2 ) \Gamma_{\mathrm{eff}}=\Gamma(c_2^2+c_3^2(1-\sqrt{1-z^2})/2) Γ eff = Γ ( c 2 2 + c 3 2 ( 1 − 1 − z 2 ) /2 ) σ x \sigma^x σ x
The resulting channel can be written explicitly as
e R ∘ L T ( ρ ) = 1 + e − 2 Γ e f f T cos ( 2 λ e f f T ) 2 ρ + 1 − e − 2 Γ e f f T cos ( 2 λ e f f T ) 2 σ x ρ σ x (126) e^{R\circ\mathcal{L}T}(\rho)=\frac{1+e^{-2\Gamma_{\mathrm{eff}}T}\cos(2\lambda_{\mathrm{eff}}T)}{2}\rho+\frac{1-e^{-2\Gamma_{\mathrm{eff}}T}\cos(2\lambda_{\mathrm{eff}}T)}{2}\sigma^x\rho\sigma^x \tag*{(126)} e R ∘ L T ( ρ ) = 2 1 + e − 2 Γ eff T cos ( 2 λ eff T ) ρ + 2 1 − e − 2 Γ eff T cos ( 2 λ eff T ) σ x ρ σ x (126) − i 2 e − 2 Γ e f f T sin ( 2 λ e f f T ) σ x ρ + i 2 e − 2 Γ e f f T sin ( 2 λ e f f T ) ρ σ x (127) -\frac{i}{2}e^{-2\Gamma_{\mathrm{eff}}T}\sin(2\lambda_{\mathrm{eff}}T)\sigma^x\rho+\frac{i}{2}e^{-2\Gamma_{\mathrm{eff}}T}\sin(2\lambda_{\mathrm{eff}}T)\rho\sigma^x \tag*{(127)} − 2 i e − 2 Γ eff T sin ( 2 λ eff T ) σ x ρ + 2 i e − 2 Γ eff T sin ( 2 λ eff T ) ρ σ x (127) Therefore, λ e f f \lambda_{\mathrm{eff}} λ eff Γ e f f \Gamma_{\mathrm{eff}} Γ eff
Γ e f f λ e f f = Γ λ c 2 2 + c 3 2 1 − 1 − z 2 2 a 2 c 2 = Γ λ c 2 + 2 c 1 c 2 z 1 − 1 − z 2 2 a 2 c 2 = Γ λ c 2 + 1 z − c 2 2 + c 2 2 ( 1 − z 2 ) + b 2 2 z 2 2 ( 1 − 1 − z 2 ) a 2 = Γ λ c 2 b 2 + − c 2 2 b 2 2 + c 2 2 b 2 2 ( 1 − z 2 ) + z 2 z 2 ( 1 − 1 − z 2 ) a 2 b 2 (128) \begin{aligned}
\frac{\Gamma_{\mathrm{eff}}}{\lambda_{\mathrm{eff}}}
=\frac{\Gamma}{\lambda}\frac{c_2^2+c_3^2\frac{1-\sqrt{1-z^2}}{2}}{a_2c_2} \\
&=\frac{\Gamma}{\lambda}\frac{c_2+\frac{2c_1c_2}{z}\frac{1-\sqrt{1-z^2}}{2}}{a_2c_2} \\
&=\frac{\Gamma}{\lambda}\frac{c_2+\frac{1}{z}\frac{-c_2^2+\sqrt{c_2^2(1-z^2)+b_2^2z^2}}{2}(1-\sqrt{1-z^2})}{a_2} \\
&=\frac{\Gamma}{\lambda}\frac{\frac{c_2}{b_2}+\frac{-\frac{c_2^2}{b_2^2}+\sqrt{\frac{c_2^2}{b_2^2}(1-z^2)+z^2}}{z^2}(1-\sqrt{1-z^2})}{\frac{a_2}{b_2}} \tag*{(128)}
\end{aligned} λ eff Γ eff = λ Γ a 2 c 2 c 2 2 + c 3 2 2 1 − 1 − z 2 = λ Γ a 2 c 2 c 2 + z 2 c 1 c 2 2 1 − 1 − z 2 = λ Γ a 2 c 2 + z 1 2 − c 2 2 + c 2 2 ( 1 − z 2 ) + b 2 2 z 2 ( 1 − 1 − z 2 ) = λ Γ b 2 a 2 b 2 c 2 + z 2 − b 2 2 c 2 2 + b 2 2 c 2 2 ( 1 − z 2 ) + z 2 ( 1 − 1 − z 2 ) (128) where we used z = 2 c 1 c 2 / c 3 2 z=2c_1c_2/c_3^2 z = 2 c 1 c 2 / c 3 2 c 1 2 + c 2 2 + c 3 2 = b 2 2 c_1^2+c_2^2+c_3^2=b_2^2 c 1 2 + c 2 2 + c 3 2 = b 2 2 c 1 c_1 c 1 0 ≤ z ≤ 1 0\le z\le1 0 ≤ z ≤ 1 0 ≤ c 2 / b 2 ≤ 1 0\le c_2/b_2\le1 0 ≤ c 2 / b 2 ≤ 1 z = c 2 / b 2 = 0 z=c_2/b_2=0 z = c 2 / b 2 = 0 c 2 = c 3 = 0 c_2=c_3=0 c 2 = c 3 = 0 c 1 = b 2 c_1=b_2 c 1 = b 2 x B x_B x B x A x_A x A
0 ≤ Γ e f f λ e f f ≤ Γ b 2 λ a 2 = Γ Δ x B λ Δ x A . (129) 0\le\frac{\Gamma_{\mathrm{eff}}}{\lambda_{\mathrm{eff}}}\le\frac{\Gamma b_2}{\lambda a_2}=\frac{\Gamma\Delta x_B}{\lambda\Delta x_A}. \tag*{(129)} 0 ≤ λ eff Γ eff ≤ λ a 2 Γ b 2 = λ Δ x A ΓΔ x B . (129) Note also that when Γ e f f / λ e f f \Gamma_{\mathrm{eff}}/\lambda_{\mathrm{eff}} Γ eff / λ eff λ e f f ∝ c 2 \lambda_{\mathrm{eff}}\propto c_2 λ eff ∝ c 2 3 λ e f f ∝ c 2 \lambda_{\mathrm{eff}}\propto c_2 λ eff ∝ c 2 Γ e f f ∝ c 2 2 \Gamma_{\mathrm{eff}}\propto c_2^2 Γ eff ∝ c 2 2 In conclusion, for a nonlocal code space (potentially equipped with nonlocal recovery), the ratio of error rate to interaction strength can be made arbitrarily small, at the price of simultaneously weakening the interaction. In other words, the signal-to-noise ratio can be magnified arbitrarily with this approach.
A natural question is whether such a code can be realized in a local fashion. This question has two aspects: whether code-state initialization can be implemented locally, and whether the recovery operation can be implemented as a separable channel.
First, suppose the code basis states are product states, i.e. ∣ 0 ˉ ⟩ = ∣ α 0 ⟩ A ⊗ ∣ β 0 ⟩ B |\bar{0}\rangle=|\alpha_0\rangle_A\otimes|\beta_0\rangle_B ∣ 0 ˉ ⟩ = ∣ α 0 ⟩ A ⊗ ∣ β 0 ⟩ B ∣ 1 ˉ ⟩ = ∣ α 1 ⟩ A ⊗ ∣ β 1 ⟩ B |\bar{1}\rangle=|\alpha_1\rangle_A\otimes|\beta_1\rangle_B ∣ 1 ˉ ⟩ = ∣ α 1 ⟩ A ⊗ ∣ β 1 ⟩ B ∣ α 0 ⟩ A = ∣ α 1 ⟩ A |\alpha_0\rangle_A=|\alpha_1\rangle_A ∣ α 0 ⟩ A = ∣ α 1 ⟩ A ∣ β 0 ⟩ B = ∣ β 1 ⟩ B |\beta_0\rangle_B=|\beta_1\rangle_B ∣ β 0 ⟩ B = ∣ β 1 ⟩ B H A \mathcal{H}_A H A ∣ β 0 ⟩ B |\beta_0\rangle_B ∣ β 0 ⟩ B H B \mathcal{H}_B H B (118) already imply ⟨ i ˉ ∣ x A ⊗ x B ∣ j ˉ ⟩ ∝ δ i j \langle\bar{i}|x_A\otimes x_B|\bar{j}\rangle\propto\delta_{ij} ⟨ i ˉ ∣ x A ⊗ x B ∣ j ˉ ⟩ ∝ δ ij (118) and Eq. (119) to hold while retaining non-trivial dynamics, the code subspace C \mathcal{C} C ∣ 0 ˉ ⟩ |\bar{0}\rangle ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ x A ⊗ x B ∣ 0 ˉ ⟩ = ⟨ 0 ˉ ∣ x A ⊗ I ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ I ⊗ x B ∣ 0 ˉ ⟩ \langle\bar{0}|x_A\otimes x_B|\bar{0}\rangle=\langle\bar{0}|x_A\otimes I|\bar{0}\rangle\langle\bar{0}|I\otimes x_B|\bar{0}\rangle ⟨ 0 ˉ ∣ x A ⊗ x B ∣ 0 ˉ ⟩ = ⟨ 0 ˉ ∣ x A ⊗ I ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ I ⊗ x B ∣ 0 ˉ ⟩ c 1 = 0 c_1=0 c 1 = 0 ⟨ x A ⊗ x B ⟩ − ⟨ x A ⟩ ⟨ x B ⟩ = 0 \langle x_A\otimes x_B\rangle-\langle x_A\rangle\langle x_B\rangle=0 ⟨ x A ⊗ x B ⟩ − ⟨ x A ⟩ ⟨ x B ⟩ = 0 Γ e f f / λ e f f = Γ c 2 / ( λ a 2 ) \Gamma_{\mathrm{eff}}/\lambda_{\mathrm{eff}}=\Gamma c_2/(\lambda a_2) Γ eff / λ eff = Γ c 2 / ( λ a 2 )
Second, we consider whether the recovery operation can be implemented by a separable channel. As discussed above, in the limit c 2 → 0 c_2 \to0 c 2 → 0 σ x \sigma^x σ x C \mathcal{C} C Γ > λ \Gamma> \lambda Γ > λ R \mathcal{R} R
In summary, given the operator matrix elements in Eq. (118) and Eq. (119) , and since C \mathcal{C} C C \mathcal{C} C R \mathcal{R} R C \mathcal{C} C
Concatenated QEC for suppressing logical σ z \sigma_z σ z
The local code discussed above generally produces both logical σ x \sigma_x σ x σ z \sigma_z σ z σ z \sigma_z σ z σ z \sigma_z σ z σ x \sigma_x σ x
As discussed above, for the Fock-state code, σ z \sigma^z σ z σ x \sigma^x σ x A A A B B B A A A B B B σ z \sigma^z σ z σ x \sigma^x σ x A A A B B B σ x \sigma^x σ x
Consider a bipartition of 2 N 2N 2 N A A A B B B i i i { ∣ 0 ˉ ⟩ A , i , ∣ 1 ˉ ⟩ A , i } \{|\bar{0}\rangle_{A,i},|\bar{1}\rangle_{A,i}\} { ∣ 0 ˉ ⟩ A , i , ∣ 1 ˉ ⟩ A , i } { ∣ 0 ˉ ⟩ B , i , ∣ 1 ˉ ⟩ B , i } \{|\bar{0}\rangle_{B,i},|\bar{1}\rangle_{B,i}\} { ∣ 0 ˉ ⟩ B , i , ∣ 1 ˉ ⟩ B , i } σ x = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ \sigma^x=|\bar{1}\rangle\langle\bar{0}|+|\bar{0}\rangle\langle\bar{1}| σ x = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ \sigma^z=|\bar{0}\rangle\langle\bar{0}|-|\bar{1}\rangle\langle\bar{1}| σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ σ A , i z ⊗ σ B , j z \sigma^z_{A,i}\otimes\sigma^z_{B,j} σ A , i z ⊗ σ B , j z σ A , i x ⊗ σ B , j x \sigma^x_{A,i}\otimes\sigma^x_{B,j} σ A , i x ⊗ σ B , j x σ z \sigma^z σ z σ x \sigma^x σ x
L ( ρ ) = − i ∑ i , j λ x [ σ A , i x ⊗ σ B , j x , ρ ] − i ∑ i , j λ z [ σ A , i z ⊗ σ B , j z , ρ ] + Γ x ∑ i ( D [ σ A , i x ] + D [ σ B , i x ] ) ( ρ ) + Γ z ∑ i ( D [ σ A , i z ] + D [ σ B , i z ] ) ( ρ ) . (130) \mathcal{L}(\rho)=-i\sum_{i,j}\lambda_x[\sigma^x_{A,i}\otimes\sigma^x_{B,j},\rho]-i\sum_{i,j}\lambda_z[\sigma^z_{A,i}\otimes\sigma^z_{B,j},\rho]
+\Gamma_x\sum_i\left(\mathcal{D}[\sigma^x_{A,i}]+\mathcal{D}[\sigma^x_{B,i}]\right)(\rho)+\Gamma_z\sum_i\left(\mathcal{D}[\sigma^z_{A,i}]+\mathcal{D}[\sigma^z_{B,i}]\right)(\rho). \tag*{(130)} L ( ρ ) = − i i , j ∑ λ x [ σ A , i x ⊗ σ B , j x , ρ ] − i i , j ∑ λ z [ σ A , i z ⊗ σ B , j z , ρ ] + Γ x i ∑ ( D [ σ A , i x ] + D [ σ B , i x ] ) ( ρ ) + Γ z i ∑ ( D [ σ A , i z ] + D [ σ B , i z ] ) ( ρ ) . (130) We focus on the regime in which the σ z \sigma^z σ z σ z \sigma^z σ z Γ z ∼ λ z \Gamma_z\sim\lambda_z Γ z ∼ λ z σ x \sigma^x σ x Γ z , λ z ≫ Γ x , λ x \Gamma_z,\lambda_z\gg\Gamma_x,\lambda_x Γ z , λ z ≫ Γ x , λ x σ x \sigma^x σ x σ x \sigma^x σ x λ x ≫ Γ x \lambda_x\gg\Gamma_x λ x ≫ Γ x σ z \sigma^z σ z σ z \sigma^z σ z σ x \sigma^x σ x
In principle, σ z \sigma^z σ z σ x \sigma^x σ x Tr σ z ( σ x ) † = 0 \operatorname{Tr}\sigma^z(\sigma^x)^\dagger=0 Tr σ z ( σ x ) † = 0 σ x \sigma^x σ x σ z \sigma^z σ z ∣ ± ⟩ = ( ∣ 0 ˉ ⟩ ± ∣ 1 ˉ ⟩ ) / 2 |\pm\rangle=(|\bar{0}\rangle\pm|\bar{1}\rangle)/\sqrt{2} ∣ ± ⟩ = ( ∣ 0 ˉ ⟩ ± ∣ 1 ˉ ⟩) / 2 A A A
∣ 0 ˉ ⟩ A = 1 2 ( ⨂ i = 1 N ∣ + ⟩ A , i + ⨂ i = 1 N ∣ − ⟩ A , i ) , ∣ 1 ˉ ⟩ A = 1 2 ( ⨂ i = 1 N ∣ + ⟩ A , i − ⨂ i = 1 N ∣ − ⟩ A , i ) . (131) |\bar{0}\rangle_A=\frac{1}{\sqrt{2}}\left(\bigotimes_{i=1}^{N}|+\rangle_{A,i}+\bigotimes_{i=1}^{N}|-\rangle_{A,i}\right),\qquad
|\bar{1}\rangle_A=\frac{1}{\sqrt{2}}\left(\bigotimes_{i=1}^{N}|+\rangle_{A,i}-\bigotimes_{i=1}^{N}|-\rangle_{A,i}\right). \tag*{(131)} ∣ 0 ˉ ⟩ A = 2 1 ( i = 1 ⨂ N ∣ + ⟩ A , i + i = 1 ⨂ N ∣ − ⟩ A , i ) , ∣ 1 ˉ ⟩ A = 2 1 ( i = 1 ⨂ N ∣ + ⟩ A , i − i = 1 ⨂ N ∣ − ⟩ A , i ) . (131) That is, we encode one logical qubit into the N N N N N N A A A Π A = ∣ 0 ˉ ⟩ A A ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ A A ⟨ 1 ˉ ∣ \Pi_A=|\bar{0}\rangle_{AA}\langle\bar{0}|+|\bar{1}\rangle_{AA}\langle\bar{1}| Π A = ∣ 0 ˉ ⟩ AA ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ AA ⟨ 1 ˉ ∣
R A ( ρ ) = Π A ρ Π A + ∑ i = 1 N Π A σ A , i z ρ σ A , i z Π A . (132) \mathcal{R}_A(\rho)=\Pi_A\rho\Pi_A+\sum_{i=1}^{N}\Pi_A\sigma^z_{A,i}\rho\sigma^z_{A,i}\Pi_A. \tag*{(132)} R A ( ρ ) = Π A ρ Π A + i = 1 ∑ N Π A σ A , i z ρ σ A , i z Π A . (132) For the N N N B B B
Suppose the state is initially supported on the code space, i.e. ρ = Π A ⊗ Π B ρ Π A ⊗ Π B \rho= \Pi_A \otimes\Pi_B \rho\Pi_A \otimes\Pi_B ρ = Π A ⊗ Π B ρ Π A ⊗ Π B
R A ⊗ R B ∘ L ( ρ ) = − i ∑ j N λ x R B [ X A ⊗ σ B , j x , ρ ] + Γ x R B ( N D [ X A ] + ∑ i D [ σ B , i x ] ) ( ρ ) + Γ z R B ( − N ρ + ∑ i D [ σ B , i z ] ( ρ ) + ∑ i ∑ j = 1 N δ i j ρ ) = − i N 2 λ x [ X A ⊗ X B , ρ ] + N Γ x ( D [ X A ] + D [ X B ] ) ( ρ ) . (133) \begin{aligned}
R_A \otimes R_B \circ\mathcal{L}(\rho)
&= -i \sum_j N\lambda_x R_B \left[X_A \otimes\sigma^x_{B,j},\rho\right] + \Gamma_x R_B \left(N\mathcal{D}[X_A] + \sum_i \mathcal{D}[\sigma^x_{B,i}]\right)(\rho) \\
&\quad+ \Gamma_z R_B \left(-N\rho+ \sum_i \mathcal{D}[\sigma^z_{B,i}](\rho) + \sum_i \sum_{j=1}^{N} \delta_{ij}\rho\right) \\
&= -iN^2\lambda_x[X_A \otimes X_B,\rho] + N\Gamma_x(\mathcal{D}[X_A] + \mathcal{D}[X_B])(\rho). \tag*{(133)}
\end{aligned} R A ⊗ R B ∘ L ( ρ ) = − i j ∑ N λ x R B [ X A ⊗ σ B , j x , ρ ] + Γ x R B ( N D [ X A ] + i ∑ D [ σ B , i x ] ) ( ρ ) + Γ z R B ( − N ρ + i ∑ D [ σ B , i z ] ( ρ ) + i ∑ j = 1 ∑ N δ ij ρ ) = − i N 2 λ x [ X A ⊗ X B , ρ ] + N Γ x ( D [ X A ] + D [ X B ]) ( ρ ) . (133) Therefore, with this second layer of error correction, it is sufficient to retain only the σ A , i x ⊗ σ B , j x \sigma^x_{A,i} \otimes\sigma^x_{B,j} σ A , i x ⊗ σ B , j x σ x \sigma^x σ x A A A σ A , i x ⊗ σ A , j x \sigma^x_{A,i} \otimes\sigma^x_{A,j} σ A , i x ⊗ σ A , j x σ A , i x ⊗ σ A , j x ∣ i ˉ ⟩ A = ∣ i ˉ ⟩ \sigma^x_{A,i} \otimes\sigma^x_{A,j}|\bar{i}\rangle_A = |\bar{i}\rangle σ A , i x ⊗ σ A , j x ∣ i ˉ ⟩ A = ∣ i ˉ ⟩ σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ = ( 1 − σ z ) σ x / 2 \sigma^+ = |\bar{1}\rangle\langle\bar{0}| = (1-\sigma^z)\sigma^x/2 σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ = ( 1 − σ z ) σ x /2 X X X
R A ( σ A , i + ρ ( σ A , i + ) † − 1 2 { ∣ 0 ˉ ⟩ A , i ⟨ 0 ˉ ∣ , ρ } ) = R A ( ( 1 − σ A , i z ) σ A , i x 2 ρ σ A , i x ( 1 − σ A , i z ) 2 − 1 2 { 1 + σ A , i z 2 , ρ } ) = 1 2 ( X A ρ X A − ρ ) . (134) \begin{aligned}
R_A\left(\sigma^+_{A,i}\rho(\sigma^+_{A,i})^\dagger-\frac{1}{2}\{|\bar{0}\rangle_{A,i}\langle\bar{0}|,\rho\}\right) \\
&=R_A\left(\frac{(1-\sigma^z_{A,i})\sigma^x_{A,i}}{2}\rho\frac{\sigma^x_{A,i}(1-\sigma^z_{A,i})}{2}-\frac{1}{2}\left\{\frac{1+\sigma^z_{A,i}}{2},\rho\right\}\right) \\
&=\frac{1}{2}(X_A\rho X_A-\rho). \tag*{(134)}
\end{aligned} R A ( σ A , i + ρ ( σ A , i + ) † − 2 1 { ∣ 0 ˉ ⟩ A , i ⟨ 0 ˉ ∣ , ρ } ) = R A ( 2 ( 1 − σ A , i z ) σ A , i x ρ 2 σ A , i x ( 1 − σ A , i z ) − 2 1 { 2 1 + σ A , i z , ρ } ) = 2 1 ( X A ρ X A − ρ ) . (134) We now study the separability classification of L \mathcal{L} L
L ( ρ ) = ∑ i , j − i [ λ x σ A , i x ⊗ σ B , j x + λ z σ A , i z ⊗ σ B , j z , ρ ] + Γ x N ( D [ σ A , i x ] + D [ σ B , j x ] ) ( ρ ) + Γ z N ( D [ σ A , i z ] + D [ σ B , j z ] ) ( ρ ) . (135) \mathcal{L}(\rho)=\sum_{i,j}-i[\lambda_x\sigma^x_{A,i}\otimes\sigma^x_{B,j}+\lambda_z\sigma^z_{A,i}\otimes\sigma^z_{B,j},\rho]+\frac{\Gamma_x}{N}\left(\mathcal{D}[\sigma^x_{A,i}]+\mathcal{D}[\sigma^x_{B,j}]\right)(\rho)+\frac{\Gamma_z}{N}\left(\mathcal{D}[\sigma^z_{A,i}]+\mathcal{D}[\sigma^z_{B,j}]\right)(\rho). \tag*{(135)} L ( ρ ) = i , j ∑ − i [ λ x σ A , i x ⊗ σ B , j x + λ z σ A , i z ⊗ σ B , j z , ρ ] + N Γ x ( D [ σ A , i x ] + D [ σ B , j x ] ) ( ρ ) + N Γ z ( D [ σ A , i z ] + D [ σ B , j z ] ) ( ρ ) . (135) By Lemma 3 , the σ x \sigma^x σ x λ x > Γ x / N \lambda_x > \Gamma_x/N λ x > Γ x / N (133) is non-separable if and only if λ x N 2 > Γ x N \lambda_xN^2 > \Gamma_xN λ x N 2 > Γ x N σ x \sigma^x σ x σ x \sigma^x σ x
Applying this protocol to the decoherence channels in App. C 3, for instance CSL decoherence channel in Eq. (164) , we obtain a clean σ x \sigma^x σ x
R A ⊗ R B ∘ L e f f , C S L ( ρ ) = N ( Γ e f f x + Γ e f f + + Γ e f f − 2 ) ( D [ X A ] + D [ X B ] ) ( ρ ) + ∑ i j Γ e f f , i j x x ( D [ X A − X B ] − D [ X A ] − D [ X B ] ) ( ρ ) , (136) R_A \otimes R_B \circ\mathcal{L}_{\mathrm{eff,CSL}}(\rho)=N\left(\Gamma^x_{\mathrm{eff}}+\frac{\Gamma^+_{\mathrm{eff}}+\Gamma^-_{\mathrm{eff}}}{2}\right)(\mathcal{D}[X_A]+\mathcal{D}[X_B])(\rho)+\sum_{ij}\Gamma^{xx}_{\mathrm{eff},ij}\left(\mathcal{D}[X_A-X_B]-\mathcal{D}[X_A]-\mathcal{D}[X_B]\right)(\rho), \tag*{(136)} R A ⊗ R B ∘ L eff , CSL ( ρ ) = N ( Γ eff x + 2 Γ eff + + Γ eff − ) ( D [ X A ] + D [ X B ]) ( ρ ) + ij ∑ Γ eff , ij xx ( D [ X A − X B ] − D [ X A ] − D [ X B ] ) ( ρ ) , (136) with the price of increasing logical error rates, since
Γ e f f x + Γ e f f + + Γ e f f − 2 ≥ Γ e f f x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − . \Gamma^x_{\mathrm{eff}}+\frac{\Gamma^+_{\mathrm{eff}}+\Gamma^-_{\mathrm{eff}}}{2}\geq\Gamma^x_{\mathrm{eff}}+\frac{\Gamma^+_{\mathrm{eff}}\Gamma^-_{\mathrm{eff}}}{\Gamma^+_{\mathrm{eff}}+\Gamma^-_{\mathrm{eff}}}. Γ eff x + 2 Γ eff + + Γ eff − ≥ Γ eff x + Γ eff + + Γ eff − Γ eff + Γ eff − . This protocol can be a cleaner alternative for separability classification in App. B 5 a. Moreover, the separability classification in Theorem 1 of this effective dynamics with concatenated QEC with η > 1 \eta> 1 η > 1 η ≤ 1 \eta\leq1 η ≤ 1
Application of the QEC framework to collapse models
This appendix provides technical and supplementary details on the application of the quantum error-correction framework developed in App. B 3 to each of the collapse models considered in the main text. We begin by deriving the logical gravitational interaction generated within the encoded subspace and discuss the validity of the approximations used in its evaluation. We then construct model-specific recovery procedures, derive the corresponding effective logical dynamics, and analyze their separability properties. Since the different models can largely be treated using the same methodology, we collect the corresponding derivations here rather than repeating similar calculations in the main text. For completeness, we also discuss a small extension of the separability analysis that proves useful for analyzing one of the effective dynamics encountered below. Then we compare the resulting classifications across the different collapse models. Finally, we analytically prove the entanglement recoverability with respect to GRW/CSL model in Thm. 5 .
Effective gravitational interaction
Before analyzing the various collapse models, we determine the logical interaction generated by the Newtonian potential within the encoded subspace introduced in App. B 3. This interaction constitutes the coherent component of the effective dynamics and serves as a common reference point for all recovery procedures considered below. As discussed in App. B 3, the QEC recovery channel has the form
R ( ρ A B ) = R A ⊗ R B ( ρ A B ) , R α ( ρ α ) = ∑ i R α i ρ α R α i † . (137) R(\rho_{AB}) = R_A \otimes R_B(\rho_{AB}), \quad R_\alpha(\rho_\alpha) = \sum_i R_\alpha^i \rho_\alpha R_\alpha^{i\dagger}. \tag*{(137)} R ( ρ A B ) = R A ⊗ R B ( ρ A B ) , R α ( ρ α ) = i ∑ R α i ρ α R α i † . (137) where R α 0 = Π α R_\alpha^0 = \Pi_\alpha R α 0 = Π α C α \mathcal{C}_\alpha C α ρ = Π A Π B ρ Π A Π B \rho= \Pi^A\Pi^B\rho\Pi^A\Pi^B ρ = Π A Π B ρ Π A Π B
R ∘ L N G ( ρ ) = i ℏ [ Π A Π B G m A m B ∣ d + x A − x B ∣ Π A Π B , ρ A B ] . (138) R \circ\mathcal{L}_{\mathrm{NG}}(\rho) = \frac{i}{\hbar}\left[\Pi^A\Pi^B \frac{Gm_A m_B}{\lvert d+x_A-x_B\rvert}\Pi^A\Pi^B,\rho_{AB}\right]. \tag*{(138)} R ∘ L NG ( ρ ) = ℏ i [ Π A Π B ∣ d + x A − x B ∣ G m A m B Π A Π B , ρ A B ] . (138) Therefore, this part of the master equation is determined by the matrix elements of the Newtonian potential. The Newtonian potential
− G m A m B ∣ d + x A − x B ∣ -\frac{Gm_A m_B}{\lvert d+x_A-x_B\rvert} − ∣ d + x A − x B ∣ G m A m B is singular when the positions of the two masses coincide, i.e. when ∣ d + x A − x B ∣ = 0 \lvert d+x_A-x_B\rvert=0 ∣ d + x A − x B ∣ = 0
f N G ( ∣ d + x A − x B ∣ ) = { − G m A m B ∣ d + x A − x B ∣ , ∣ d + x A − x B ∣ > 2 r 0 , 0 , ∣ d + x A − x B ∣ ≤ 2 r 0 . (139) f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)=
\begin{cases}
-\dfrac{Gm_A m_B}{\lvert d+x_A-x_B\rvert}, & \lvert d+x_A-x_B\rvert>2r_0,\\
0, & \lvert d+x_A-x_B\rvert\leq2r_0.
\end{cases} \tag*{(139)} f NG (∣ d + x A − x B ∣) = ⎩ ⎨ ⎧ − ∣ d + x A − x B ∣ G m A m B , 0 , ∣ d + x A − x B ∣ > 2 r 0 , ∣ d + x A − x B ∣ ≤ 2 r 0 . (139) We then compute the nonlocal matrix elements of this regularized potential. By symmetry of the potential and of the wavefunctions under exchange of the two masses, the following four off-diagonal matrix elements are equal:
ℏ λ e f f x x = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B = A ⟨ 1 ˉ ∣ B ⟨ 0 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B = A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B . (140) \hbar\lambda_{\mathrm{eff}}^{xx}
= {}_A\langle\bar{0}|{}_B\langle\bar{0}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{1}\rangle_A|\bar{1}\rangle_B
= {}_A\langle\bar{0}|{}_B\langle\bar{1}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{1}\rangle_A|\bar{0}\rangle_B
= {}_A\langle\bar{1}|{}_B\langle\bar{0}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{0}\rangle_A|\bar{1}\rangle_B
= {}_A\langle\bar{1}|{}_B\langle\bar{1}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{0}\rangle_A|\bar{0}\rangle_B. \tag*{(140)} ℏ λ eff xx = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B = A ⟨ 1 ˉ ∣ B ⟨ 0 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B = A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B . (140) where the superscript indicates that this matrix element contributes to the effective σ A x σ B x \sigma_A^x\sigma_B^x σ A x σ B x σ A x σ B z \sigma_A^x\sigma_B^z σ A x σ B z σ A z σ B x \sigma_A^z\sigma_B^x σ A z σ B x σ A z σ B z \sigma_A^z\sigma_B^z σ A z σ B z
2 ℏ λ e f f x z = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = − 2 ℏ λ e f f z x 2\hbar\lambda_{\mathrm{eff}}^{xz}
= {}_A\langle\bar{0}|{}_B\langle\bar{0}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{1}\rangle_A|\bar{0}\rangle_B
- {}_A\langle\bar{0}|{}_B\langle\bar{1}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{1}\rangle_A|\bar{1}\rangle_B
= -2\hbar\lambda_{\mathrm{eff}}^{zx} 2ℏ λ eff x z = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = − 2ℏ λ eff z x 4 ℏ λ e f f z z = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B − A ⟨ 1 ˉ ∣ B ⟨ 0 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B + A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B (141) 4\hbar\lambda_{\mathrm{eff}}^{zz}
= {}_A\langle\bar{0}|{}_B\langle\bar{0}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{0}\rangle_A|\bar{0}\rangle_B
- {}_A\langle\bar{0}|{}_B\langle\bar{1}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{0}\rangle_A|\bar{1}\rangle_B
- {}_A\langle\bar{1}|{}_B\langle\bar{0}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{1}\rangle_A|\bar{0}\rangle_B
+ {}_A\langle\bar{1}|{}_B\langle\bar{1}|f_{\mathrm{NG}}(\lvert d+x_A-x_B\rvert)|\bar{1}\rangle_A|\bar{1}\rangle_B \tag*{(141)} 4ℏ λ eff z z = A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B − A ⟨ 0 ˉ ∣ B ⟨ 1 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 0 ˉ ⟩ A ∣ 1 ˉ ⟩ B − A ⟨ 1 ˉ ∣ B ⟨ 0 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 1 ˉ ⟩ A ∣ 0 ˉ ⟩ B + A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ f NG (∣ d + x A − x B ∣) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B (141) Consequently, the effective error-corrected unitary dynamics can be written as
L e f f , N G ( ρ ) = R ∘ L N G ( ρ ) = − i [ λ e f f x x σ A x ⊗ σ B x + λ e f f x z σ A x ⊗ σ B z + λ e f f z x σ A z ⊗ σ B x + λ e f f z z σ A z ⊗ σ B z , ρ ] + ⋯ (142) \mathcal{L}_{\mathrm{eff,NG}}(\rho)=R\circ\mathcal{L}_{\mathrm{NG}}(\rho)=-i\left[\lambda_{\mathrm{eff}}^{xx}\sigma_A^x\otimes\sigma_B^x+\lambda_{\mathrm{eff}}^{xz}\sigma_A^x\otimes\sigma_B^z+\lambda_{\mathrm{eff}}^{zx}\sigma_A^z\otimes\sigma_B^x+\lambda_{\mathrm{eff}}^{zz}\sigma_A^z\otimes\sigma_B^z,\rho\right]+\cdots \tag*{(142)} L eff , NG ( ρ ) = R ∘ L NG ( ρ ) = − i [ λ eff xx σ A x ⊗ σ B x + λ eff x z σ A x ⊗ σ B z + λ eff z x σ A z ⊗ σ B x + λ eff z z σ A z ⊗ σ B z , ρ ] + ⋯ (142) where we have omitted additional local Hamiltonian contributions, since they are irrelevant for the separability classification. We defer the numerical evaluation of λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx λ e f f x z \lambda_{\mathrm{eff}}^{xz} λ eff x z λ e f f z x \lambda_{\mathrm{eff}}^{zx} λ eff z x λ e f f z z \lambda_{\mathrm{eff}}^{zz} λ eff z z
The effective interaction calculation above relies on a regularized Newtonian potential and assumes that the two masses never contact each other throughout the evolution. In this subsection, we quantify the contact probability between the particles and identify the parameter regime in which the regularized interaction provides a consistent description.
Without loss of generality, consider the two masses prepared in Fock states ∣ n ⟩ A |n\rangle_A ∣ n ⟩ A ∣ m ⟩ B |m\rangle_B ∣ m ⟩ B ∣ m ⟩ B |m\rangle_B ∣ m ⟩ B
B ⟨ m ∣ x B ⟩ B = ψ m ( x B ) = 1 m ! ( 2 π x 0 2 ) 1 / 4 e − x B 2 4 x 0 2 H e m ( x B x 0 ) , (143) {}_B\langle m|x_B\rangle_B=\psi_m(x_B)=\frac{1}{\sqrt{m!}(2\pi x_0^2)^{1/4}}e^{-\frac{x_B^2}{4x_0^2}}H_{e_m}\left(\frac{x_B}{x_0}\right), \tag*{(143)} B ⟨ m ∣ x B ⟩ B = ψ m ( x B ) = m ! ( 2 π x 0 2 ) 1/4 1 e − 4 x 0 2 x B 2 H e m ( x 0 x B ) , (143) where H e p H_{e_p} H e p x 0 x_0 x 0 ∣ n ⟩ A |n\rangle_A ∣ n ⟩ A A ⟨ n ∣ x A ⟩ A = ψ n ( x A ) {}_A\langle n|x_A\rangle_A=\psi_n(x_A) A ⟨ n ∣ x A ⟩ A = ψ n ( x A )
The probability that the two masses are separated by a distance no more than δ d \delta d δ d
P ( δ d ) = ∫ ∣ ψ m ( x B ) ∣ 2 ∣ ψ n ( x A ) ∣ 2 Θ ( δ d − ∣ d + x A − x B ∣ ) d x B d x A = A ⟨ n ∣ B ⟨ m ∣ 1 ∣ d + x A − x B ∣ ≤ δ d ∣ n ⟩ A ∣ m ⟩ B , (144) P(\delta d)=\int|\psi_m(x_B)|^2|\psi_n(x_A)|^2\Theta(\delta d-|d+x_A-x_B|)\,dx_Bdx_A={}_A\langle n|{}_B\langle m|\mathbb{1}_{|d+x_A-x_B|\leq\delta d}|n\rangle_A|m\rangle_B, \tag*{(144)} P ( δ d ) = ∫ ∣ ψ m ( x B ) ∣ 2 ∣ ψ n ( x A ) ∣ 2 Θ ( δ d − ∣ d + x A − x B ∣ ) d x B d x A = A ⟨ n ∣ B ⟨ m ∣ 1 ∣ d + x A − x B ∣ ≤ δ d ∣ n ⟩ A ∣ m ⟩ B , (144) where Θ ( x ) = 1 \Theta(x)=1 Θ ( x ) = 1 x > 0 x>0 x > 0 Θ ( x ) = 0 \Theta(x)=0 Θ ( x ) = 0 x ≤ 0 x\leq0 x ≤ 0 δ d \delta d δ d δ d = 2 r 0 \delta d=2r_0 δ d = 2 r 0
In the numerics, we consider separations d = 10 x 0 d=10x_0 d = 10 x 0 15 x 0 15x_0 15 x 0 20 x 0 20x_0 20 x 0 r 0 ≈ 0.238 x 0 r_0\approx0.238x_0 r 0 ≈ 0.238 x 0 ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = ∣ 2 N + 1 ⟩ A ∣ 2 N + 1 ⟩ B |\bar{1}\rangle_A|\bar{1}\rangle_B=|2N+1\rangle_A|2N+1\rangle_B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = ∣2 N + 1 ⟩ A ∣2 N + 1 ⟩ B P ( 2 r 0 ) ≤ 10 − 5 P(2r_0)\leq10^{-5} P ( 2 r 0 ) ≤ 1 0 − 5 N N N Figure 5 . In particular, for d = 20 x 0 d=20x_0 d = 20 x 0 N ≤ 9 N\leq9 N ≤ 9 N → ∞ N\to\infty N → ∞ d d d x 0 x_0 x 0 N N N
Figure 5. The contact probability P ( 2 r 0 ) P(2r_0) P ( 2 r 0 ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = ∣ 2 N + 1 ⟩ A ∣ 2 N + 1 ⟩ B \lvert\bar{1}\rangle_A\lvert\bar{1}\rangle_B = \lvert2N+1\rangle_A\lvert2N+1\rangle_B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = ∣ 2 N + 1 ⟩ A ∣ 2 N + 1 ⟩ B P ( 2 r 0 ) ≤ 10 − 5 P(2r_0) \le10^{-5} P ( 2 r 0 ) ≤ 1 0 − 5 N N N
Recovery procedures and effective dynamics for different collapse models
We now apply the quantum error-correction framework to the collapse and decoherence models considered in the main text and in App. B3. For each model, we construct an appropriate recovery procedure, derive the resulting effective logical dynamics, and analyze the corresponding separability properties with respect to the separability parameter η \eta η
To that end, we explicitly calculate the effective logical error rates for the various models introduced in App. A1. These effective error rates can be extracted from the matrix elements of the effective error-corrected Lindbladian, see App. B4.
Specifically, the error-corrected Lindbladian is
L e f f , e r r ( ρ ) = R ∘ L e r r ( ρ ) = ∑ i , j R A i R B j L e r r ( ρ ) R A i † R B j † . (145) \mathcal{L}_{\mathrm{eff,err}}(\rho)=R\circ\mathcal{L}_{\mathrm{err}}(\rho)=\sum_{i,j}R_A^iR_B^j\mathcal{L}_{\mathrm{err}}(\rho)R_A^{i\dagger}R_B^{j\dagger}. \tag*{(145)} L eff , err ( ρ ) = R ∘ L err ( ρ ) = i , j ∑ R A i R B j L err ( ρ ) R A i † R B j † . (145) Since the recovery channel R R R C = C A ⊗ C B \mathcal{C}=\mathcal{C}_A\otimes\mathcal{C}_B C = C A ⊗ C B
KTM model
We begin with the KTM model. Since this dynamics is constructed from local measurement and feedback, it provides a useful benchmark for validating both the recovery procedure and the separability framework developed in the main text.
Consider the recovery procedure for L K T M \mathcal{L}_{\mathrm{KTM}} L KTM (27) in App. A1. Here we choose the local Fock code space with code distance 0, i.e. C = C A ⊗ C B \mathcal{C}=\mathcal{C}_A\otimes\mathcal{C}_B C = C A ⊗ C B C α = { ∣ 0 ˉ ⟩ α = ∣ 0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣ 1 ⟩ α } \mathcal{C}_\alpha=\{\lvert\bar{0}\rangle_\alpha=\lvert0\rangle_\alpha,\lvert\bar{1}\rangle_\alpha=\lvert1\rangle_\alpha\} C α = {∣ 0 ˉ ⟩ α = ∣ 0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣ 1 ⟩ α } Π α = ∑ i = 0 , 1 ∣ i ˉ ⟩ α α ⟨ i ˉ ∣ \Pi_\alpha=\sum_{i=0,1}\lvert\bar{i}\rangle_\alpha{}_\alpha\langle\bar{i}\rvert Π α = ∑ i = 0 , 1 ∣ i ˉ ⟩ α α ⟨ i ˉ ∣
x A ∣ 0 ˉ ⟩ A = x 0 ∣ 1 ⟩ , x A ∣ 1 ˉ ⟩ A = x 0 ( ∣ 0 ⟩ A + 2 ∣ 2 ⟩ A ) . (146) x_A\lvert\bar{0}\rangle_A=x_0\lvert1\rangle,\qquad x_A\lvert\bar{1}\rangle_A=x_0(\lvert0\rangle_A+\sqrt{2}\lvert2\rangle_A). \tag*{(146)} x A ∣ 0 ˉ ⟩ A = x 0 ∣ 1 ⟩ , x A ∣ 1 ˉ ⟩ A = x 0 (∣ 0 ⟩ A + 2 ∣ 2 ⟩ A ) . (146) Thus, the codeword ∣ 1 ˉ ⟩ A \lvert\bar{1}\rangle_A ∣ 1 ˉ ⟩ A ∣ 2 ⟩ A \lvert2\rangle_A ∣ 2 ⟩ A C A \mathcal{C}_A C A
R α ( ρ ) = R α 0 ρ R α 0 † + R α 1 ρ R α 1 † = Π α ρ Π α + ∣ 1 ˉ ⟩ α α ⟨ 2 ∣ ρ ∣ 2 ⟩ α α ⟨ 1 ˉ ∣ , (147) R_\alpha(\rho)=R^0_\alpha\rho R^{0\dagger}_\alpha+R^1_\alpha\rho R^{1\dagger}_\alpha=\Pi_\alpha\rho\Pi_\alpha+\lvert\bar{1}\rangle_\alpha{}_\alpha\langle2\rvert\rho\lvert2\rangle_\alpha{}_\alpha\langle\bar{1}\rvert, \tag*{(147)} R α ( ρ ) = R α 0 ρ R α 0 † + R α 1 ρ R α 1 † = Π α ρ Π α + ∣ 1 ˉ ⟩ α α ⟨ 2 ∣ ρ ∣ 2 ⟩ α α ⟨ 1 ˉ ∣ , (147) where α ∈ { A , B } \alpha\in\{A,B\} α ∈ { A , B }
R ∘ L K T M ( ρ ) = − i 2 G m A m B x 0 2 ℏ d 3 [ σ A x ⊗ σ B x , ρ ] + 2 G m A m B x 0 2 ℏ d 3 ( D [ σ A x ] + 1 2 D [ σ A z ] + D [ σ B x ] + 1 2 D [ σ B z ] ) ( ρ ) (148) \mathcal{R}\circ\mathcal{L}_{\mathrm{KTM}}(\rho)=-i\frac{2Gm_A m_B x_0^2}{\hbar d^3}[\sigma_A^x\otimes\sigma_B^x,\rho]+\frac{2Gm_A m_B x_0^2}{\hbar d^3}\left(\mathcal{D}[\sigma_A^x]+\frac{1}{2}\mathcal{D}[\sigma_A^z]+\mathcal{D}[\sigma_B^x]+\frac{1}{2}\mathcal{D}[\sigma_B^z]\right)(\rho) \tag*{(148)} R ∘ L KTM ( ρ ) = − i ℏ d 3 2 G m A m B x 0 2 [ σ A x ⊗ σ B x , ρ ] + ℏ d 3 2 G m A m B x 0 2 ( D [ σ A x ] + 2 1 D [ σ A z ] + D [ σ B x ] + 2 1 D [ σ B z ] ) ( ρ ) (148) where we have imposed that the state is supported on the code space, i.e. Π A ⊗ Π B ρ Π A ⊗ Π B = ρ \Pi_A\otimes\Pi_B\rho\Pi_A\otimes\Pi_B=\rho Π A ⊗ Π B ρ Π A ⊗ Π B = ρ
σ α x = ∣ 0 ˉ ⟩ α α ⟨ 1 ˉ ∣ + ∣ 1 ˉ ⟩ α α ⟨ 0 ˉ ∣ , σ α z = ∣ 0 ˉ ⟩ α α ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ α α ⟨ 1 ˉ ∣ . (149) \sigma_\alpha^x=\lvert\bar{0}\rangle_\alpha{}_\alpha\langle\bar{1}\rvert+\lvert\bar{1}\rangle_\alpha{}_\alpha\langle\bar{0}\rvert,\qquad\sigma_\alpha^z=\lvert\bar{0}\rangle_\alpha{}_\alpha\langle\bar{0}\rvert-\lvert\bar{1}\rangle_\alpha{}_\alpha\langle\bar{1}\rvert. \tag*{(149)} σ α x = ∣ 0 ˉ ⟩ α α ⟨ 1 ˉ ∣ + ∣ 1 ˉ ⟩ α α ⟨ 0 ˉ ∣ , σ α z = ∣ 0 ˉ ⟩ α α ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ α α ⟨ 1 ˉ ∣ . (149) We also assume that the KTM error rate is set to its minimum. Note that although the leakage component has been corrected, logical errors within the code space remain.
We now determine the separability classification of the error-corrected generator. Direct application of Thm. 1 implies that the effective Lindbladian is separable, since its corresponding η = 1 \eta=1 η = 1 L K T M \mathcal{L}_{\mathrm{KTM}} L KTM
GRW model
We next analyze the effective logical dynamics generated by the GRW collapse model. The primary objective is to determine how the effective interaction and logical error rates scale with the code distance and whether error correction can enhance the visibility of non-separable dynamics.
As discussed in App. B 3, the QEC Kraus operators R α i R_{\alpha}^{i} R α i x α i x_{\alpha}^{i} x α i N N N C α = { ∣ 0 ˉ ⟩ α = ∣ 0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣ 2 N + 1 ⟩ α } \mathcal{C}_{\alpha}=\{|\bar{0}\rangle_{\alpha}=|0\rangle_{\alpha},|\bar{1}\rangle_{\alpha}=|2N+1\rangle_{\alpha}\} C α = { ∣ 0 ˉ ⟩ α = ∣0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣2 N + 1 ⟩ α }
R 0 = Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ , R i = { ∣ 0 ˉ ⟩ ⟨ i ∣ + ∣ 1 ˉ ⟩ ( c i ⟨ 2 N + 1 − i ∣ + d i ⟨ 2 N + 1 + i ∣ ) , 1 ≤ i ≤ N , ∣ 1 ˉ ⟩ ( d i − N ⟨ 3 N + 1 − i ∣ − c i − N ⟨ N + 1 + i ∣ ) , N + 1 ≤ i ≤ 2 N , ∣ 1 ˉ ⟩ ⟨ i + N + 1 ∣ , i > 2 N , (150) R^{0}=\Pi=|\bar{0}\rangle\langle\bar{0}|+|\bar{1}\rangle\langle\bar{1}|,\qquad R^{i}=\begin{cases}
|\bar{0}\rangle\langle i|+|\bar{1}\rangle(c_{i}\langle2N+1-i|+d_{i}\langle2N+1+i|), & 1\leq i\leq N,\\
|\bar{1}\rangle(d_{i-N}\langle3N+1-i|-c_{i-N}\langle N+1+i|), & N+1\leq i\leq2N,\\
|\bar{1}\rangle\langle i+N+1|, & i>2N,
\end{cases} \tag*{(150)} R 0 = Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ , R i = ⎩ ⎨ ⎧ ∣ 0 ˉ ⟩ ⟨ i ∣ + ∣ 1 ˉ ⟩ ( c i ⟨ 2 N + 1 − i ∣ + d i ⟨ 2 N + 1 + i ∣ ) , ∣ 1 ˉ ⟩ ( d i − N ⟨ 3 N + 1 − i ∣ − c i − N ⟨ N + 1 + i ∣ ) , ∣ 1 ˉ ⟩ ⟨ i + N + 1∣ , 1 ≤ i ≤ N , N + 1 ≤ i ≤ 2 N , i > 2 N , (150) where
c k = ⟨ k ∣ L G R W ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣ 2 N + 1 − k ⟩ ( ⟨ k ∣ L G R W ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣ 2 N + 1 − k ⟩ ) 2 + ( ⟨ k ∣ L G R W ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣ 2 N + 1 + k ⟩ ) 2 c_{k}=\frac{\langle k|\mathcal{L}_{\mathrm{GRW}}(|\bar{0}\rangle\langle\bar{1}|)|2N+1-k\rangle}{\sqrt{\left(\langle k|\mathcal{L}_{\mathrm{GRW}}(|\bar{0}\rangle\langle\bar{1}|)|2N+1-k\rangle\right)^{2}+\left(\langle k|\mathcal{L}_{\mathrm{GRW}}(|\bar{0}\rangle\langle\bar{1}|)|2N+1+k\rangle\right)^{2}}} c k = ( ⟨ k ∣ L GRW ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣2 N + 1 − k ⟩ ) 2 + ( ⟨ k ∣ L GRW ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣2 N + 1 + k ⟩ ) 2 ⟨ k ∣ L GRW ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣2 N + 1 − k ⟩ d k = ⟨ k ∣ L G R W ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣ 2 N + 1 + k ⟩ ( ⟨ k ∣ L G R W ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣ 2 N + 1 − k ⟩ ) 2 + ( ⟨ k ∣ L G R W ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣ 2 N + 1 + k ⟩ ) 2 . (151) d_{k}=\frac{\langle k|\mathcal{L}_{\mathrm{GRW}}(|\bar{0}\rangle\langle\bar{1}|)|2N+1+k\rangle}{\sqrt{\left(\langle k|\mathcal{L}_{\mathrm{GRW}}(|\bar{0}\rangle\langle\bar{1}|)|2N+1-k\rangle\right)^{2}+\left(\langle k|\mathcal{L}_{\mathrm{GRW}}(|\bar{0}\rangle\langle\bar{1}|)|2N+1+k\rangle\right)^{2}}}. \tag*{(151)} d k = ( ⟨ k ∣ L GRW ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣2 N + 1 − k ⟩ ) 2 + ( ⟨ k ∣ L GRW ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣2 N + 1 + k ⟩ ) 2 ⟨ k ∣ L GRW ( ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ ) ∣2 N + 1 + k ⟩ . (151) Note that for i > 2 N i>2N i > 2 N R i R^{i} R i
We then calculate the matrix elements of the effective Lindbladian for a single mass:
⟨ i ˉ ∣ R ∘ L G R W ( ∣ j ˉ ⟩ ⟨ k ˉ ∣ ) ∣ l ˉ ⟩ = n × λ G R W ( − δ i j δ k l + ∑ m ∫ d x 1 d x 2 e − ( x 1 − x 2 ) 2 4 r C 2 ⟨ i ˉ ∣ R m ∣ x 1 ⟩ ⟨ x 1 ∣ j ˉ ⟩ ⟨ k ˉ ∣ x 2 ⟩ ⟨ x 2 ∣ R m † ∣ l ˉ ⟩ ) , (152) \langle\bar{i}|\mathcal{R}\circ\mathcal{L}_{\mathrm{GRW}}(|\bar{j}\rangle\langle\bar{k}|)|\bar{l}\rangle=n\times\lambda_{\mathrm{GRW}}\left(-\delta_{ij}\delta_{kl}+\sum_{m}\int dx_{1}dx_{2}e^{-\frac{(x_{1}-x_{2})^{2}}{4r_{C}^{2}}}\langle\bar{i}|R_{m}|x_{1}\rangle\langle x_{1}|\bar{j}\rangle\langle\bar{k}|x_{2}\rangle\langle x_{2}|R_{m}^{\dagger}|\bar{l}\rangle\right), \tag*{(152)} ⟨ i ˉ ∣ R ∘ L GRW ( ∣ j ˉ ⟩ ⟨ k ˉ ∣ ) ∣ l ˉ ⟩ = n × λ GRW ( − δ ij δ k l + m ∑ ∫ d x 1 d x 2 e − 4 r C 2 ( x 1 − x 2 ) 2 ⟨ i ˉ ∣ R m ∣ x 1 ⟩ ⟨ x 1 ∣ j ˉ ⟩ ⟨ k ˉ ∣ x 2 ⟩ ⟨ x 2 ∣ R m † ∣ l ˉ ⟩ ) , (152) by performing the integration over the wavefunctions. These integrals can be evaluated analytically because they involve only Gaussian factors multiplied by polynomials. We define the logical operators σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ \sigma^{+}=|\bar{1}\rangle\langle\bar{0}| σ + = ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ σ − = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ \sigma^{-}=|\bar{0}\rangle\langle\bar{1}| σ − = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ σ x = σ + + σ − \sigma^{x}=\sigma^{+}+\sigma^{-} σ x = σ + + σ − σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ \sigma^{z}=|\bar{0}\rangle\langle\bar{0}|-|\bar{1}\rangle\langle\bar{1}| σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ ρ \rho ρ
L e f f , G R W α ( ρ ) = R α ∘ L G R W α ( ρ ) = Γ e f f + D [ σ α + ] ( ρ ) + Γ e f f − D [ σ α − ] ( ρ ) + Γ e f f x D [ σ α x ] ( ρ ) + Γ e f f z D [ σ α z ] ( ρ ) . (153) \mathcal{L}_{\mathrm{eff},\mathrm{GRW}}^{\alpha}(\rho)=\mathcal{R}_{\alpha}\circ\mathcal{L}_{\mathrm{GRW}}^{\alpha}(\rho)=\Gamma_{\mathrm{eff}}^{+}\mathcal{D}[\sigma_{\alpha}^{+}](\rho)+\Gamma_{\mathrm{eff}}^{-}\mathcal{D}[\sigma_{\alpha}^{-}](\rho)+\Gamma_{\mathrm{eff}}^{x}\mathcal{D}[\sigma_{\alpha}^{x}](\rho)+\Gamma_{\mathrm{eff}}^{z}\mathcal{D}[\sigma_{\alpha}^{z}](\rho). \tag*{(153)} L eff , GRW α ( ρ ) = R α ∘ L GRW α ( ρ ) = Γ eff + D [ σ α + ] ( ρ ) + Γ eff − D [ σ α − ] ( ρ ) + Γ eff x D [ σ α x ] ( ρ ) + Γ eff z D [ σ α z ] ( ρ ) . (153) We now illustrate these effective interaction and error rates with a concrete example. We first take r C = 2 × 10 − 6 m r_{C}=2\times10^{-6}\,\mathrm{m} r C = 2 × 1 0 − 6 m λ G R W = 10 − 8 s − 1 \lambda_{\mathrm{GRW}}=10^{-8}\,\mathrm{s}^{-1} λ GRW = 1 0 − 8 s − 1 [57 , 58 ] ). We set x 0 = 2 × 10 − 8 m x_{0}=2\times10^{-8}\,\mathrm{m} x 0 = 2 × 1 0 − 8 m m A = m B = 10 − 21 k g m_{A}=m_{B}=10^{-21}\,\mathrm{kg} m A = m B = 1 0 − 21 kg d = 4 × 10 − 7 m d=4\times10^{-7}\,\mathrm{m} d = 4 × 1 0 − 7 m ρ ≈ 2.2 × 10 3 k g / m 3 \rho\approx2.2\times10^{3}\,\mathrm{kg}/\mathrm{m}^{3} ρ ≈ 2.2 × 1 0 3 kg / m 3 r 0 = ( 3 m A / 4 π ρ ) 1 / 3 ≈ 4.8 × 10 − 9 m r_{0}=(3m_{A}/4\pi\rho)^{1/3}\approx4.8\times10^{-9}\,\mathrm{m} r 0 = ( 3 m A /4 π ρ ) 1/3 ≈ 4.8 × 1 0 − 9 m 9.98 × 10 − 26 k g 9.98\times10^{-26}\,\mathrm{kg} 9.98 × 1 0 − 26 kg n ≈ 6 × 10 5 n\approx6\times10^{5} n ≈ 6 × 1 0 5 ( 4 r C 2 G m A m B ) / ( ℏ λ G R W n d min 3 ) ≈ 1.9 × 10 − 3 (4r_{C}^{2}Gm_{A}m_{B})/(\hbar\lambda_{\mathrm{GRW}}nd_{\min}^{3})\approx1.9\times10^{-3} ( 4 r C 2 G m A m B ) / ( ℏ λ GRW n d m i n 3 ) ≈ 1.9 × 1 0 − 3 d min = 2 r 0 d_{\min}=2r_{0} d m i n = 2 r 0 N N N Figure 6 a. We observe that the σ α − \sigma_{\alpha}^{-} σ α − N N N σ A x ⊗ σ B x \sigma_{A}^{x}\otimes\sigma_{B}^{x} σ A x ⊗ σ B x
Figure 6. The effective interaction rate λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx λ e f f x z \lambda_{\mathrm{eff}}^{xz} λ eff x z λ e f f z z \lambda_{\mathrm{eff}}^{zz} λ eff z z Γ e f f x \Gamma_{\mathrm{eff}}^{x} Γ eff x Γ e f f − \Gamma_{\mathrm{eff}}^{-} Γ eff − Γ e f f + \Gamma_{\mathrm{eff}}^{+} Γ eff + Γ e f f z \Gamma_{\mathrm{eff}}^{z} Γ eff z N N N r C r_{\mathrm{C}} r C λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx N N N Γ e f f x \Gamma_{\mathrm{eff}}^{x} Γ eff x Γ e f f − \Gamma_{\mathrm{eff}}^{-} Γ eff − Γ e f f + \Gamma_{\mathrm{eff}}^{+} Γ eff + N ≥ 5 N \ge5 N ≥ 5 r C r_{\mathrm{C}} r C
η G R W = ∥ ( λ e f f x x Γ e f f x x − λ e f f z z Γ e f f z Γ e f f z λ e f f x z Γ e f f x Γ e f f x λ e f f z z Γ e f f z z ) ∥ 2 , (154) \eta_{\mathrm{GRW}}=\left\|\begin{pmatrix}
\frac{\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^{xx}} & -\frac{\lambda_{\mathrm{eff}}^{zz}}{\sqrt{\Gamma_{\mathrm{eff}}^{z}\Gamma_{\mathrm{eff}}^{z}}}\\
\frac{\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{x}\Gamma_{\mathrm{eff}}^{x}}} & \frac{\lambda_{\mathrm{eff}}^{zz}}{\Gamma_{\mathrm{eff}}^{zz}}
\end{pmatrix}\right\|_{2}, \tag*{(154)} η GRW = Γ eff xx λ eff xx Γ eff x Γ eff x λ eff x z − Γ eff z Γ eff z λ eff z z Γ eff z z λ eff z z 2 , (154) where Γ ˉ e f f x = Γ e f f x + + Γ e f f − Γ e f f + / ( Γ e f f + + Γ e f f − ) \bar{\Gamma}_{\mathrm{eff}}^{x}=\Gamma_{\mathrm{eff}}^{x+}+\Gamma_{\mathrm{eff}}^{-}\Gamma_{\mathrm{eff}}^{+}/(\Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}) Γ ˉ eff x = Γ eff x + + Γ eff − Γ eff + / ( Γ eff + + Γ eff − ) σ α z \sigma_{\alpha}^{z} σ α z ∣ λ e f f x z ∣ / Γ ˉ e f f x Γ ˉ e f f z , ∣ λ e f f x z ∣ / Γ e f f z ≪ ∣ λ e f f x x ∣ / Γ ˉ e f f x |\lambda_{\mathrm{eff}}^{xz}|/\sqrt{\bar{\Gamma}_{\mathrm{eff}}^{x}\bar{\Gamma}_{\mathrm{eff}}^{z}},|\lambda_{\mathrm{eff}}^{xz}|/\Gamma_{\mathrm{eff}}^{z}\ll|\lambda_{\mathrm{eff}}^{xx}|/\bar{\Gamma}_{\mathrm{eff}}^{x} ∣ λ eff x z ∣/ Γ ˉ eff x Γ ˉ eff z , ∣ λ eff x z ∣/ Γ eff z ≪ ∣ λ eff xx ∣/ Γ ˉ eff x L e f f , N G \mathcal{L}_{\mathrm{eff},\mathrm{NG}} L eff , NG L e f f , G R W A + L e f f , G R W B \mathcal{L}_{\mathrm{eff},\mathrm{GRW}}^{A}+\mathcal{L}_{\mathrm{eff},\mathrm{GRW}}^{B} L eff , GRW A + L eff , GRW B η G R W \eta_{\mathrm{GRW}} η GRW ∣ λ e f f x x ∣ / Γ ˉ e f f x |\lambda_{\mathrm{eff}}^{xx}|/\bar{\Gamma}_{\mathrm{eff}}^{x} ∣ λ eff xx ∣/ Γ ˉ eff x η > 1 \eta>1 η > 1 N ≥ 5 N\geq5 N ≥ 5 η < 1 \eta<1 η < 1 N ≤ 4 N\leq4 N ≤ 4 Figure 10 . Moreover, when N ≥ 5 N\geq5 N ≥ 5 ∣ λ e f f x x ∣ > Γ e f f x + + Γ e f f + |\lambda_{\mathrm{eff}}^{xx}|>\Gamma_{\mathrm{eff}}^{x+}+\Gamma_{\mathrm{eff}}^{+} ∣ λ eff xx ∣ > Γ eff x + + Γ eff +
Figure 10. Separability parameter η \eta η N N N η > 1 \eta> 1 η > 1 N N N N = 0 N = 0 N = 0
For the parameters above, note that r C = 100 x 0 r_{C}=100x_{0} r C = 100 x 0 d = 20 x 0 d=20x_{0} d = 20 x 0 r C r_{C} r C r C r_{C} r C d d d x 0 x_{0} x 0 r C = x 0 r_{C}=x_{0} r C = x 0 Figure 6 b. In this case, the error rates still decrease exponentially with N N N N N N
Note that, in contrast to analyses that apply a leading-order separability condition directly to the uncorrected master equation, the separability classification in our setting is governed by how the effective rates scale with the code distance N N N (51) , then it is non-separable if λ G R W n / 2 r C 2 < 2 G m A m B / ℏ d 3 \lambda_{\mathrm{GRW}}n/2r_{\mathrm{C}}^{2} < 2Gm_{\mathrm{A}}m_{\mathrm{B}}/\hbar d^{3} λ GRW n /2 r C 2 < 2 G m A m B /ℏ d 3 λ G R W \lambda_{\mathrm{GRW}} λ GRW λ G R W \lambda_{\mathrm{GRW}} λ GRW Figure 6 a upward without changing their scaling with N N N r C r_{\mathrm{C}} r C
CSL model
We now perform the same analysis for the CSL model. In contrast to the GRW case, the recovered dynamics contains additional nonlocal-like contributions, leading to a richer effective logical structure and additional entries in the separability parameter.
We first simplify the CSL model in a coarse-grained description. We consider two solid spherical masses of equal radius r 0 r_0 r 0 m c m_{\mathrm{c}} m c A A A
H A = span { ∣ x A ⟩ ≡ ∏ i ∣ 1 x A + x i , y i , z i ⟩ = ∏ i a † ( x A + x i , y i , z i ) ∣ v a c ⟩ } x A (155) \mathcal{H}_{A} = \operatorname{span}\left\{|x_{A}\rangle\equiv\prod_i |1_{x_A+x_i,y_i,z_i}\rangle= \prod_i a^\dagger(x_A+x_i,y_i,z_i)|\mathrm{vac}\rangle\right\}_{x_A} \tag*{(155)} H A = span { ∣ x A ⟩ ≡ i ∏ ∣ 1 x A + x i , y i , z i ⟩ = i ∏ a † ( x A + x i , y i , z i ) ∣ vac ⟩ } x A (155) where x i = ( x i , y i , z i ) x_i=(x_i,y_i,z_i) x i = ( x i , y i , z i ) i i i x A = ( x A , 0 , 0 ) x_A=(x_A,0,0) x A = ( x A , 0 , 0 ) a † ( x , y , z ) a^\dagger(x,y,z) a † ( x , y , z ) B B B d d d A A A M ( x ) M(x) M ( x )
M ( x ) ∣ x A ⟩ ∣ x B ⟩ = ∑ k m k ψ k † ( x ) ψ k ( x ) ∏ i ∣ 1 x A + x i , y i , z i ⟩ ∏ j ∣ 1 x B + x j , y j , z j ⟩ = m c ( ∑ i δ ( x − d x ^ − x A − x i ) + ∑ j δ ( x − x B − x j ) ) ∣ x A ⟩ ∣ x B ⟩ ≈ ρ 0 ( ρ x A ( x ) + ρ x B ( x ) ) ∣ x A ⟩ ∣ x B ⟩ (156) \begin{aligned}
M(x)|x_A\rangle|x_B\rangle= \sum_k m_k\psi_k^\dagger(x)\psi_k(x)\prod_i |1_{x_A+x_i,y_i,z_i}\rangle\prod_j |1_{x_B+x_j,y_j,z_j}\rangle\\
&= m_{\mathrm{c}}\left(\sum_i\delta(x-d\hat{x}-x_A-x_i)+\sum_j\delta(x-x_B-x_j)\right)|x_A\rangle|x_B\rangle\\
&\approx\rho_0\left(\rho_{x_A}(x)+\rho_{x_B}(x)\right)|x_A\rangle|x_B\rangle \tag*{(156)}
\end{aligned} M ( x ) ∣ x A ⟩ ∣ x B ⟩ = k ∑ m k ψ k † ( x ) ψ k ( x ) i ∏ ∣ 1 x A + x i , y i , z i ⟩ j ∏ ∣ 1 x B + x j , y j , z j ⟩ = m c ( i ∑ δ ( x − d x ^ − x A − x i ) + j ∑ δ ( x − x B − x j ) ) ∣ x A ⟩ ∣ x B ⟩ ≈ ρ 0 ( ρ x A ( x ) + ρ x B ( x ) ) ∣ x A ⟩ ∣ x B ⟩ (156) where ρ x A ( x ) = { 1 , ∣ x − d x ^ − x A ∣ ≤ r 0 , 0 , otherwise , \rho_{x_A}(x)=\begin{cases}1, & |x-d\hat{x}-x_A|\le r_0,\\0, & \text{otherwise},\end{cases} ρ x A ( x ) = { 1 , 0 , ∣ x − d x ^ − x A ∣ ≤ r 0 , otherwise , ρ x B ( x ) = { 1 , ∣ x − x B ∣ ≤ r 0 , 0 , otherwise . \rho_{x_B}(x)=\begin{cases}1, & |x-x_B|\le r_0,\\0, & \text{otherwise}.\end{cases} ρ x B ( x ) = { 1 , 0 , ∣ x − x B ∣ ≤ r 0 , otherwise .
ρ x A ( x ) = { 1 , ∣ x − d x ^ − x A ∣ ≤ r 0 , 0 , otherwise , ρ x B ( x ) = { 1 , ∣ x − x B ∣ ≤ r 0 , 0 , otherwise . (157) \rho_{x_A}(x)=\begin{cases}1, & |x-d\hat{x}-x_A|\le r_0,\\0, & \text{otherwise},\end{cases}
\qquad
\rho_{x_B}(x)=\begin{cases}1, & |x-x_B|\le r_0,\\0, & \text{otherwise}.\end{cases} \tag*{(157)} ρ x A ( x ) = { 1 , 0 , ∣ x − d x ^ − x A ∣ ≤ r 0 , otherwise , ρ x B ( x ) = { 1 , 0 , ∣ x − x B ∣ ≤ r 0 , otherwise . (157) where ρ 0 \rho_0 ρ 0
With this setting, we compute the CSL Lindbladian matrix elements in the positional basis of H A ⊗ H B \mathcal{H}_A \otimes\mathcal{H}_B H A ⊗ H B
L C S L ( ρ ) = − λ C S L π 2 ρ 0 2 r C 6 m 0 2 ∫ d x A d x B d y A d y B ρ x A x B , y A y B ∣ x A ⟩ ⟨ y A ∣ ⊗ ∣ x B ⟩ ⟨ y B ∣ × [ 2 f r 0 / r C ( 0 ) − f r 0 / r C ( ∣ x A − y A ∣ r C ) − f r 0 / r C ( ∣ x B − y B ∣ r C ) + f r 0 / r C ( ∣ d + x A − x B ∣ r C ) + f r 0 / r C ( ∣ d + y A − y B ∣ r C ) − f r 0 / r C ( ∣ d + x A − y B ∣ r C ) − f r 0 / r C ( ∣ d + y A − x B ∣ r C ) ] , (158) \mathcal{L}_{\mathrm{CSL}}(\rho)=-\frac{\lambda_{\mathrm{CSL}}\pi^2\rho_0^2r_{\mathrm{C}}^6}{m_0^2}\int\mathrm{d}x_A\mathrm{d}x_B\mathrm{d}y_A\mathrm{d}y_B\rho_{x_Ax_B,y_Ay_B}|x_A\rangle\langle y_A|\otimes|x_B\rangle\langle y_B|
\begin{aligned}
&\times\left[2f_{r_0/r_{\mathrm{C}}}(0)-f_{r_0/r_{\mathrm{C}}}\left(\frac{|x_A-y_A|}{r_{\mathrm{C}}}\right)-f_{r_0/r_{\mathrm{C}}}\left(\frac{|x_B-y_B|}{r_{\mathrm{C}}}\right)+f_{r_0/r_{\mathrm{C}}}\left(\frac{|d+x_A-x_B|}{r_{\mathrm{C}}}\right)\right.\\
&\left.\quad+f_{r_0/r_{\mathrm{C}}}\left(\frac{|d+y_A-y_B|}{r_{\mathrm{C}}}\right)-f_{r_0/r_{\mathrm{C}}}\left(\frac{|d+x_A-y_B|}{r_{\mathrm{C}}}\right)-f_{r_0/r_{\mathrm{C}}}\left(\frac{|d+y_A-x_B|}{r_{\mathrm{C}}}\right)\right],
\end{aligned} \tag*{(158)} L CSL ( ρ ) = − m 0 2 λ CSL π 2 ρ 0 2 r C 6 ∫ d x A d x B d y A d y B ρ x A x B , y A y B ∣ x A ⟩ ⟨ y A ∣ ⊗ ∣ x B ⟩ ⟨ y B ∣ × [ 2 f r 0 / r C ( 0 ) − f r 0 / r C ( r C ∣ x A − y A ∣ ) − f r 0 / r C ( r C ∣ x B − y B ∣ ) + f r 0 / r C ( r C ∣ d + x A − x B ∣ ) + f r 0 / r C ( r C ∣ d + y A − y B ∣ ) − f r 0 / r C ( r C ∣ d + x A − y B ∣ ) − f r 0 / r C ( r C ∣ d + y A − x B ∣ ) ] , (158) where the function f b ( a ) f_b(a) f b ( a )
f b ( a ) = 1 3 a ( 4 a e − a 2 / 4 ( a 2 − 12 b 2 + 10 ) − 4 e − a 2 / 4 − b 2 ( a ( a 2 − 8 b 2 + 10 ) cosh ( a b ) + 2 ( a 2 + 6 ) b sinh ( a b ) ) + π ( a 4 − 12 ( a 2 + 2 ) b 2 + 12 a 2 + 12 ) ( 2 Erf ( a 2 ) − Erf ( a 2 − b ) − Erf ( a 2 + b ) ) − π 16 a b 3 ( Erf ( a 2 − b ) − Erf ( a 2 + b ) ) ) , (159) \begin{aligned}
f_b(a)=\frac{1}{3a}\Bigg(&4ae^{-a^2/4}\left(a^2-12b^2+10\right)-4e^{-a^2/4-b^2}\left(a\left(a^2-8b^2+10\right)\cosh(ab)+2\left(a^2+6\right)b\sinh(ab)\right)\\
&+\sqrt{\pi}\left(a^4-12\left(a^2+2\right)b^2+12a^2+12\right)\left(2\operatorname{Erf}\left(\frac{a}{2}\right)-\operatorname{Erf}\left(\frac{a}{2}-b\right)-\operatorname{Erf}\left(\frac{a}{2}+b\right)\right)\\
&-\sqrt{\pi}16ab^3\left(\operatorname{Erf}\left(\frac{a}{2}-b\right)-\operatorname{Erf}\left(\frac{a}{2}+b\right)\right)\Bigg), \tag*{(159)}
\end{aligned} f b ( a ) = 3 a 1 ( 4 a e − a 2 /4 ( a 2 − 12 b 2 + 10 ) − 4 e − a 2 /4 − b 2 ( a ( a 2 − 8 b 2 + 10 ) cosh ( ab ) + 2 ( a 2 + 6 ) b sinh ( ab ) ) + π ( a 4 − 12 ( a 2 + 2 ) b 2 + 12 a 2 + 12 ) ( 2 Erf ( 2 a ) − Erf ( 2 a − b ) − Erf ( 2 a + b ) ) − π 16 a b 3 ( Erf ( 2 a − b ) − Erf ( 2 a + b ) ) ) , (159) and Erf ( x ) = 2 π ∫ 0 x e − t 2 d t \operatorname{Erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\mathrm{d}t Erf ( x ) = π 2 ∫ 0 x e − t 2 d t a , b ≈ 0 a,b\approx0 a , b ≈ 0 f b ( a ) ∼ b 6 f_b(a)\sim b^6 f b ( a ) ∼ b 6 ρ 0 2 r C 6 ( r 0 / r C ) 6 ∼ ρ 0 2 r 0 6 \rho_0^2r_{\mathrm{C}}^6(r_0/r_{\mathrm{C}})^6\sim\rho_0^2r_0^6 ρ 0 2 r C 6 ( r 0 / r C ) 6 ∼ ρ 0 2 r 0 6 f b ( a ) f_b(a) f b ( a ) f b ( − a ) = f b ( a ) f_b(-a)=f_b(a) f b ( − a ) = f b ( a ) f b ( a ) f_b(a) f b ( a ) a ≪ 1 a\ll1 a ≪ 1 f b ′ ′ ( 0 ) a 2 / 2 f_b''(0)a^2/2 f b ′′ ( 0 ) a 2 /2 [59 ] . We therefore introduce a Gaussian approximation f b ′ ( a ) f_b'(a) f b ′ ( a )
f b ′ ( a ) = 32 3 ( π b 3 Erf ( b ) − 3 b 2 + ( b 2 − 2 ) e − b 2 + 2 ) exp [ − a 2 ( 4 b 6 3 ( π b 3 Erf ( b ) − 3 b 2 + ( b 2 − 2 ) e − b 2 + 2 ) ) − 2 3 ] . (160) f_b'(a)=\frac{32}{3}\left(\sqrt{\pi}b^3\operatorname{Erf}(b)-3b^2+(b^2-2)e^{-b^2}+2\right)\exp\left[-a^2\left(\frac{4b^6}{3\left(\sqrt{\pi}b^3\operatorname{Erf}(b)-3b^2+(b^2-2)e^{-b^2}+2\right)}\right)^{-\frac{2}{3}}\right]. \tag*{(160)} f b ′ ( a ) = 3 32 ( π b 3 Erf ( b ) − 3 b 2 + ( b 2 − 2 ) e − b 2 + 2 ) exp − a 2 ( 3 ( π b 3 Erf ( b ) − 3 b 2 + ( b 2 − 2 ) e − b 2 + 2 ) 4 b 6 ) − 3 2 . (160) Using this approximation, the matrix elements of the approximated CSL Lindbladian L C S L ′ ( ρ ) \mathcal{L}'_{\mathrm{CSL}}(\rho) L CSL ′ ( ρ ) f r 0 / r C ′ ( ∣ d + x A − x B ∣ r C ) f'_{r_0/r_{\mathrm{C}}}\left(\frac{|d+x_A-x_B|}{r_{\mathrm{C}}}\right) f r 0 / r C ′ ( r C ∣ d + x A − x B ∣ ) f r 0 / r C ′ ( d / r C ) f'_{r_0/r_{\mathrm{C}}}(d/r_{\mathrm{C}}) f r 0 / r C ′ ( d / r C ) exp ( − O ( d 2 / r C 2 ) ) \exp(-O(d^2/r_{\mathrm{C}}^2)) exp ( − O ( d 2 / r C 2 ))
As in the GRW case, we define local Kraus operators as
R 0 = Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ , R i = { ∣ 0 ˉ ⟩ ⟨ i ∣ + ∣ 1 ˉ ⟩ ( c i ⟨ 2 N + 1 − i ∣ + d i ⟨ 2 N + 1 + i ∣ ) , 1 ≤ i ≤ N , ∣ 1 ˉ ⟩ ( d i − N ⟨ 3 N + 1 − i ∣ − c i − N ⟨ N + 1 + i ∣ ) , N + 1 ≤ i ≤ 2 N , ∣ 1 ˉ ⟩ ⟨ i + N + 1 ∣ , i > 2 N , (161) R_0=\Pi=|\bar{0}\rangle\langle\bar{0}|+|\bar{1}\rangle\langle\bar{1}|,\qquad
R_i=
\begin{cases}
|\bar{0}\rangle\langle i|+|\bar{1}\rangle\left(c_i\langle2N+1-i|+d_i\langle2N+1+i|\right), & 1\leq i\leq N,\\
|\bar{1}\rangle\left(d_{i-N}\langle3N+1-i|-c_{i-N}\langle N+1+i|\right), & N+1\leq i\leq2N,\\
|\bar{1}\rangle\langle i+N+1|, & i>2N,
\end{cases} \tag*{(161)} R 0 = Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ , R i = ⎩ ⎨ ⎧ ∣ 0 ˉ ⟩ ⟨ i ∣ + ∣ 1 ˉ ⟩ ( c i ⟨ 2 N + 1 − i ∣ + d i ⟨ 2 N + 1 + i ∣ ) , ∣ 1 ˉ ⟩ ( d i − N ⟨ 3 N + 1 − i ∣ − c i − N ⟨ N + 1 + i ∣ ) , ∣ 1 ˉ ⟩ ⟨ i + N + 1∣ , 1 ≤ i ≤ N , N + 1 ≤ i ≤ 2 N , i > 2 N , (161) where
c k ∝ ∫ d x 1 d x 2 f r 0 / r C ′ ( ∣ x 1 − x 2 ∣ r C ) ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣ 2 N + 1 − k ⟩ , c_k\propto\int\mathrm{d}x_1\mathrm{d}x_2f'_{r_0/r_{\mathrm{C}}}\left(\frac{|x_1-x_2|}{r_{\mathrm{C}}}\right)\langle k|x_1\rangle\langle x_1|\bar{0}\rangle\langle\bar{1}|x_2\rangle\langle x_2|2N+1-k\rangle, c k ∝ ∫ d x 1 d x 2 f r 0 / r C ′ ( r C ∣ x 1 − x 2 ∣ ) ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣2 N + 1 − k ⟩ , d k ∝ ∫ d x 1 d x 2 f r 0 / r C ′ ( ∣ x 1 − x 2 ∣ r C ) ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣ 2 N + 1 + k ⟩ , d_k\propto\int\mathrm{d}x_1\mathrm{d}x_2f'_{r_0/r_{\mathrm{C}}}\left(\frac{|x_1-x_2|}{r_{\mathrm{C}}}\right)\langle k|x_1\rangle\langle x_1|\bar{0}\rangle\langle\bar{1}|x_2\rangle\langle x_2|2N+1+k\rangle, d k ∝ ∫ d x 1 d x 2 f r 0 / r C ′ ( r C ∣ x 1 − x 2 ∣ ) ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣2 N + 1 + k ⟩ , c k 2 + d k 2 = 1. (162) c_k^2+d_k^2=1. \tag*{(162)} c k 2 + d k 2 = 1. (162) We then compute the matrix elements of the effective error-corrected Lindbladian in the full two-mass space:
A ⟨ i ˉ ∣ B ⟨ j ˉ ∣ R ∘ L C S L ′ ( ∣ k ˉ ⟩ A ∣ l ˉ ⟩ B A ⟨ m ˉ ∣ B ⟨ n ˉ ∣ ) ∣ p ˉ ⟩ A ∣ q ˉ ⟩ B . (163) {}_A\langle\bar{i}|{}_B\langle\bar{j}|R\circ\mathcal{L}'_{\mathrm{CSL}}\left(|\bar{k}\rangle_A|\bar{l}\rangle_B{}_A\langle\bar{m}|{}_B\langle\bar{n}|\right)|\bar{p}\rangle_A|\bar{q}\rangle_B. \tag*{(163)} A ⟨ i ˉ ∣ B ⟨ j ˉ ∣ R ∘ L CSL ′ ( ∣ k ˉ ⟩ A ∣ l ˉ ⟩ B A ⟨ m ˉ ∣ B ⟨ n ˉ ∣ ) ∣ p ˉ ⟩ A ∣ q ˉ ⟩ B . (163) These integrals again reduce to Gaussian-polynomial integrals and can be computed analytically. By symmetry, the effective error dynamics for the two masses takes the form, for ρ \rho ρ
L e f f , C S L ( ρ ) = R ∘ L C S L ( ρ ) = Γ e f f + ( D [ σ A + ] + D [ σ B + ] ) ( ρ ) + Γ e f f − ( D [ σ A − ] + D [ σ B − ] ) ( ρ ) + Γ e f f x x ( D [ σ A x ] + D [ σ B x ] ) ( ρ ) + Γ e f f z z ( D [ σ A z ] + D [ σ B z ] ) ( ρ ) + Γ e f f z z ( D [ σ A z − σ B z ] − D [ σ A z ] − D [ σ B z ] ) ( ρ ) + Γ e f f x x ( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ] ) ( ρ ) + Γ e f f x z ( D [ σ A x − σ B z ] − D [ σ A x ] − D [ σ B z ] − D [ σ A z − σ B x ] + D [ σ A z ] + D [ σ B x ] ) ( ρ ) . (164) \begin{aligned}
\mathcal{L}_{\mathrm{eff,CSL}}(\rho) ={}& \mathcal{R}\circ\mathcal{L}_{\mathrm{CSL}}(\rho)\\
={}&\Gamma_{\mathrm{eff}}^{+}\left(\mathcal{D}[\sigma_A^{+}]+\mathcal{D}[\sigma_B^{+}]\right)(\rho)+\Gamma_{\mathrm{eff}}^{-}\left(\mathcal{D}[\sigma_A^{-}]+\mathcal{D}[\sigma_B^{-}]\right)(\rho)+\Gamma_{\mathrm{eff}}^{xx}\left(\mathcal{D}[\sigma_A^{x}]+\mathcal{D}[\sigma_B^{x}]\right)(\rho)+\Gamma_{\mathrm{eff}}^{zz}\left(\mathcal{D}[\sigma_A^{z}]+\mathcal{D}[\sigma_B^{z}]\right)(\rho)\\
&+\Gamma_{\mathrm{eff}}^{zz}\left(\mathcal{D}[\sigma_A^{z}-\sigma_B^{z}]-\mathcal{D}[\sigma_A^{z}]-\mathcal{D}[\sigma_B^{z}]\right)(\rho)+\Gamma_{\mathrm{eff}}^{xx}\left(\mathcal{D}[\sigma_A^{x}-\sigma_B^{x}]-\mathcal{D}[\sigma_A^{x}]-\mathcal{D}[\sigma_B^{x}]\right)(\rho)\\
&+\Gamma_{\mathrm{eff}}^{xz}\left(\mathcal{D}[\sigma_A^{x}-\sigma_B^{z}]-\mathcal{D}[\sigma_A^{x}]-\mathcal{D}[\sigma_B^{z}]-\mathcal{D}[\sigma_A^{z}-\sigma_B^{x}]+\mathcal{D}[\sigma_A^{z}]+\mathcal{D}[\sigma_B^{x}]\right)(\rho). \tag*{(164)}
\end{aligned} L eff , CSL ( ρ ) = = R ∘ L CSL ( ρ ) Γ eff + ( D [ σ A + ] + D [ σ B + ] ) ( ρ ) + Γ eff − ( D [ σ A − ] + D [ σ B − ] ) ( ρ ) + Γ eff xx ( D [ σ A x ] + D [ σ B x ] ) ( ρ ) + Γ eff z z ( D [ σ A z ] + D [ σ B z ] ) ( ρ ) + Γ eff z z ( D [ σ A z − σ B z ] − D [ σ A z ] − D [ σ B z ] ) ( ρ ) + Γ eff xx ( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ] ) ( ρ ) + Γ eff x z ( D [ σ A x − σ B z ] − D [ σ A x ] − D [ σ B z ] − D [ σ A z − σ B x ] + D [ σ A z ] + D [ σ B x ] ) ( ρ ) . (164) This structure is distinct from the term arising from D [ σ A x − i σ B x ] \mathcal{D}[\sigma_A^{x}-i\sigma_B^{x}] D [ σ A x − i σ B x ]
η C S L = ∥ ( Γ e f f x x − i λ e f f x x Γ e f f + − Γ e f f x z − i λ e f f x z Γ e f f + Γ e f f − Γ e f f x z − i λ e f f x z Γ e f f + Γ e f f − − Γ e f f z z − i λ e f f z z Γ e f f − ) ∥ 2 . (165) \eta_{\mathrm{CSL}}=\left\|\begin{pmatrix}
\dfrac{\Gamma_{\mathrm{eff}}^{xx}-i\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^{+}}-\dfrac{\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{+}\Gamma_{\mathrm{eff}}^{-}}}\\
\dfrac{\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{+}\Gamma_{\mathrm{eff}}^{-}}}-\dfrac{\Gamma_{\mathrm{eff}}^{zz}-i\lambda_{\mathrm{eff}}^{zz}}{\Gamma_{\mathrm{eff}}^{-}}
\end{pmatrix}\right\|_2 . \tag*{(165)} η CSL = Γ eff + Γ eff xx − i λ eff xx − Γ eff + Γ eff − Γ eff x z − i λ eff x z Γ eff + Γ eff − Γ eff x z − i λ eff x z − Γ eff − Γ eff z z − i λ eff z z 2 . (165) As an example, we take r C = 2 × 10 − 6 m r_{\mathrm{C}}=2\times10^{-6}\,\mathrm{m} r C = 2 × 1 0 − 6 m λ C S L = 10 − 8 s − 1 \lambda_{\mathrm{CSL}}=10^{-8}\,\mathrm{s}^{-1} λ CSL = 1 0 − 8 s − 1 m 0 = 1.6 × 10 − 27 k g m_0=1.6\times10^{-27}\,\mathrm{kg} m 0 = 1.6 × 1 0 − 27 kg [57 , 58 ] ). We choose x 0 = 2 × 10 − 8 m x_0=2\times10^{-8}\,\mathrm{m} x 0 = 2 × 1 0 − 8 m m A = m B = 10 − 21 k g m_A=m_B=10^{-21}\,\mathrm{kg} m A = m B = 1 0 − 21 kg d = 4 × 10 − 7 m d=4\times10^{-7}\,\mathrm{m} d = 4 × 1 0 − 7 m ρ ≈ 2.2 × 10 3 k g / m 3 \rho\approx2.2\times10^{3}\,\mathrm{kg}/\mathrm{m}^{3} ρ ≈ 2.2 × 1 0 3 kg / m 3 r 0 = ( 3 m A / 4 π ρ ) 1 / 3 ≈ 4.8 × 10 − 9 m r_0=(3m_A/4\pi\rho)^{1/3}\approx4.8\times10^{-9}\,\mathrm{m} r 0 = ( 3 m A /4 π ρ ) 1/3 ≈ 4.8 × 1 0 − 9 m ( 4 r C 2 G m A m B m 0 2 ) / ( ℏ λ C S L m A m B d min 3 ) ≈ 3.0 × 10 − 9 (4r_{\mathrm{C}}^{2}Gm_Am_Bm_0^{2})/(\hbar\lambda_{\mathrm{CSL}}m_Am_Bd_{\min}^{3})\approx3.0\times10^{-9} ( 4 r C 2 G m A m B m 0 2 ) / ( ℏ λ CSL m A m B d m i n 3 ) ≈ 3.0 × 1 0 − 9 d min = 2 r 0 d_{\min}=2r_0 d m i n = 2 r 0 N N N Figure 7 a. We observe that the σ x \sigma^x σ x L e f f , N G + L e f f , C S L \mathcal{L}_{\mathrm{eff,NG}}+\mathcal{L}_{\mathrm{eff,CSL}} L eff , NG + L eff , CSL η C S L > 1 \eta_{\mathrm{CSL}}>1 η CSL > 1 η C S L \eta_{\mathrm{CSL}} η CSL σ x \sigma^x σ x
Figure 7. The effective error rates Γ e f f x \Gamma_{\mathrm{eff}}^{x} Γ eff x Γ e f f − \Gamma_{\mathrm{eff}}^{-} Γ eff − Γ e f f + \Gamma_{\mathrm{eff}}^{+} Γ eff + Γ e f f z \Gamma_{\mathrm{eff}}^{z} Γ eff z Γ e f f z z \Gamma_{\mathrm{eff}}^{zz} Γ eff z z Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx Γ e f f x z \Gamma_{\mathrm{eff}}^{xz} Γ eff x z λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx λ e f f x z \lambda_{\mathrm{eff}}^{xz} λ eff x z λ e f f z z \lambda_{\mathrm{eff}}^{zz} λ eff z z N N N r C r_C r C λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx N N N σ x \sigma^x σ x N ≥ 8 N \ge8 N ≥ 8 r C r_C r C λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx
( λ e f f x x ) 2 + ( Γ e f f x x ) 2 > Γ ˉ e f f x = Γ e f f x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − . (166) \sqrt{\left(\lambda_{\mathrm{eff}}^{xx}\right)^2+\left(\Gamma_{\mathrm{eff}}^{xx}\right)^2}>\bar{\Gamma}_{\mathrm{eff}}^{x}=\Gamma_{\mathrm{eff}}^{x}+\frac{\Gamma_{\mathrm{eff}}^{+}\Gamma_{\mathrm{eff}}^{-}}{\Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}} . \tag*{(166)} ( λ eff xx ) 2 + ( Γ eff xx ) 2 > Γ ˉ eff x = Γ eff x + Γ eff + + Γ eff − Γ eff + Γ eff − . (166) This is satisfied for N ≥ 8 N\geq8 N ≥ 8 N ≤ 7 N\leq7 N ≤ 7 η < 1 \eta<1 η < 1 Figure 10 . Note that for the parameters here, when N = 0 N=0 N = 0 η C S L \eta_{\mathrm{CSL}} η CSL Γ e f f x x / Γ e f f x \Gamma_{\mathrm{eff}}^{xx}/\Gamma_{\mathrm{eff}}^{x} Γ eff xx / Γ eff x λ e f f x x ≪ Γ e f f x x \lambda_{\mathrm{eff}}^{xx}\ll\Gamma_{\mathrm{eff}}^{xx} λ eff xx ≪ Γ eff xx η e f f ≈ 0.97 \eta_{\mathrm{eff}}\approx0.97 η eff ≈ 0.97 L e f f , C S L \mathcal{L}_{\mathrm{eff,CSL}} L eff , CSL Γ e f f x x ≤ Γ ˉ e f f x \Gamma_{\mathrm{eff}}^{xx}\leq\bar{\Gamma}_{\mathrm{eff}}^{x} Γ eff xx ≤ Γ ˉ eff x N = 0 N=0 N = 0 η C S L \eta_{\mathrm{CSL}} η CSL 1 ≤ N ≤ 6 1\leq N\leq6 1 ≤ N ≤ 6 σ z \sigma^z σ z ∣ Γ e f f z z − i λ e f f z z ∣ / Γ e f f z z \left|\Gamma_{\mathrm{eff}}^{zz}-i\lambda_{\mathrm{eff}}^{zz}\right|/\Gamma_{\mathrm{eff}}^{zz} ∣ Γ eff z z − i λ eff z z ∣ / Γ eff z z N ≥ 8 N\geq8 N ≥ 8 ( λ e f f x x ) 2 + ( Γ e f f x x ) 2 > Γ e f f x + Γ e f f + \sqrt{\left(\lambda_{\mathrm{eff}}^{xx}\right)^2+\left(\Gamma_{\mathrm{eff}}^{xx}\right)^2}>\Gamma_{\mathrm{eff}}^{x}+\Gamma_{\mathrm{eff}}^{+} ( λ eff xx ) 2 + ( Γ eff xx ) 2 > Γ eff x + Γ eff +
As in the GRW discussion, error correction becomes ineffective if the localization length r C r_{\mathrm{C}} r C r C = x 0 = 2 × 10 − 8 m r_{\mathrm{C}}=x_0=2\times10^{-8}\,\mathrm{m} r C = x 0 = 2 × 1 0 − 8 m Figure 7 b. In this regime, the interaction rate λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx N N N σ x \sigma^x σ x Γ e f f x \Gamma_{\mathrm{eff}}^{x} Γ eff x η C S L < 1 \eta_{\mathrm{CSL}}<1 η CSL < 1
Compared with the GRW model, the CSL model introduces an additional nonlocal-like contribution proportional to Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx Figure 7 b. This does not mean the CSL collapse mechanism itself is fundamentally nonlocal. Positivity of the CSL generator implies ∣ Γ e f f x x ∣ ≤ Γ e f f x |\Gamma_{\mathrm{eff}}^{xx}|\leq\Gamma_{\mathrm{eff}}^{x} ∣ Γ eff xx ∣ ≤ Γ eff x (158) . The CSL model is therefore separable by itself, but it can enhance the non-separable features introduced by the Newtonian interaction.
DP model
Finally, we investigate the DP model within the same framework. As will become apparent, the DP model differs qualitatively from the GRW and CSL cases because the effective interaction and the dominant error channels originate from the same underlying kernel.
Without loss of generality, we assume the motion is effectively one-dimensional along the relative axis connecting the two equilibrium positions, and we denote the displacements from equilibrium by operators x A x_A x A x B x_B x B (21) reduces to the following form in the x x x
L D P ( ρ ) = G ℏ ∫ d x A d x B d y A d y B ρ x A x B ; y A y B ∣ x A x B ⟩ ⟨ y A y B ∣ ( m A 2 ∣ x A − y A ∣ Erf ( ∣ x A − y A ∣ 2 σ ) + m B 2 ∣ x B − y B ∣ Erf ( ∣ x B − y B ∣ 2 σ ) − m A 2 π σ − m B 2 π σ + m A m B ∣ d + x A − y B ∣ Erf ( ∣ d + x A − y B ∣ 2 σ ) + m A m B ∣ d + y A − x B ∣ Erf ( ∣ d + y A − x B ∣ 2 σ ) − ( 1 − i ) m A m B ∣ d + x A − x B ∣ Erf ( ∣ d + x A − x B ∣ 2 σ ) − ( 1 + i ) m A m B ∣ d + y A − y B ∣ Erf ( ∣ d + y A − y B ∣ 2 σ ) ) . (167) \mathcal{L}_{\mathrm{DP}}(\rho) = \frac{G}{\hbar} \int\mathrm{d}x_A\mathrm{d}x_B\mathrm{d}y_A\mathrm{d}y_B \rho_{x_Ax_B;y_Ay_B} \lvert x_Ax_B\rangle\langle y_Ay_B\rvert\left( \frac{m_A^2}{\lvert x_A-y_A\rvert}\operatorname{Erf}\left(\frac{\lvert x_A-y_A\rvert}{2\sigma}\right) + \frac{m_B^2}{\lvert x_B-y_B\rvert}\operatorname{Erf}\left(\frac{\lvert x_B-y_B\rvert}{2\sigma}\right) - \frac{m_A^2}{\sqrt{\pi}\sigma} - \frac{m_B^2}{\sqrt{\pi}\sigma} + \frac{m_A m_B}{\lvert d+x_A-y_B\rvert}\operatorname{Erf}\left(\frac{\lvert d+x_A-y_B\rvert}{2\sigma}\right) + \frac{m_A m_B}{\lvert d+y_A-x_B\rvert}\operatorname{Erf}\left(\frac{\lvert d+y_A-x_B\rvert}{2\sigma}\right) - \frac{(1-i)m_A m_B}{\lvert d+x_A-x_B\rvert}\operatorname{Erf}\left(\frac{\lvert d+x_A-x_B\rvert}{2\sigma}\right) - \frac{(1+i)m_A m_B}{\lvert d+y_A-y_B\rvert}\operatorname{Erf}\left(\frac{\lvert d+y_A-y_B\rvert}{2\sigma}\right) \right). \tag*{(167)} L DP ( ρ ) = ℏ G ∫ d x A d x B d y A d y B ρ x A x B ; y A y B ∣ x A x B ⟩ ⟨ y A y B ∣ ( ∣ x A − y A ∣ m A 2 Erf ( 2 σ ∣ x A − y A ∣ ) + ∣ x B − y B ∣ m B 2 Erf ( 2 σ ∣ x B − y B ∣ ) − π σ m A 2 − π σ m B 2 + ∣ d + x A − y B ∣ m A m B Erf ( 2 σ ∣ d + x A − y B ∣ ) + ∣ d + y A − x B ∣ m A m B Erf ( 2 σ ∣ d + y A − x B ∣ ) − ∣ d + x A − x B ∣ ( 1 − i ) m A m B Erf ( 2 σ ∣ d + x A − x B ∣ ) − ∣ d + y A − y B ∣ ( 1 + i ) m A m B Erf ( 2 σ ∣ d + y A − y B ∣ ) ) . (167) In the original DP formulation, the unitary interaction term also includes the regime where the two masses contact, so we keep this expression as the starting point for our effective dynamics analysis.
As recently noted in [34 ] , the DP model can display nonlocal characteristics in the large-smearing regime d < 0.850872 σ d < 0.850872\sigma d < 0.850872 σ N = 0 N=0 N = 0 (167) the nonlocal interaction and the dominant errors share the same distance-dependent kernel. Consequently, increasing the code distance N N N
We now compute the effective error-corrected dynamics for the DP model. We define the local Kraus operators similarly to the CSL construction:
R 0 = Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ , R i = { ∣ 0 ˉ ⟩ ⟨ i ∣ + ∣ 1 ˉ ⟩ ( c i ⟨ 2 N + 1 − i ∣ + d i ⟨ 2 N + 1 + i ∣ ) , 1 ≤ i ≤ N , ∣ 1 ˉ ⟩ ( d i − N ⟨ 3 N + 1 − i ∣ − c i − N ⟨ N + 1 + i ∣ ) , N + 1 ≤ i ≤ 2 N , ∣ 1 ˉ ⟩ ⟨ i + N + 1 ∣ , i > 2 N . (168) R_0=\Pi=|\bar{0}\rangle\langle\bar{0}|+|\bar{1}\rangle\langle\bar{1}|,\qquad R_i=\begin{cases}
|\bar{0}\rangle\langle i|+|\bar{1}\rangle(c_i\langle2N+1-i|+d_i\langle2N+1+i|), & 1\le i\le N,\\
|\bar{1}\rangle(d_{i-N}\langle3N+1-i|-c_{i-N}\langle N+1+i|), & N+1\le i\le2N,\\
|\bar{1}\rangle\langle i+N+1|, & i>2N.
\end{cases} \tag*{(168)} R 0 = Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ , R i = ⎩ ⎨ ⎧ ∣ 0 ˉ ⟩ ⟨ i ∣ + ∣ 1 ˉ ⟩ ( c i ⟨ 2 N + 1 − i ∣ + d i ⟨ 2 N + 1 + i ∣ ) , ∣ 1 ˉ ⟩ ( d i − N ⟨ 3 N + 1 − i ∣ − c i − N ⟨ N + 1 + i ∣ ) , ∣ 1 ˉ ⟩ ⟨ i + N + 1∣ , 1 ≤ i ≤ N , N + 1 ≤ i ≤ 2 N , i > 2 N . (168) where
c k ∝ ∫ d x 1 d x 2 Erf ( ∣ x 1 − x 2 ∣ 2 σ ) ∣ x 1 − x 2 ∣ ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣ 2 N + 1 − k ⟩ , c_k\propto\int dx_1dx_2\frac{\operatorname{Erf}\left(\frac{|x_1-x_2|}{2\sigma}\right)}{|x_1-x_2|}\langle k|x_1\rangle\langle x_1|\bar{0}\rangle\langle\bar{1}|x_2\rangle\langle x_2|2N+1-k\rangle, c k ∝ ∫ d x 1 d x 2 ∣ x 1 − x 2 ∣ Erf ( 2 σ ∣ x 1 − x 2 ∣ ) ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣2 N + 1 − k ⟩ , d k ∝ ∫ d x 1 d x 2 Erf ( ∣ x 1 − x 2 ∣ 2 σ ) ∣ x 1 − x 2 ∣ ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣ 2 N + 1 + k ⟩ , d_k\propto\int dx_1dx_2\frac{\operatorname{Erf}\left(\frac{|x_1-x_2|}{2\sigma}\right)}{|x_1-x_2|}\langle k|x_1\rangle\langle x_1|\bar{0}\rangle\langle\bar{1}|x_2\rangle\langle x_2|2N+1+k\rangle, d k ∝ ∫ d x 1 d x 2 ∣ x 1 − x 2 ∣ Erf ( 2 σ ∣ x 1 − x 2 ∣ ) ⟨ k ∣ x 1 ⟩ ⟨ x 1 ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ x 2 ⟩ ⟨ x 2 ∣2 N + 1 + k ⟩ , c k 2 + d k 2 = 1. (169) c_k^2+d_k^2=1. \tag*{(169)} c k 2 + d k 2 = 1. (169) We then compute the matrix elements of the effective error-corrected Lindbladian in the full two-mass space. By the same symmetry arguments as in the CSL case, the effective two-mass dynamics takes the form, for ρ \rho ρ
L e f f , D P ( ρ ) = R ∘ L D P ( ρ ) = − i [ λ e f f x x σ A x ⊗ σ B x + λ e f f x z σ A x ⊗ σ B z − λ e f f x z σ A z ⊗ σ B x + λ e f f z z σ A z ⊗ σ B z , ρ ] + Γ e f f + ( D [ σ A + ] + D [ σ B + ] ) ( ρ ) + Γ e f f − ( D [ σ A − ] + D [ σ B − ] ) ( ρ ) + Γ e f f x ( D [ σ A x ] + D [ σ B x ] ) ( ρ ) + Γ e f f z ( D [ σ A z ] + D [ σ B z ] ) ( ρ ) + Γ e f f z z ( D [ σ A z − σ B z ] − D [ σ A z ] − D [ σ B z ] ) ( ρ ) + Γ e f f x x ( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ] ) ( ρ ) + Γ e f f x z ( D [ σ A x − σ B z ] − D [ σ A x ] − D [ σ B z ] − D [ σ A z − σ B x ] + D [ σ A z ] + D [ σ B x ] ) ( ρ ) . (170) \begin{aligned}
\mathcal{L}_{\mathrm{eff},\mathrm{DP}}(\rho)&=R\circ\mathcal{L}_{\mathrm{DP}}(\rho)\\
&=-i[\lambda_{\mathrm{eff}}^{xx}\sigma_A^x\otimes\sigma_B^x+\lambda_{\mathrm{eff}}^{xz}\sigma_A^x\otimes\sigma_B^z-\lambda_{\mathrm{eff}}^{xz}\sigma_A^z\otimes\sigma_B^x+\lambda_{\mathrm{eff}}^{zz}\sigma_A^z\otimes\sigma_B^z,\rho]\\
&\quad+\Gamma_{\mathrm{eff}}^+(\mathcal{D}[\sigma_A^+]+\mathcal{D}[\sigma_B^+])(\rho)+\Gamma_{\mathrm{eff}}^-(\mathcal{D}[\sigma_A^-]+\mathcal{D}[\sigma_B^-])(\rho)\\
&\quad+\Gamma_{\mathrm{eff}}^x(\mathcal{D}[\sigma_A^x]+\mathcal{D}[\sigma_B^x])(\rho)+\Gamma_{\mathrm{eff}}^z(\mathcal{D}[\sigma_A^z]+\mathcal{D}[\sigma_B^z])(\rho)\\
&\quad+\Gamma_{\mathrm{eff}}^{zz}(\mathcal{D}[\sigma_A^z-\sigma_B^z]-\mathcal{D}[\sigma_A^z]-\mathcal{D}[\sigma_B^z])(\rho)\\
&\quad+\Gamma_{\mathrm{eff}}^{xx}(\mathcal{D}[\sigma_A^x-\sigma_B^x]-\mathcal{D}[\sigma_A^x]-\mathcal{D}[\sigma_B^x])(\rho)\\
&\quad+\Gamma_{\mathrm{eff}}^{xz}(\mathcal{D}[\sigma_A^x-\sigma_B^z]-\mathcal{D}[\sigma_A^x]-\mathcal{D}[\sigma_B^z]\\
&\qquad-\mathcal{D}[\sigma_A^z-\sigma_B^x]+\mathcal{D}[\sigma_A^z]+\mathcal{D}[\sigma_B^x])(\rho). \tag*{(170)}
\end{aligned} L eff , DP ( ρ ) = R ∘ L DP ( ρ ) = − i [ λ eff xx σ A x ⊗ σ B x + λ eff x z σ A x ⊗ σ B z − λ eff x z σ A z ⊗ σ B x + λ eff z z σ A z ⊗ σ B z , ρ ] + Γ eff + ( D [ σ A + ] + D [ σ B + ]) ( ρ ) + Γ eff − ( D [ σ A − ] + D [ σ B − ]) ( ρ ) + Γ eff x ( D [ σ A x ] + D [ σ B x ]) ( ρ ) + Γ eff z ( D [ σ A z ] + D [ σ B z ]) ( ρ ) + Γ eff z z ( D [ σ A z − σ B z ] − D [ σ A z ] − D [ σ B z ]) ( ρ ) + Γ eff xx ( D [ σ A x − σ B x ] − D [ σ A x ] − D [ σ B x ]) ( ρ ) + Γ eff x z ( D [ σ A x − σ B z ] − D [ σ A x ] − D [ σ B z ] − D [ σ A z − σ B x ] + D [ σ A z ] + D [ σ B x ]) ( ρ ) . (170) In the DP case, the effective coherent coupling is tied to the same kernel responsible for the nonlocal-like σ A x − σ B x \sigma_A^x-\sigma_B^x σ A x − σ B x
λ e f f x x = Γ e f f x x , λ e f f x z = Γ e f f x z , λ e f f z z = Γ e f f z z . (171) \lambda_{\mathrm{eff}}^{xx}=\Gamma_{\mathrm{eff}}^{xx},\qquad
\lambda_{\mathrm{eff}}^{xz}=\Gamma_{\mathrm{eff}}^{xz},\qquad
\lambda_{\mathrm{eff}}^{zz}=\Gamma_{\mathrm{eff}}^{zz}. \tag*{(171)} λ eff xx = Γ eff xx , λ eff x z = Γ eff x z , λ eff z z = Γ eff z z . (171) Following Thm. 1, the non-separability is determined by the separability parameter
η D P = ∥ ( 1 − i ) ( λ e f f x x Γ ˉ e f f x − λ e f f x z Γ e f f z Γ e f f x λ e f f x z Γ e f f z Γ e f f x λ e f f x x Γ e f f x ) ∥ 2 = 2 ∥ ( λ e f f x x Γ ˉ e f f x − λ e f f x z Γ e f f z Γ e f f x λ e f f x z Γ e f f z Γ e f f x λ e f f x x Γ e f f x ) ∥ 2 . (172) \eta_{\mathrm{DP}}=\left\|(1-i)\begin{pmatrix}
\frac{\lambda_{\mathrm{eff}}^{xx}}{\bar{\Gamma}_{\mathrm{eff}}^x}&-\frac{\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^z\Gamma_{\mathrm{eff}}^x}}\\
\frac{\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^z\Gamma_{\mathrm{eff}}^x}}&\frac{\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^x}
\end{pmatrix}\right\|_2
=\sqrt{2}\left\|\begin{pmatrix}
\frac{\lambda_{\mathrm{eff}}^{xx}}{\bar{\Gamma}_{\mathrm{eff}}^x}&-\frac{\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^z\Gamma_{\mathrm{eff}}^x}}\\
\frac{\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^z\Gamma_{\mathrm{eff}}^x}}&\frac{\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^x}
\end{pmatrix}\right\|_2. \tag*{(172)} η DP = ( 1 − i ) Γ ˉ eff x λ eff xx Γ eff z Γ eff x λ eff x z − Γ eff z Γ eff x λ eff x z Γ eff x λ eff xx 2 = 2 Γ ˉ eff x λ eff xx Γ eff z Γ eff x λ eff x z − Γ eff z Γ eff x λ eff x z Γ eff x λ eff xx 2 . (172) A simple sufficient condition for non-separability is η x = ( λ e f f x x ) 2 + ( Γ e f f x x ) 2 / Γ ˉ e f f x = 2 ∣ λ e f f x x ∣ / Γ ˉ e f f x > 1 \eta_x=\sqrt{(\lambda_{\mathrm{eff}}^{xx})^2+(\Gamma_{\mathrm{eff}}^{xx})^2}/\bar{\Gamma}_{\mathrm{eff}}^x=\sqrt{2}|\lambda_{\mathrm{eff}}^{xx}|/\bar{\Gamma}_{\mathrm{eff}}^x>1 η x = ( λ eff xx ) 2 + ( Γ eff xx ) 2 / Γ ˉ eff x = 2 ∣ λ eff xx ∣/ Γ ˉ eff x > 1 η D P < 1 \eta_{\mathrm{DP}}<1 η DP < 1
As an example, we take x 0 = 2 × 10 − 8 m x_0=2\times10^{-8}\,\mathrm{m} x 0 = 2 × 1 0 − 8 m m A = m B = 10 − 21 k g m_A=m_B=10^{-21}\,\mathrm{kg} m A = m B = 1 0 − 21 kg d = 4 × 10 − 7 m d=4\times10^{-7}\,\mathrm{m} d = 4 × 1 0 − 7 m σ = 2 × 10 − 6 m \sigma=2\times10^{-6}\,\mathrm{m} σ = 2 × 1 0 − 6 m Figure 8 a. We find that the σ x \sigma^x σ x N ≥ 1 N\ge1 N ≥ 1 N = 0 N=0 N = 0 N = 0 N=0 N = 0 ( λ e f f x x ) 2 + ( Γ e f f x x ) 2 / ( Γ e f f x + Γ e f f − ) > 1 \sqrt{(\lambda_{\mathrm{eff}}^{xx})^2+(\Gamma_{\mathrm{eff}}^{xx})^2}/(\Gamma_{\mathrm{eff}}^x+\Gamma_{\mathrm{eff}}^-) > 1 ( λ eff xx ) 2 + ( Γ eff xx ) 2 / ( Γ eff x + Γ eff − ) > 1 N ≥ 1 N\ge1 N ≥ 1 λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx N N N η D P ≪ 1 \eta_{\mathrm{DP}}\ll1 η DP ≪ 1
Figure 8. The effective error rates Γ e f f x \Gamma_{\mathrm{eff}}^{x} Γ eff x Γ e f f − \Gamma_{\mathrm{eff}}^{-} Γ eff − Γ e f f + \Gamma_{\mathrm{eff}}^{+} Γ eff + Γ e f f z \Gamma_{\mathrm{eff}}^{z} Γ eff z Γ e f f z z \Gamma_{\mathrm{eff}}^{zz} Γ eff z z Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx Γ e f f x z \Gamma_{\mathrm{eff}}^{xz} Γ eff x z λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx λ e f f x z \lambda_{\mathrm{eff}}^{xz} λ eff x z λ e f f z z \lambda_{\mathrm{eff}}^{zz} λ eff z z N N N σ \sigma σ λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx σ x \sigma^{x} σ x N ≥ 1 N \ge1 N ≥ 1 N = 0 N=0 N = 0 σ \sigma σ λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx σ x \sigma^{x} σ x N N N d ≫ 2 σ d \gg2\sigma d ≫ 2 σ
As a comparison, we also evaluate a small smearing radius σ = r 0 ≈ 4.8 × 10 − 9 m \sigma=r_0\approx4.8\times10^{-9}\,\mathrm{m} σ = r 0 ≈ 4.8 × 1 0 − 9 m Figure 8 b. In this case, the σ x \sigma^x σ x N = 0 N=0 N = 0
We now identify parameters where the DP model becomes non-separable. We restrict to N = 0 N=0 N = 0 x 0 x_0 x 0 d d d σ \sigma σ η \eta η x 0 / σ x_0/\sigma x 0 / σ d / σ d/\sigma d / σ Figure 9 . The maximal value of this ratio is 2 \sqrt{2} 2 ∣ Γ e f f x x ∣ ≤ Γ ˉ e f f x |\Gamma_{\mathrm{eff}}^{xx}|\le\bar{\Gamma}_{\mathrm{eff}}^x ∣ Γ eff xx ∣ ≤ Γ ˉ eff x λ e f f x x = Γ e f f x x \lambda_{\mathrm{eff}}^{xx}=\Gamma_{\mathrm{eff}}^{xx} λ eff xx = Γ eff xx ∣ λ e f f x x ∣ ≤ ∣ Γ ˉ e f f x ∣ |\lambda_{\mathrm{eff}}^{xx}|\le|\bar{\Gamma}_{\mathrm{eff}}^x| ∣ λ eff xx ∣ ≤ ∣ Γ ˉ eff x ∣ Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx D [ x A − x B ] \mathcal{D}[x_A-x_B] D [ x A − x B ] x 0 / σ → 0 x_0/\sigma\to0 x 0 / σ → 0 d / σ ≲ 0.850872 d/\sigma\lesssim0.850872 d / σ ≲ 0.850872 [34 ] . On the other hand, the non-separability increases as x 0 x_0 x 0 x 0 x_0 x 0
Figure 9. The separability parameter η \eta η ( x 0 / σ , d / σ ) (x_0/\sigma,d/\sigma) ( x 0 / σ , d / σ ) η \eta η
Comparison of the effective dynamics and separability classifications
The preceding sections derived the effective logical interactions and error channels associated with the KTM, GRW, CSL, and DP models. For convenience, we compare here the resulting effective dynamics and separability classifications. This subsection provides a compact overview of the main conclusions of the appendix and highlights the qualitative differences between the various models. Readers primarily interested in the final numerical results may use this subsection as a summary of the effective dynamics obtained after error correction.
The relevant separability parameter η \eta η
η = ∥ ( Γ e f f x x − i λ e f f x x Γ e f f x Γ e f f x z − i λ e f f x z Γ e f f x Γ e f f z Γ e f f z x − i λ e f f z x Γ e f f z Γ e f f x Γ e f f z z − i λ e f f z z Γ e f f z ) ∥ 2 (173) \eta= \left\| \begin{pmatrix}
\dfrac{\Gamma_{\mathrm{eff}}^{xx}-i\lambda_{\mathrm{eff}}^{xx}}{\Gamma_{\mathrm{eff}}^{x}} & \dfrac{\Gamma_{\mathrm{eff}}^{xz}-i\lambda_{\mathrm{eff}}^{xz}}{\sqrt{\Gamma_{\mathrm{eff}}^{x}\Gamma_{\mathrm{eff}}^{z}}} \\
\dfrac{\Gamma_{\mathrm{eff}}^{zx}-i\lambda_{\mathrm{eff}}^{zx}}{\sqrt{\Gamma_{\mathrm{eff}}^{z}\Gamma_{\mathrm{eff}}^{x}}} & \dfrac{\Gamma_{\mathrm{eff}}^{zz}-i\lambda_{\mathrm{eff}}^{zz}}{\Gamma_{\mathrm{eff}}^{z}}
\end{pmatrix} \right\|_{2} \tag*{(173)} η = Γ eff x Γ eff xx − i λ eff xx Γ eff z Γ eff x Γ eff z x − i λ eff z x Γ eff x Γ eff z Γ eff x z − i λ eff x z Γ eff z Γ eff z z − i λ eff z z 2 (173) such that η > 1 \eta> 1 η > 1 Figure 10 provides a visual comparison of the corresponding behavior as a function of the code distance. More specifically, these models have different additional properties for simplifying the separability parameter, as summarized in Table I .
Table I. Comparison of the effective dynamics and separability classifications for the models considered in this work.
A central distinction emerges between the GRW/CSL models and the DP model. In the GRW and CSL cases, the dominant correctable error channels and the effective interaction originate from different physical kernels. As a result, increasing the code distance suppresses the effective logical error rates more rapidly than the effective interaction rate, allowing the effective dynamics to become non-separable. In contrast, for the DP model the effective interaction and the dominant error channels originate from the same kernel and are therefore suppressed simultaneously by the QEC procedure. Consequently, increasing the code distance does not improve the effective separability classification for the encoding considered here. The KTM model remains separable by construction, providing a consistency check of the framework.
The comparison presented in Table I and Figure 10 shows that error correction affects the different collapse models in qualitatively distinct ways. While the GRW and CSL models admit parameter regimes in which the effective dynamics becomes non-separable after encoding, the DP model does not exhibit the same enhancement because the effective interaction and dominant error channels remain intrinsically linked. This distinction ultimately determines whether QEC can improve the visibility of non-separable dynamics in the corresponding effective description.
Analytical bounds on local recoverability
In this section we establish analytically the qualitative distinction between the collapse models observed numerically above. For GRW and CSL noise, we show that the separability parameter can be made arbitrarily large by increasing the code distance, while simultaneously keeping the probability of particle contact exponentially small. By contrast, for the Diòsi–Penrose model we derive a finite upper bound on the separability parameter under local encoding and recovery, showing that the unbounded enhancement available for GRW and CSL is not possible. We also clarify why increasing the code distance is essential: at fixed finite code distance, varying only the geometric parameters does not in general produce the same asymptotic enhancement.
Establishing whether entanglement generation can be retained while keeping the contact probability of the two particles below a prescribed threshold, is physically important because configurations with center-to-center separation smaller approaching 2 r 0 2r_0 2 r 0 N N N
Unbounded recoverability for GRW and CSL
We first study the GRW case; the CSL case follows from an equivalent analysis.
Theorem 5. Let us consider the bipartite Fock state code C α = { ∣ 0 ˉ ⟩ α = ∣ 0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣ 2 N + 1 ⟩ α } \mathcal{C}_{\alpha}=\{|\bar{0}\rangle_{\alpha}=|0\rangle_{\alpha},|\bar{1}\rangle_{\alpha}=|2N+1\rangle_{\alpha}\} C α = { ∣ 0 ˉ ⟩ α = ∣0 ⟩ α , ∣ 1 ˉ ⟩ α = ∣2 N + 1 ⟩ α } R R R (150) . The effective error-corrected Lindbladian is then described by Eq. (142) and Eq. (153) . We denote k = 2 N + 1 k=2N+1 k = 2 N + 1 λ G R W \lambda_{\mathrm{GRW}} λ GRW r C r_C r C
x 0 = x 0 , k = y 0 k − 1 / 2 , d = d k = 2 r 0 + l k γ , 0 < γ < 1 4 , (174) x_0=x_{0,k}=y_0k^{-1/2},\qquad d=d_k=2r_0+lk^\gamma,\qquad0<\gamma<\frac{1}{4}, \tag*{(174)} x 0 = x 0 , k = y 0 k − 1/2 , d = d k = 2 r 0 + l k γ , 0 < γ < 4 1 , (174) where y 0 y_0 y 0 r 0 r_0 r 0 l l l k k k 2 r 0 2r_0 2 r 0 o k < C 0 exp ( − C 1 k 1 + 2 γ ) o_k<C_0\exp\left(-C_1k^{1+2\gamma}\right) o k < C 0 exp ( − C 1 k 1 + 2 γ ) C 0 , C 1 > 0 C_0,C_1>0 C 0 , C 1 > 0 log ( ∣ Γ e f f x ∣ / Γ e f f x ) ≥ ( 1 2 − 2 γ ) k log k + C 2 k \log\left(\left|\Gamma_{\mathrm{eff}}^{x}\right|/\Gamma_{\mathrm{eff}}^{x}\right)\geq\left(\frac{1}{2}-2\gamma\right)k\log k+C_2k log ( ∣ Γ eff x ∣ / Γ eff x ) ≥ ( 2 1 − 2 γ ) k log k + C 2 k C 2 C_2 C 2 k → ∞ k\to\infty k → ∞ Γ e f f x = Γ e f f x + ( Γ e f f + + Γ e f f − ) / 2 \Gamma_{\mathrm{eff}}^{x}=\Gamma_{\mathrm{eff}}^{x}+\left(\Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}\right)/2 Γ eff x = Γ eff x + ( Γ eff + + Γ eff − ) /2 Γ e f f x ≥ Γ ~ e f f x = Γ e f f x + Γ e f f + Γ e f f − / ( Γ e f f + + Γ e f f − ) \Gamma_{\mathrm{eff}}^{x}\geq\widetilde{\Gamma}_{\mathrm{eff}}^{x}=\Gamma_{\mathrm{eff}}^{x}+\Gamma_{\mathrm{eff}}^{+}\Gamma_{\mathrm{eff}}^{-}/\left(\Gamma_{\mathrm{eff}}^{+}+\Gamma_{\mathrm{eff}}^{-}\right) Γ eff x ≥ Γ eff x = Γ eff x + Γ eff + Γ eff − / ( Γ eff + + Γ eff − )
Note that x 0 x_0 x 0 k k k ω = ω k = ( ℏ / 2 m y 0 2 ) k \omega=\omega_k=(\hbar/2m y_0^2)k ω = ω k = ( ℏ/2 m y 0 2 ) k k k k k → ∞ k\to\infty k → ∞ k → ∞ k\to\infty k → ∞ ⟨ k ∣ x 2 ∣ k ⟩ = 2 + 1 / k y 0 \sqrt{\langle k|x^2|k\rangle}=\sqrt{2+1/k}y_0 ⟨ k ∣ x 2 ∣ k ⟩ = 2 + 1/ k y 0 k k k k k k ∣ k ⟩ |k\rangle ∣ k ⟩ O ( 1 ) O(1) O ( 1 )
Following Thm. 1, ∣ Γ e f f x ∣ / Γ e f f x |\Gamma_{\mathrm{eff}}^{x}|/\Gamma_{\mathrm{eff}}^{x} ∣ Γ eff x ∣/ Γ eff x η \eta η Γ e f f x ≥ min ( Γ e f f x + Γ e f f + , Γ e f f x + Γ e f f − ) \Gamma_{\mathrm{eff}}^{x}\geq\min(\Gamma_{\mathrm{eff}}^{x}+\Gamma_{\mathrm{eff}}^{+},\Gamma_{\mathrm{eff}}^{x}+\Gamma_{\mathrm{eff}}^{-}) Γ eff x ≥ min ( Γ eff x + Γ eff + , Γ eff x + Γ eff − ) N N N
Proof. The probability of the two particles getting closer than 2 r 0 2r_0 2 r 0 contact probability , is given by:
o k = sup ρ ∈ L ( C α k ⊗ C β k ) Tr ρ 1 ∣ d + x A − x B ∣ ≤ 2 r 0 (175) o_k=\sup_{\rho\in\mathcal{L}(\mathcal{C}_{\alpha}^{k}\otimes\mathcal{C}_{\beta}^{k})}\operatorname{Tr}\rho\mathbb{1}_{|d+x_A-x_B|\leq2r_0} \tag*{(175)} o k = ρ ∈ L ( C α k ⊗ C β k ) sup Tr ρ 1 ∣ d + x A − x B ∣ ≤ 2 r 0 (175) see also Eq. (144) , and we have assumed ρ ≥ 0 \rho\geq0 ρ ≥ 0 Tr ρ = 1 \operatorname{Tr}\rho=1 Tr ρ = 1 ∣ x A ∣ , ∣ x B ∣ < l k γ / 2 |x_A|,|x_B|<lk^\gamma/2 ∣ x A ∣ , ∣ x B ∣ < l k γ /2 ∣ d + x A − x B ∣ > 2 r 0 |d+x_A-x_B|>2r_0 ∣ d + x A − x B ∣ > 2 r 0
o k ≤ 2 sup ∣ ψ ⟩ ∈ C α k ⟨ ψ ∣ 1 ∣ x α ∣ ≥ l k γ / 2 ∣ ψ ⟩ = 2 Pr ∣ k ⟩ α ( ∣ x α ∣ ≥ l k γ / 2 ) o_k\leq2\sup_{|\psi\rangle\in\mathcal{C}_{\alpha}^{k}}\langle\psi|\mathbb{1}_{|x_\alpha|\geq lk^\gamma/2}|\psi\rangle=2\operatorname{Pr}_{|k\rangle_\alpha}(|x_\alpha|\geq lk^\gamma/2) o k ≤ 2 ∣ ψ ⟩ ∈ C α k sup ⟨ ψ ∣ 1 ∣ x α ∣ ≥ l k γ /2 ∣ ψ ⟩ = 2 Pr ∣ k ⟩ α ( ∣ x α ∣ ≥ l k γ /2 ) Note that ( l k γ / 2 ) / x 0 , k = l / ( 2 y 0 ) k 1 / 2 + γ (lk^\gamma/2)/x_{0,k}=l/(2y_0)k^{1/2+\gamma} ( l k γ /2 ) / x 0 , k = l / ( 2 y 0 ) k 1/2 + γ O ( k ) O(\sqrt{k}) O ( k ) k k k ∣ x α ∣ > l k γ / 2 |x_\alpha|>lk^\gamma/2 ∣ x α ∣ > l k γ /2
o k ≤ 2 Pr ∣ k ⟩ α ( ∣ x α ∣ ≥ l k γ / 2 ) ≤ C 0 exp [ − C 1 k 1 + 2 γ ] , (176) o_k \leq2\Pr_{|k\rangle_\alpha}(|x_\alpha|\geq lk^\gamma/2)\leq C_0\exp\left[-C_1k^{1+2\gamma}\right], \tag*{(176)} o k ≤ 2 ∣ k ⟩ α Pr ( ∣ x α ∣ ≥ l k γ /2 ) ≤ C 0 exp [ − C 1 k 1 + 2 γ ] , (176) where C 0 C_0 C 0 C 1 C_1 C 1 k k k
Then let us estimate the scaling of the effective interaction rate. We define the normal modes as b = ( a A + a B ) / 2 b=(a_A+a_B)/\sqrt{2} b = ( a A + a B ) / 2 c = ( a A − a B ) / 2 c=(a_A-a_B)/\sqrt{2} c = ( a A − a B ) / 2
A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = ⟨ v a c ∣ f N G ( ( b † 2 − c † 2 ) k 2 k k ! ) ∣ v a c ⟩ B = c ⟨ 0 ∣ f N G ( ∣ d + x A − x B ∣ ) ( − 1 ) k ( 2 k ) ! 2 k k ! ∣ 2 k ⟩ c , (177) {}_A\langle\bar{0}|{}_B\langle\bar{0}|f_{\mathrm{NG}}(|d+x_A-x_B|)|\bar{1}\rangle_A|\bar{1}\rangle_B
=\langle\mathrm{vac}|f_{\mathrm{NG}}\left(\frac{(b^{\dagger2}-c^{\dagger2})^k}{2^k k!}\right)|\mathrm{vac}\rangle_B
={}_c\langle0|f_{\mathrm{NG}}(|d+x_A-x_B|)(-1)^k\frac{\sqrt{(2k)!}}{2^k k!}|2k\rangle_c, \tag*{(177)} A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f NG ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = ⟨ vac ∣ f NG ( 2 k k ! ( b † 2 − c † 2 ) k ) ∣ vac ⟩ B = c ⟨ 0∣ f NG ( ∣ d + x A − x B ∣ ) ( − 1 ) k 2 k k ! ( 2 k )! ∣2 k ⟩ c , (177) where ∣ 2 k ⟩ c |2k\rangle_c ∣2 k ⟩ c c c c Π k = 1 ∣ x A − x B ∣ ≤ l k γ / 2 \Pi_k=\mathbf{1}_{|x_A-x_B|\leq lk^\gamma/2} Π k = 1 ∣ x A − x B ∣ ≤ l k γ /2 d + x A − x B > 2 r 0 > 0 d+x_A-x_B>2r_0>0 d + x A − x B > 2 r 0 > 0
Π k f N G ( ∣ d + x A − x B ∣ ) Π k = Π k f N G ( d + x A − x B ) Π k , \Pi_kf_{\mathrm{NG}}(|d+x_A-x_B|)\Pi_k=\Pi_kf_{\mathrm{NG}}(d+x_A-x_B)\Pi_k, Π k f NG ( ∣ d + x A − x B ∣ ) Π k = Π k f NG ( d + x A − x B ) Π k , namely we can remove the absolute value. The residual outside this subspace is controlled as follows:
∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ [ f N G ( ∣ d + x A − x B ∣ ) − Π k f N G ( ∣ d + x A − x B ∣ ) Π k ] ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ = ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) f N G ( ∣ d + x A − x B ∣ ) ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ ≤ ∥ f N G ∥ ∞ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ≤ G m A m B 2 r 0 A ⟨ k ∣ B ⟨ k ∣ ( 1 − Π k ) ∣ k ⟩ A ∣ k ⟩ B ≤ C 2 exp ( − C 3 k 1 + 2 γ ) , (178) \begin{aligned}
&\left|{}_A\langle\bar{0}|{}_B\langle\bar{0}|\left[f_{\mathrm{NG}}(|d+x_A-x_B|)-\Pi_kf_{\mathrm{NG}}(|d+x_A-x_B|)\Pi_k\right]|\bar{1}\rangle_A|\bar{1}\rangle_B\right|\\
&=\left|{}_A\langle\bar{0}|{}_B\langle\bar{0}|(1-\Pi_k)f_{\mathrm{NG}}(|d+x_A-x_B|)(1-\Pi_k)|\bar{1}\rangle_A|\bar{1}\rangle_B\right|\\
&\leq\|f_{\mathrm{NG}}\|_\infty\sqrt{{}_A\langle\bar{0}|{}_B\langle\bar{0}|(1-\Pi_k)|\bar{0}\rangle_A|\bar{0}\rangle_B{}_A\langle\bar{1}|{}_B\langle\bar{1}|(1-\Pi_k)|\bar{1}\rangle_A|\bar{1}\rangle_B}\\
&\leq\frac{Gm_Am_B}{2r_0}\sqrt{{}_A\langle k|{}_B\langle k|(1-\Pi_k)|k\rangle_A|k\rangle_B}\\
&\leq C_2\exp\left(-C_3k^{1+2\gamma}\right), \tag*{(178)}
\end{aligned} ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ [ f NG ( ∣ d + x A − x B ∣ ) − Π k f NG ( ∣ d + x A − x B ∣ ) Π k ] ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ = ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) f NG ( ∣ d + x A − x B ∣ ) ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ ≤ ∥ f NG ∥ ∞ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) ∣ 0 ˉ ⟩ A ∣ 0 ˉ ⟩ B A ⟨ 1 ˉ ∣ B ⟨ 1 ˉ ∣ ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ≤ 2 r 0 G m A m B A ⟨ k ∣ B ⟨ k ∣ ( 1 − Π k ) ∣ k ⟩ A ∣ k ⟩ B ≤ C 2 exp ( − C 3 k 1 + 2 γ ) , (178) where for the first inequality we have used ∣ f N G ∣ ≤ ∥ f N G ∥ ∞ |f_{\mathrm{NG}}|\leq\|f_{\mathrm{NG}}\|_\infty ∣ f NG ∣ ≤ ∥ f NG ∥ ∞ o k o_k o k C 2 , C 3 C_2,C_3 C 2 , C 3
At the same time, let us also consider the complementary residual
∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) 1 d + x A − x B ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ . (179) \left|{}_A\langle\bar{0}|{}_B\langle\bar{0}|(1-\Pi_k)\frac{1}{d+x_A-x_B}(1-\Pi_k)|\bar{1}\rangle_A|\bar{1}\rangle_B\right|. \tag*{(179)} A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) d + x A − x B 1 ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B . (179) In the region that x A − x B > l k γ / 2 x_A-x_B>lk^\gamma/2 x A − x B > l k γ /2 − d k + 2 x 0 , k < x A − x B < − l k γ / 2 -d_k+\sqrt{2}x_{0,k}<x_A-x_B<-lk^\gamma/2 − d k + 2 x 0 , k < x A − x B < − l k γ /2 x A − x B < − d − 2 x 0 , k x_A-x_B<-d-\sqrt{2}x_{0,k} x A − x B < − d − 2 x 0 , k ∣ 1 / ( d + x A − x B ) ∣ ≤ 1 / 2 x 0 , k |1/(d+x_A-x_B)|\leq1/\sqrt{2}x_{0,k} ∣1/ ( d + x A − x B ) ∣ ≤ 1/ 2 x 0 , k x A − x B = − d x_A-x_B=-d x A − x B = − d
( 2 k ) ! 2 k k ! ∫ − d / 2 x 0 + 1 − d / 2 x 0 − 1 d s c ⟨ 0 ∣ s ⟩ 1 d + 2 x 0 s ⟨ s ∣ 2 k ⟩ c = ( 2 k ) ! 2 k k ! ∫ − 1 1 d s h k ( − d 2 x 0 + s ) 2 x 0 s , where h k ( s ) = H e 2 k ( s ) 2 ( 2 k ) ! π e − s 2 / 2 2 ( 2 k ) ! π (180) \frac{\sqrt{(2k)!}}{2^k k!}\int_{-d/\sqrt{2}x_{0}+1}^{-d/\sqrt{2}x_{0}-1}ds\,{}_c\langle0|s\rangle\frac{1}{d+\sqrt{2}x_{0}s}\langle s|2k\rangle_c
=\frac{\sqrt{(2k)!}}{2^k k!}\int_{-1}^{1}ds\,\frac{h_k\left(-\frac{d}{\sqrt{2}x_0}+s\right)}{\sqrt{2}x_0s},
\quad\text{where }h_k(s)=\frac{\mathrm{He}_{2k}(s)}{\sqrt{2(2k)!\pi}}\frac{e^{-s^2/2}}{\sqrt{2(2k)!\pi}} \tag*{(180)} 2 k k ! ( 2 k )! ∫ − d / 2 x 0 + 1 − d / 2 x 0 − 1 d s c ⟨ 0∣ s ⟩ d + 2 x 0 s 1 ⟨ s ∣2 k ⟩ c = 2 k k ! ( 2 k )! ∫ − 1 1 d s 2 x 0 s h k ( − 2 x 0 d + s ) , where h k ( s ) = 2 ( 2 k )! π He 2 k ( s ) 2 ( 2 k )! π e − s 2 /2 (180) where H e m \mathrm{He}_m He m 2 sup ∣ s ∣ ≤ 1 ∣ h k ′ ( − d / 2 x 0 + s ) ∣ / x 0 \sqrt{2}\sup_{|s|\leq1}|h_k'(-d/\sqrt{2}x_0+s)|/x_0 2 sup ∣ s ∣ ≤ 1 ∣ h k ′ ( − d / 2 x 0 + s ) ∣/ x 0 ( 2 k ) ! ≤ 2 k k ! \sqrt{(2k)!}\leq2^k k! ( 2 k )! ≤ 2 k k ! d k / 2 x 0 , k ∝ k 1 / 2 + γ d_k/\sqrt{2}x_{0,k}\propto k^{1/2+\gamma} d k / 2 x 0 , k ∝ k 1/2 + γ h k ′ ( − d / 2 x 0 + s ) h_k'(-d/\sqrt{2}x_0+s) h k ′ ( − d / 2 x 0 + s )
log ∣ h k ′ ( − d / 2 x 0 + s ) ∣ ≤ − d k 2 4 x 0 , k 2 + O ( k log d k 2 x 0 , k ) ≤ log ( C 4 ) − C 5 k 1 + 2 γ \log|h_k'(-d/\sqrt{2}x_0+s)|\leq-\frac{d_k^2}{4x_{0,k}^2}+O\left(k\log\frac{d_k}{\sqrt{2}x_{0,k}}\right)\leq\log(C_4)-C_5k^{1+2\gamma} log ∣ h k ′ ( − d / 2 x 0 + s ) ∣ ≤ − 4 x 0 , k 2 d k 2 + O ( k log 2 x 0 , k d k ) ≤ log ( C 4 ) − C 5 k 1 + 2 γ with constants C 4 , C 5 C_4,C_5 C 4 , C 5 ∣ s ∣ ≤ 1 |s|\leq1 ∣ s ∣ ≤ 1 x A − x B > l k γ / 2 x_A-x_B>lk^\gamma/2 x A − x B > l k γ /2 x A − x B < − d − 2 x 0 , k x_A-x_B<-d-\sqrt{2}x_{0,k} x A − x B < − d − 2 x 0 , k
In this situation, the effective interaction is bounded as
∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f N G ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ = ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ Π k f N G ( ∣ d + x A − x B ∣ ) Π k ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B + A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) f N G ( ∣ d + x A − x B ∣ ) ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ ≥ ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ Π k f N G ( d + x A − x B ) Π k ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ − C 2 e − C 3 k 1 + 2 γ ≥ ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ G m A m B d + x A − x B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ − C 2 e − C 3 k 1 + 2 γ − G m A m B 2 x 0 , k C 4 e − C 5 k 1 + 2 γ . (181) \begin{aligned}
&\left|{}_A\langle\bar{0}|{}_B\langle\bar{0}|f_{\mathrm{NG}}(|d+x_A-x_B|)|\bar{1}\rangle_A|\bar{1}\rangle_B\right|\\
&=\left|{}_A\langle\bar{0}|{}_B\langle\bar{0}|\Pi_kf_{\mathrm{NG}}(|d+x_A-x_B|)\Pi_k|\bar{1}\rangle_A|\bar{1}\rangle_B
+{}_A\langle\bar{0}|{}_B\langle\bar{0}|(1-\Pi_k)f_{\mathrm{NG}}(|d+x_A-x_B|)(1-\Pi_k)|\bar{1}\rangle_A|\bar{1}\rangle_B\right|\\
&\geq\left|{}_A\langle\bar{0}|{}_B\langle\bar{0}|\Pi_kf_{\mathrm{NG}}(d+x_A-x_B)\Pi_k|\bar{1}\rangle_A|\bar{1}\rangle_B\right|-C_2e^{-C_3k^{1+2\gamma}}\\
&\geq\left|{}_A\langle\bar{0}|{}_B\langle\bar{0}|\frac{Gm_Am_B}{d+x_A-x_B}|\bar{1}\rangle_A|\bar{1}\rangle_B\right|-C_2e^{-C_3k^{1+2\gamma}}-\frac{Gm_Am_B}{\sqrt{2}x_{0,k}}C_4e^{-C_5k^{1+2\gamma}}. \tag*{(181)}
\end{aligned} ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ f NG ( ∣ d + x A − x B ∣ ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ = ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ Π k f NG ( ∣ d + x A − x B ∣ ) Π k ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B + A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ ( 1 − Π k ) f NG ( ∣ d + x A − x B ∣ ) ( 1 − Π k ) ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ ≥ ∣ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ Π k f NG ( d + x A − x B ) Π k ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B ∣ − C 2 e − C 3 k 1 + 2 γ ≥ A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ d + x A − x B G m A m B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B − C 2 e − C 3 k 1 + 2 γ − 2 x 0 , k G m A m B C 4 e − C 5 k 1 + 2 γ . (181) Then let us evaluate the matrix element of 1 / ( d + x A − x B ) 1/(d+x_A-x_B) 1/ ( d + x A − x B )
1 d + 2 x 0 ( c + c † ) = 1 2 x 0 lim ϵ → 0 + ∫ 0 ∞ e − ϵ u sin u ( d / 2 x 0 + ( c + c † ) ) d u . \frac{1}{d+\sqrt{2}x_0(c+c^\dagger)}=\frac{1}{\sqrt{2}x_0}\lim_{\epsilon\to0^+}\int_0^\infty e^{-\epsilon u}\sin u\left(d/\sqrt{2}x_0+(c+c^\dagger)\right)\,\mathrm{d}u. d + 2 x 0 ( c + c † ) 1 = 2 x 0 1 ϵ → 0 + lim ∫ 0 ∞ e − ϵ u sin u ( d / 2 x 0 + ( c + c † ) ) d u . Here and below the inverse should be understood in a principal value sense. Since ∣ 0 ⟩ c \lvert0\rangle_c ∣ 0 ⟩ c ∣ 2 k ⟩ c \lvert2k\rangle_c ∣ 2 k ⟩ c sin u ( c + c † ) \sin u(c+c^\dagger) sin u ( c + c † ) 0 0 0
c ⟨ 0 ∣ sin u ( d / 2 x 0 + ( c + c † ) ) ∣ 2 k ⟩ c = c ⟨ 0 ∣ sin u d / 2 x 0 cos u ( c + c † ) ∣ 2 k ⟩ c = sin u d / 2 x 0 c ⟨ 0 ∣ e i u ( c + c † ) ∣ 2 k ⟩ c = sin u d 2 x 0 ( − 1 ) k u 2 k e − u 2 / 2 ( 2 k ) ! . (182) \begin{aligned}
{}_c\langle0\lvert\sin u\left(d/\sqrt{2}x_0+(c+c^\dagger)\right)\rvert2k\rangle_c={}_c\langle0\lvert\sin ud/\sqrt{2}x_0\cos u(c+c^\dagger)\rvert2k\rangle_c \\
&=\sin ud/\sqrt{2}x_0{}_c\langle0\lvert e^{iu(c+c^\dagger)}\rvert2k\rangle_c \\
&=\sin\frac{ud}{\sqrt{2}x_0}(-1)^k\frac{u^{2k}e^{-u^2/2}}{\sqrt{(2k)!}}. \tag*{(182)}
\end{aligned} c ⟨ 0 ∣ sin u ( d / 2 x 0 + ( c + c † ) ) ∣ 2 k ⟩ c = c ⟨ 0 ∣ sin u d / 2 x 0 cos u ( c + c † )∣ 2 k ⟩ c = sin u d / 2 x 0 c ⟨ 0 ∣ e i u ( c + c † ) ∣ 2 k ⟩ c = sin 2 x 0 u d ( − 1 ) k ( 2 k )! u 2 k e − u 2 /2 . (182) The effective interaction rate then reads as
A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ 1 d + x A − x B ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = 1 2 k k ! 1 2 x 0 , k lim ϵ → 0 + ∫ 0 ∞ e − ϵ u u 2 k e − u 2 / 2 sin u d k 2 x 0 , k d u = 1 2 k 2 k ! x 0 , k ∫ 0 ∞ u 2 k e − u 2 / 2 sin u d k 2 x 0 , k d u , (183) \begin{aligned}
{}_A\langle\bar{0}\lvert{}_B\langle\bar{0}\lvert\frac{1}{d+x_A-x_B}\rvert\bar{1}\rangle_A\rvert\bar{1}\rangle_B=\frac{1}{2^k k!}\frac{1}{\sqrt{2}x_{0,k}}\lim_{\epsilon\to0^+}\int_0^\infty e^{-\epsilon u}u^{2k}e^{-u^2/2}\sin\frac{ud_k}{\sqrt{2}x_{0,k}}\,\mathrm{d}u \\
&=\frac{1}{2^k\sqrt{2k!}x_{0,k}}\int_0^\infty u^{2k}e^{-u^2/2}\sin\frac{ud_k}{\sqrt{2}x_{0,k}}\,\mathrm{d}u, \tag*{(183)}
\end{aligned} A ⟨ 0 ˉ ∣ B ⟨ 0 ˉ ∣ d + x A − x B 1 ∣ 1 ˉ ⟩ A ∣ 1 ˉ ⟩ B = 2 k k ! 1 2 x 0 , k 1 ϵ → 0 + lim ∫ 0 ∞ e − ϵ u u 2 k e − u 2 /2 sin 2 x 0 , k u d k d u = 2 k 2 k ! x 0 , k 1 ∫ 0 ∞ u 2 k e − u 2 /2 sin 2 x 0 , k u d k d u , (183) where the limit and the integral are interchangeable in the second equality, since the Gaussian integral is finite. Let us denote v k = d k / ( 2 x 0 , k ) v_k=d_k/(\sqrt{2}x_{0,k}) v k = d k / ( 2 x 0 , k ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) [ 0 + i v k , ∞ + i v k ) [0+iv_k,\infty+iv_k) [ 0 + i v k , ∞ + i v k )
H k ( v ) = ∫ 0 ∞ u 2 k e − u 2 / 2 sin ( v u ) d u = ( − 1 ) k I k ( v ) + J k ( v ) , H_k(v)=\int_0^\infty u^{2k}e^{-u^2/2}\sin(vu)\,\mathrm{d}u=(-1)^kI_k(v)+J_k(v), H k ( v ) = ∫ 0 ∞ u 2 k e − u 2 /2 sin ( v u ) d u = ( − 1 ) k I k ( v ) + J k ( v ) , where I k ( v ) = ∫ 0 v x 2 k e x 2 / 2 − v x d x , J k ( v ) = e − v 2 / 2 Im ∫ 0 ∞ ( x + i v ) 2 k e − x 2 / 2 d x . (184) \text{where }\ I_k(v)=\int_0^v x^{2k}e^{x^2/2-vx}\,\mathrm{d}x,\qquad J_k(v)=e^{-v^2/2}\operatorname{Im}\int_0^\infty(x+iv)^{2k}e^{-x^2/2}\,\mathrm{d}x. \tag*{(184)} where I k ( v ) = ∫ 0 v x 2 k e x 2 /2 − v x d x , J k ( v ) = e − v 2 /2 Im ∫ 0 ∞ ( x + i v ) 2 k e − x 2 /2 d x . (184) The first integral has a lower bound as follows:
I k ( v ) = ∫ 0 v x 2 k e x 2 / 2 − v x d x = 1 v 2 k + 1 ∫ 0 v 2 t 2 k e t 2 / ( 2 v 2 ) − t d t ≥ 1 v 2 k + 1 ∫ 0 v 2 t 2 k e − t d t = ( 2 k ) ! v 2 k + 1 − 1 v 2 k + 1 ∫ v 2 ∞ t 2 k e − t d t ≥ ( 2 k ) ! v 2 k + 1 − e − v 2 / 2 v 2 k + 1 ∫ 0 ∞ t 2 k e − t / 2 d t = ( 2 k ) ! v 2 k + 1 ( 1 − 2 2 k + 1 e − v 2 / 2 ) . (185) \begin{aligned}
I_k(v)=\int_0^v x^{2k}e^{x^2/2-vx}\,\mathrm{d}x \\
&=\frac{1}{v^{2k+1}}\int_0^{v^2}t^{2k}e^{t^2/(2v^2)-t}\,\mathrm{d}t \\
&\ge\frac{1}{v^{2k+1}}\int_0^{v^2}t^{2k}e^{-t}\,\mathrm{d}t \\
&=\frac{(2k)!}{v^{2k+1}}-\frac{1}{v^{2k+1}}\int_{v^2}^\infty t^{2k}e^{-t}\,\mathrm{d}t \\
&\ge\frac{(2k)!}{v^{2k+1}}-\frac{e^{-v^2/2}}{v^{2k+1}}\int_0^\infty t^{2k}e^{-t/2}\,\mathrm{d}t \\
&=\frac{(2k)!}{v^{2k+1}}\left(1-2^{2k+1}e^{-v^2/2}\right). \tag*{(185)}
\end{aligned} I k ( v ) = ∫ 0 v x 2 k e x 2 /2 − v x d x = v 2 k + 1 1 ∫ 0 v 2 t 2 k e t 2 / ( 2 v 2 ) − t d t ≥ v 2 k + 1 1 ∫ 0 v 2 t 2 k e − t d t = v 2 k + 1 ( 2 k )! − v 2 k + 1 1 ∫ v 2 ∞ t 2 k e − t d t ≥ v 2 k + 1 ( 2 k )! − v 2 k + 1 e − v 2 /2 ∫ 0 ∞ t 2 k e − t /2 d t = v 2 k + 1 ( 2 k )! ( 1 − 2 2 k + 1 e − v 2 /2 ) . (185) Since v ∝ k γ + 1 / 2 v\propto k^{\gamma+1/2} v ∝ k γ + 1/2 k k k v 2 ≥ 4 k v^2\ge4k v 2 ≥ 4 k 2 2 k + 1 e − v 2 / 2 ≤ 2 ( 2 / e ) 2 k ≪ 1 2^{2k+1}e^{-v^2/2}\le2(2/e)^{2k}\ll1 2 2 k + 1 e − v 2 /2 ≤ 2 ( 2/ e ) 2 k ≪ 1
∣ J k ( v ) ∣ ≤ e − v 2 / 2 ∫ 0 ∞ ( x 2 + v 2 ) k e − x 2 / 2 d x = e − v 2 / 2 v 2 k ∫ 0 ∞ ( x 2 / v 2 + 1 ) k e − x 2 / 2 d x < e − v 2 / 2 v 2 k ∫ 0 ∞ e k x 2 / v 2 e − x 2 / 2 d x ≤ π e − v 2 / 2 v 2 k . (186) \begin{aligned}
\lvert J_k(v)\rvert\le e^{-v^2/2}\int_0^\infty(x^2+v^2)^k e^{-x^2/2}\,\mathrm{d}x \\
&=e^{-v^2/2}v^{2k}\int_0^\infty(x^2/v^2+1)^k e^{-x^2/2}\,\mathrm{d}x \\
&<e^{-v^2/2}v^{2k}\int_0^\infty e^{kx^2/v^2}e^{-x^2/2}\,\mathrm{d}x \\
&\le\sqrt{\pi}e^{-v^2/2}v^{2k}. \tag*{(186)}
\end{aligned} ∣ J k ( v )∣ ≤ e − v 2 /2 ∫ 0 ∞ ( x 2 + v 2 ) k e − x 2 /2 d x = e − v 2 /2 v 2 k ∫ 0 ∞ ( x 2 / v 2 + 1 ) k e − x 2 /2 d x < e − v 2 /2 v 2 k ∫ 0 ∞ e k x 2 / v 2 e − x 2 /2 d x ≤ π e − v 2 /2 v 2 k . (186) Therefore, their ratio
log ∣ J k I k ∣ ≤ log π v 4 k + 1 e − v 2 / 2 ( 2 k ) ! ( 1 − 2 2 k + 1 e − v 2 / 2 ) = − l 2 4 y 0 2 k 1 + 2 γ − r 0 l y 0 2 k 1 + γ + 4 γ k log ( k ) + O ( k ) , (187) \log\left|\frac{J_k}{I_k}\right| \le\log\sqrt{\pi}\frac{v^{4k+1}e^{-v^2/2}}{(2k)!\left(1-2^{2k+1}e^{-v^2/2}\right)}=-\frac{l^2}{4y_0^2}k^{1+2\gamma}-\frac{r_0l}{y_0^2}k^{1+\gamma}+4\gamma k\log(k)+O(k), \tag*{(187)} log I k J k ≤ log π ( 2 k )! ( 1 − 2 2 k + 1 e − v 2 /2 ) v 4 k + 1 e − v 2 /2 = − 4 y 0 2 l 2 k 1 + 2 γ − y 0 2 r 0 l k 1 + γ + 4 γ k log ( k ) + O ( k ) , (187) goes to − ∞ -\infty − ∞ k k k C 6 C_6 C 6 ∣ H k ( v k ) ∣ ≥ C 6 ( 2 k ) ! / v k 2 k + 1 |H_k(v_k)| \ge C_6(2k)!/v_k^{2k+1} ∣ H k ( v k ) ∣ ≥ C 6 ( 2 k )! / v k 2 k + 1
∣ λ e f f ∣ ≥ G m A m B ℏ 2 k 2 k ! x 0 , k × C 6 ( 2 k ) ! v k 2 k + 1 − C 2 ℏ e − C 3 k 1 + 2 γ − G m A m B 2 ℏ x 0 , k C 4 e − C 5 k 1 + 2 γ ≥ C 6 G m A m B ℏ ( 2 k ) ! x 0 , k 2 k k ! d k 2 k + 1 − C 2 ℏ e − C 3 k 1 + 2 γ − G m A m B 2 x 0 , k ℏ C 4 e − C 5 k 1 + 2 γ , (188) \begin{aligned}
|\lambda_{\mathrm{eff}}| &\ge\frac{Gm_A m_B}{\hbar2^k\sqrt{2k!}x_{0,k}}\times C_6\frac{(2k)!}{v_k^{2k+1}}-\frac{C_2}{\hbar}e^{-C_3k^{1+2\gamma}}-\frac{Gm_A m_B}{\sqrt{2}\hbar x_{0,k}}C_4e^{-C_5k^{1+2\gamma}}\\
&\ge C_6\frac{Gm_A m_B}{\hbar}\frac{(2k)!x_{0,k}^{2k}}{k!d_k^{2k+1}}-\frac{C_2}{\hbar}e^{-C_3k^{1+2\gamma}}-\frac{Gm_A m_B}{\sqrt{2}x_{0,k}\hbar}C_4e^{-C_5k^{1+2\gamma}}, \tag*{(188)}
\end{aligned} ∣ λ eff ∣ ≥ ℏ 2 k 2 k ! x 0 , k G m A m B × C 6 v k 2 k + 1 ( 2 k )! − ℏ C 2 e − C 3 k 1 + 2 γ − 2 ℏ x 0 , k G m A m B C 4 e − C 5 k 1 + 2 γ ≥ C 6 ℏ G m A m B k ! d k 2 k + 1 ( 2 k )! x 0 , k 2 k − ℏ C 2 e − C 3 k 1 + 2 γ − 2 x 0 , k ℏ G m A m B C 4 e − C 5 k 1 + 2 γ , (188) Therefore, we conclude the scaling of the effective interaction rate as
log ∣ λ e f f ∣ ≥ log ( 2 k ) ! k ! + log x 0 , k 2 k d k 2 k + 1 + O ( 1 ) ≥ − 2 γ k log ( k ) − C 7 k , (189) \log|\lambda_{\mathrm{eff}}| \ge\log\frac{(2k)!}{k!}+\log\frac{x_{0,k}^{2k}}{d_k^{2k+1}}+O(1)\ge-2\gamma k\log(k)-C_7k, \tag*{(189)} log ∣ λ eff ∣ ≥ log k ! ( 2 k )! + log d k 2 k + 1 x 0 , k 2 k + O ( 1 ) ≥ − 2 γ k log ( k ) − C 7 k , (189) where C 7 C_7 C 7
Finally, let us study the scaling behavior of the effective error rate. Equivalently, one can rewrite the dimensionless GRW decoherence channel with respect to mode a a a
Φ κ k ( ρ ) = ∫ − ∞ ∞ d η π κ k e − η 2 / κ k e i η ( a + a † ) ρ e − i η ( a + a † ) , where κ k = x 0 , k 2 r C 2 = y 0 2 r C 2 1 k . (190) \Phi_{\kappa_k}(\rho)=\int_{-\infty}^{\infty}\frac{\mathrm{d}\eta}{\sqrt{\pi\kappa_k}}e^{-\eta^2/\kappa_k}e^{i\eta(a+a^\dagger)}\rho e^{-i\eta(a+a^\dagger)},\qquad\text{where}\quad\kappa_k=\frac{x_{0,k}^2}{r_C^2}=\frac{y_0^2}{r_C^2}\frac{1}{k}. \tag*{(190)} Φ κ k ( ρ ) = ∫ − ∞ ∞ π κ k d η e − η 2 / κ k e i η ( a + a † ) ρ e − i η ( a + a † ) , where κ k = r C 2 x 0 , k 2 = r C 2 y 0 2 k 1 . (190) Then the transition probability ⟨ m ∣ Φ κ k ( ∣ l ⟩ ⟨ l ∣ ) ∣ m ⟩ \langle m|\Phi_{\kappa_k}(|l\rangle\langle l|)|m\rangle ⟨ m ∣ Φ κ k ( ∣ l ⟩ ⟨ l ∣ ) ∣ m ⟩ (152) can be calculated with the generation functional as follows:
F ( s , t ) = ∑ m , l = 0 ∞ ⟨ m ∣ Φ κ k ( ∣ l ⟩ ⟨ l ∣ ) ∣ m ⟩ s m t l = ∫ − ∞ ∞ ∑ m , l = 0 ∞ d η π κ k e − η 2 / κ k Tr [ s m ∣ m ⟩ ⟨ m ∣ e i η ( a + a † ) ∣ l ⟩ ⟨ l ∣ e − i η ( a + a † ) t l ] = ∫ − ∞ ∞ d η π κ k e − η 2 / κ k Tr [ s a † a e i η ( a + a † ) t a † a e − i η ( a + a † ) ] = ∫ − ∞ ∞ d η π κ k e − η 2 / κ k 1 1 − s t exp [ − η 2 ( 1 − s ) ( 1 − t ) 1 − s t ] = 1 ( 1 − s t ) [ 1 − s t + κ k ( 1 − s ) ( 1 − t ) ] . (191) \begin{aligned}
F(s,t)&=\sum_{m,l=0}^{\infty}\langle m|\Phi_{\kappa_k}(|l\rangle\langle l|)|m\rangle s^m t^l\\
&=\int_{-\infty}^{\infty}\sum_{m,l=0}^{\infty}\frac{\mathrm{d}\eta}{\sqrt{\pi\kappa_k}}e^{-\eta^2/\kappa_k}\operatorname{Tr}[s^m|m\rangle\langle m|e^{i\eta(a+a^\dagger)}|l\rangle\langle l|e^{-i\eta(a+a^\dagger)}t^l]\\
&=\int_{-\infty}^{\infty}\frac{\mathrm{d}\eta}{\sqrt{\pi\kappa_k}}e^{-\eta^2/\kappa_k}\operatorname{Tr}[s^{a^\dagger a}e^{i\eta(a+a^\dagger)}t^{a^\dagger a}e^{-i\eta(a+a^\dagger)}]\\
&=\int_{-\infty}^{\infty}\frac{\mathrm{d}\eta}{\sqrt{\pi\kappa_k}}e^{-\eta^2/\kappa_k}\frac{1}{1-st}\exp\left[-\eta^2\frac{(1-s)(1-t)}{1-st}\right]\\
&=\frac{1}{\sqrt{(1-st)[1-st+\kappa_k(1-s)(1-t)]}}. \tag*{(191)}
\end{aligned} F ( s , t ) = m , l = 0 ∑ ∞ ⟨ m ∣ Φ κ k ( ∣ l ⟩ ⟨ l ∣ ) ∣ m ⟩ s m t l = ∫ − ∞ ∞ m , l = 0 ∑ ∞ π κ k d η e − η 2 / κ k Tr [ s m ∣ m ⟩ ⟨ m ∣ e i η ( a + a † ) ∣ l ⟩ ⟨ l ∣ e − i η ( a + a † ) t l ] = ∫ − ∞ ∞ π κ k d η e − η 2 / κ k Tr [ s a † a e i η ( a + a † ) t a † a e − i η ( a + a † ) ] = ∫ − ∞ ∞ π κ k d η e − η 2 / κ k 1 − s t 1 exp [ − η 2 1 − s t ( 1 − s ) ( 1 − t ) ] = ( 1 − s t ) [ 1 − s t + κ k ( 1 − s ) ( 1 − t )] 1 . (191) The error probability of logical ∣ 0 ˉ ⟩ |\bar{0}\rangle ∣ 0 ˉ ⟩ ∣ 1 ˉ ⟩ |\bar{1}\rangle ∣ 1 ˉ ⟩
a k = ∑ m = N + 1 ∞ ⟨ m ∣ Φ κ k ( ∣ 0 ⟩ ⟨ 0 ∣ ) ∣ m ⟩ = ∑ m = N + 1 ∞ 1 m ! ( d d s ) m F ( s , t ) ∣ s = t = 0 = ∑ m = N + 1 ∞ ( 2 m ) ! m ! m ! 4 m κ k m ( 1 + κ k ) m + 1 / 2 ≤ ∑ m = N + 1 ∞ ( 1 + 1 ) 2 m 4 m κ k m ( 1 + κ k ) m + 1 / 2 = 1 + κ k ( κ k 1 + κ k ) N + 1 ≤ C a κ k N + 1 , (192) \begin{aligned}
a_k&=\sum_{m=N+1}^{\infty}\langle m|\Phi_{\kappa_k}(|0\rangle\langle0|)|m\rangle\\
&=\sum_{m=N+1}^{\infty}\frac{1}{m!}\left(\frac{\mathrm{d}}{\mathrm{d}s}\right)^mF(s,t)\bigg|_{s=t=0}\\
&=\sum_{m=N+1}^{\infty}\frac{(2m)!}{m!m!4^m}\frac{\kappa_k^m}{(1+\kappa_k)^{m+1/2}}\\
&\le\sum_{m=N+1}^{\infty}\frac{(1+1)^{2m}}{4^m}\frac{\kappa_k^m}{(1+\kappa_k)^{m+1/2}}\\
&=\sqrt{1+\kappa_k}\left(\frac{\kappa_k}{1+\kappa_k}\right)^{N+1}\\
&\le C_a\kappa_k^{N+1}, \tag*{(192)}
\end{aligned} a k = m = N + 1 ∑ ∞ ⟨ m ∣ Φ κ k ( ∣0 ⟩ ⟨ 0∣ ) ∣ m ⟩ = m = N + 1 ∑ ∞ m ! 1 ( d s d ) m F ( s , t ) s = t = 0 = m = N + 1 ∑ ∞ m ! m ! 4 m ( 2 m )! ( 1 + κ k ) m + 1/2 κ k m ≤ m = N + 1 ∑ ∞ 4 m ( 1 + 1 ) 2 m ( 1 + κ k ) m + 1/2 κ k m = 1 + κ k ( 1 + κ k κ k ) N + 1 ≤ C a κ k N + 1 , (192) where C a C_a C a n λ G R W n\lambda_{\mathrm{GRW}} n λ GRW ∣ 1 ˉ ⟩ |\bar{1}\rangle ∣ 1 ˉ ⟩ ∣ 0 ˉ ⟩ |\bar{0}\rangle ∣ 0 ˉ ⟩ 0 < s < 1 0<s<1 0 < s < 1 s m ≥ s N s^m\ge s^N s m ≥ s N m ≤ N m\le N m ≤ N
b k = ∑ m = 0 N ⟨ m ∣ Φ κ k ( ∣ k ⟩ ⟨ k ∣ ) ∣ m ⟩ ≤ s − N ∑ m = 0 N ⟨ m ∣ Φ κ k ( ∣ k ⟩ ⟨ k ∣ ) ∣ m ⟩ s m < s − N ∑ m = 0 ∞ ⟨ m ∣ Φ κ k ( ∣ k ⟩ ⟨ k ∣ ) ∣ m ⟩ s m = s − N 1 k ! ( d d t ) k F ( s , t ) ∣ t = 0 ≤ s − N t − k F ( s , t ) (193) \begin{aligned}
b_k &= \sum_{m=0}^{N}\langle m|\Phi_{\kappa_k}(|k\rangle\langle k|)|m\rangle\\
&\le s^{-N}\sum_{m=0}^{N}\langle m|\Phi_{\kappa_k}(|k\rangle\langle k|)|m\rangle s^m \\
&<s^{-N}\sum_{m=0}^{\infty}\langle m|\Phi_{\kappa_k}(|k\rangle\langle k|)|m\rangle s^m \\
&=s^{-N}\frac{1}{k!}\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^k F(s,t)\bigg|_{t=0} \\
&\le s^{-N}t^{-k}F(s,t) \tag*{(193)}
\end{aligned} b k = m = 0 ∑ N ⟨ m ∣ Φ κ k ( ∣ k ⟩ ⟨ k ∣ ) ∣ m ⟩ ≤ s − N m = 0 ∑ N ⟨ m ∣ Φ κ k ( ∣ k ⟩ ⟨ k ∣ ) ∣ m ⟩ s m < s − N m = 0 ∑ ∞ ⟨ m ∣ Φ κ k ( ∣ k ⟩ ⟨ k ∣ ) ∣ m ⟩ s m = s − N k ! 1 ( d t d ) k F ( s , t ) t = 0 ≤ s − N t − k F ( s , t ) (193) where for the inequalities we have used the fact that all the Taylor expansion coefficients are positive. Let us take s = κ k s=\kappa_k s = κ k t = 1 / 4 κ k t=1/4\kappa_k t = 1/4 κ k k k k s < 1 s<1 s < 1 b k b_k b k
b k ≤ κ k − N ( 4 κ k ) k 1 3 4 [ 3 4 + κ k ( 1 − κ k ) ( 1 − 1 4 κ k ) ] = 4 2 N + 1 κ k N + 1 ( 8 3 + O ( 1 / k ) ) . (194) b_k\le\kappa_k^{-N}(4\kappa_k)^k\frac{1}{\sqrt{\frac{3}{4}\left[\frac{3}{4}+\kappa_k(1-\kappa_k)\left(1-\frac{1}{4\kappa_k}\right)\right]}}=4^{2N+1}\kappa_k^{N+1}\left(\sqrt{\frac{8}{3}}+O(1/k)\right). \tag*{(194)} b k ≤ κ k − N ( 4 κ k ) k 4 3 [ 4 3 + κ k ( 1 − κ k ) ( 1 − 4 κ k 1 ) ] 1 = 4 2 N + 1 κ k N + 1 ( 3 8 + O ( 1/ k ) ) . (194) In other words, the leading contribution from both a k a_k a k b k b_k b k κ k k / 2 \kappa_k^{k/2} κ k k /2 (152) is
log ( Γ e f f x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − ) ≤ log ( Γ e f f x + Γ e f f + + Γ e f f − 2 ) = log a k + b k 2 + log n λ G R W ≤ − k 2 log k + O ( k ) . (195) \log\left(\Gamma_{\mathrm{eff}}^x+\frac{\Gamma_{\mathrm{eff}}^+\Gamma_{\mathrm{eff}}^-}{\Gamma_{\mathrm{eff}}^++\Gamma_{\mathrm{eff}}^-}\right)\le\log\left(\Gamma_{\mathrm{eff}}^x+\frac{\Gamma_{\mathrm{eff}}^++\Gamma_{\mathrm{eff}}^-}{2}\right)=\log\frac{a_k+b_k}{2}+\log n\lambda_{\mathrm{GRW}}\le-\frac{k}{2}\log k+O(k). \tag*{(195)} log ( Γ eff x + Γ eff + + Γ eff − Γ eff + Γ eff − ) ≤ log ( Γ eff x + 2 Γ eff + + Γ eff − ) = log 2 a k + b k + log n λ GRW ≤ − 2 k log k + O ( k ) . (195) In summary, we have shown that the contact probability o k ≤ C 0 exp ( − C 1 k 1 + 2 γ ) o_k\le C_0\exp(-C_1k^{1+2\gamma}) o k ≤ C 0 exp ( − C 1 k 1 + 2 γ ) k k k
log ∣ λ e f f x x ∣ Γ e f f x x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − ≥ log ∣ λ e f f x x ∣ Γ e f f x x + Γ e f f + + Γ e f f − 2 ≥ ( 1 2 − 2 γ ) k log k + O ( k ) . (196) \log\frac{|\lambda_{\mathrm{eff}}^{xx}|}{\Gamma_{\mathrm{eff}}^{xx}+\frac{\Gamma_{\mathrm{eff}}^+\Gamma_{\mathrm{eff}}^-}{\Gamma_{\mathrm{eff}}^++\Gamma_{\mathrm{eff}}^-}}\ge\log\frac{|\lambda_{\mathrm{eff}}^{xx}|}{\Gamma_{\mathrm{eff}}^{xx}+\frac{\Gamma_{\mathrm{eff}}^++\Gamma_{\mathrm{eff}}^-}{2}}\ge\left(\frac{1}{2}-2\gamma\right)k\log k+O(k). \tag*{(196)} log Γ eff xx + Γ eff + + Γ eff − Γ eff + Γ eff − ∣ λ eff xx ∣ ≥ log Γ eff xx + 2 Γ eff + + Γ eff − ∣ λ eff xx ∣ ≥ ( 2 1 − 2 γ ) k log k + O ( k ) . (196) where for simplicity, we have omitted the constant G m A m B / ( n × ℏ λ G R W ) Gm_Am_B/(n\times\hbar\lambda_{\mathrm{GRW}}) G m A m B / ( n × ℏ λ GRW ) γ < 1 / 4 \gamma<1/4 γ < 1/4 k k k
Note that the scaling behaviour of the proof here is completely different from the result given in App. A 3. There the leading contribution of the effective interaction rate comes from the contribution in the contact limit, i.e., the regime ∣ d + x A − x B ∣ ≈ 2 r 0 |d+x_A-x_B|\approx2r_0 ∣ d + x A − x B ∣ ≈ 2 r 0 d d d
For the CSL model, a similar derivation can also be implemented to show the same result. The only qualitative difference is in the nonlocal correlated error Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx d k / r C = O ( k γ ) d_k/r_C=O(k^\gamma) d k / r C = O ( k γ ) Γ e f f x x \Gamma_{\mathrm{eff}}^{xx} Γ eff xx λ e f f x x \lambda_{\mathrm{eff}}^{xx} λ eff xx
Necessity of increasing the code distance
Here, let us show that this scaling control is not possible if we fix N N N d d d x 0 x_0 x 0 x 0 → 0 x_0\to0 x 0 → 0 d − 2 r 0 ≥ δ d d-2r_0\ge\delta d d − 2 r 0 ≥ δ d x 0 x_0 x 0 ≲ exp ( − O ( δ d 2 / x 0 2 ) ) \lesssim\exp(-O(\delta d^2/x_0^2)) ≲ exp ( − O ( δ d 2 / x 0 2 )) x 0 → 0 x_0\to0 x 0 → 0
∣ λ e f f x x ∣ ≤ C 0 , N x 0 4 N + 2 + C 1 , N e − C 2 , N δ d 2 x 0 2 . (197) |\lambda_{\mathrm{eff}}^{xx}|\le C_{0,N}x_0^{4N+2}+C_{1,N}e^{-C_{2,N}\frac{\delta d^2}{x_0^2}}. \tag*{(197)} ∣ λ eff xx ∣ ≤ C 0 , N x 0 4 N + 2 + C 1 , N e − C 2 , N x 0 2 δ d 2 . (197) where C i , N C_{i,N} C i , N N N N ∣ x A − x B ∣ ≤ δ d / 2 |x_A-x_B| \le\delta d/2 ∣ x A − x B ∣ ≤ δ d /2 ∣ x A − x B ∣ ≥ δ d / 2 |x_A-x_B| \ge\delta d/2 ∣ x A − x B ∣ ≥ δ d /2 x 0 → 0 x_0 \to0 x 0 → 0 O ( x 0 4 N + 2 ) O(x_0^{4N+2}) O ( x 0 4 N + 2 )
On the other hand, the leading contribution of the error rate is
a k = ⟨ N + 1 ∣ Φ κ ( ∣ 0 ⟩ ⟨ 0 ∣ ) ∣ N + 1 ⟩ + O ( ( x 0 r C ) 2 N + 4 ) = ( 2 N + 2 ) ! [ ( N + 1 ) ! ] 2 2 2 N + 2 ( x 0 r C ) 2 N + 2 + O ( ( x 0 r C ) 2 N + 4 ) , a_k = \langle N+1|\Phi_\kappa(|0\rangle\langle0|)|N+1\rangle+ O\left(\left(\frac{x_0}{r_C}\right)^{2N+4}\right) = \frac{(2N+2)!}{[(N+1)!]^2 2^{2N+2}}\left(\frac{x_0}{r_C}\right)^{2N+2} + O\left(\left(\frac{x_0}{r_C}\right)^{2N+4}\right), a k = ⟨ N + 1∣ Φ κ ( ∣0 ⟩ ⟨ 0∣ ) ∣ N + 1 ⟩ + O ( ( r C x 0 ) 2 N + 4 ) = [( N + 1 )! ] 2 2 2 N + 2 ( 2 N + 2 )! ( r C x 0 ) 2 N + 2 + O ( ( r C x 0 ) 2 N + 4 ) , b k = ⟨ N ∣ Φ κ ( ∣ k ⟩ ⟨ k ∣ ) ∣ N ⟩ + O ( ( x 0 r C ) 2 N + 4 ) = ( 2 N + 2 ) ! ( 2 N + 1 ) ! [ ( N + 1 ) ! ] 3 N ! 2 2 N + 2 ( x 0 r C ) 2 N + 2 + O ( ( x 0 r C ) 2 N + 4 ) , b_k = \langle N|\Phi_\kappa(|k\rangle\langle k|)|N\rangle+ O\left(\left(\frac{x_0}{r_C}\right)^{2N+4}\right) = \frac{(2N+2)!(2N+1)!}{[(N+1)!]^3N!2^{2N+2}}\left(\frac{x_0}{r_C}\right)^{2N+2} + O\left(\left(\frac{x_0}{r_C}\right)^{2N+4}\right), b k = ⟨ N ∣ Φ κ ( ∣ k ⟩ ⟨ k ∣ ) ∣ N ⟩ + O ( ( r C x 0 ) 2 N + 4 ) = [( N + 1 )! ] 3 N ! 2 2 N + 2 ( 2 N + 2 )! ( 2 N + 1 )! ( r C x 0 ) 2 N + 2 + O ( ( r C x 0 ) 2 N + 4 ) , Γ e f f r x = c N Re ⟨ N + 1 ∣ Φ κ ( ∣ 0 ⟩ ⟨ k ∣ ) ∣ N ⟩ + O ( ( x 0 r C ) 2 N + 4 ) = c N ( 2 N + 2 ) ! [ ( N + 1 ) ! ] 5 / 2 2 2 N + 2 ( 2 N + 1 ) ! N ! ( x 0 r C ) 2 N + 2 + O ( ( x 0 r C ) 2 N + 4 ) . (198) \Gamma_{\mathrm{eff}}^{\mathrm{rx}} = c_N \operatorname{Re}\langle N+1|\Phi_\kappa(|0\rangle\langle k|)|N\rangle+ O\left(\left(\frac{x_0}{r_C}\right)^{2N+4}\right) = c_N\frac{(2N+2)!}{[(N+1)!]^{5/2}2^{2N+2}}\sqrt{\frac{(2N+1)!}{N!}}\left(\frac{x_0}{r_C}\right)^{2N+2} + O\left(\left(\frac{x_0}{r_C}\right)^{2N+4}\right). \tag*{(198)} Γ eff rx = c N Re ⟨ N + 1∣ Φ κ ( ∣0 ⟩ ⟨ k ∣ ) ∣ N ⟩ + O ( ( r C x 0 ) 2 N + 4 ) = c N [( N + 1 )! ] 5/2 2 2 N + 2 ( 2 N + 2 )! N ! ( 2 N + 1 )! ( r C x 0 ) 2 N + 2 + O ( ( r C x 0 ) 2 N + 4 ) . (198) where c N c_N c N (151) . Note that
c N = c N ( 0 ) + O ( ( x 0 r C ) 2 ) = 1 1 + ( 3 N + 1 ) ! ( N + 1 ) ! [ ( 2 N + 1 ) ! ] 2 + O ( ( x 0 r C ) 2 ) < 1. (199) c_N = c_N^{(0)} + O\left(\left(\frac{x_0}{r_C}\right)^2\right) = \frac{1}{\sqrt{1+\frac{(3N+1)!(N+1)!}{[(2N+1)!]^2}}} + O\left(\left(\frac{x_0}{r_C}\right)^2\right) < 1. \tag*{(199)} c N = c N ( 0 ) + O ( ( r C x 0 ) 2 ) = 1 + [( 2 N + 1 )! ] 2 ( 3 N + 1 )! ( N + 1 )! 1 + O ( ( r C x 0 ) 2 ) < 1. (199) As a result, the effective error rate
Γ e f f r x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − = a k b k − ( Γ e f f r x ) 2 a k + b k − 2 Γ e f f r x = C 2 , N x 0 2 N + 2 + O ( x 0 2 N + 4 ) ≥ C 2 , N ′ x 0 2 N + 2 . (200) \Gamma_{\mathrm{eff}}^{\mathrm{rx}}+\frac{\Gamma_{\mathrm{eff}}^+\Gamma_{\mathrm{eff}}^-}{\Gamma_{\mathrm{eff}}^++\Gamma_{\mathrm{eff}}^-} = \frac{a_kb_k-(\Gamma_{\mathrm{eff}}^{\mathrm{rx}})^2}{a_k+b_k-2\Gamma_{\mathrm{eff}}^{\mathrm{rx}}} = C_{2,N}x_0^{2N+2}+O(x_0^{2N+4}) \ge C'_{2,N}x_0^{2N+2}. \tag*{(200)} Γ eff rx + Γ eff + + Γ eff − Γ eff + Γ eff − = a k + b k − 2 Γ eff rx a k b k − ( Γ eff rx ) 2 = C 2 , N x 0 2 N + 2 + O ( x 0 2 N + 4 ) ≥ C 2 , N ′ x 0 2 N + 2 . (200) Here C 2 , N ∝ [ 1 − ( c N ( 0 ) ) 2 ] > 0 C_{2,N}\propto[1-(c_N^{(0)})^2]>0 C 2 , N ∝ [ 1 − ( c N ( 0 ) ) 2 ] > 0
Therefore, we conclude that
∣ λ x x ∣ Γ e f f r x + Γ e f f + Γ e f f − Γ e f f + + Γ e f f − ≤ C 3 , N x 0 2 N + C 4 , N x 0 − 2 N − 2 e − C 2 , N δ d 2 / x 0 2 . (201) \frac{|\lambda^{xx}|}{\Gamma_{\mathrm{eff}}^{\mathrm{rx}}+\frac{\Gamma_{\mathrm{eff}}^+\Gamma_{\mathrm{eff}}^-}{\Gamma_{\mathrm{eff}}^++\Gamma_{\mathrm{eff}}^-}} \le C_{3,N}x_0^{2N}+C_{4,N}x_0^{-2N-2}e^{-C_{2,N}\delta d^2/x_0^2}. \tag*{(201)} Γ eff rx + Γ eff + + Γ eff − Γ eff + Γ eff − ∣ λ xx ∣ ≤ C 3 , N x 0 2 N + C 4 , N x 0 − 2 N − 2 e − C 2 , N δ d 2 / x 0 2 . (201) where C i , N C_{i,N} C i , N N N N x 0 → 0 x_0 \to0 x 0 → 0 N ≥ 1 N \ge1 N ≥ 1 N = 0 N=0 N = 0 x 0 > r C x_0>r_C x 0 > r C N N N λ G R W \lambda_{\mathrm{GRW}} λ GRW
Finite recoverability bound for the DP model
On the other hand, the DP model has an upper bound on the separability parameter η \eta η
∫ d 3 k ( 2 π ) 3 4 π e − σ 2 ∣ k ∣ 2 ∣ k ∣ 2 e i k ⋅ ( x − y ) = ∑ ν ∫ d 3 k f ν ( x ) f ν ( y ) . (202) \int\frac{\mathrm{d}^3k}{(2\pi)^3}\frac{4\pi e^{-\sigma^2|\mathbf{k}|^2}}{|\mathbf{k}|^2}e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{y})}=\sum_\nu\int\mathrm{d}^3k f_\nu(\mathbf{x})f_\nu(\mathbf{y}). \tag*{(202)} ∫ ( 2 π ) 3 d 3 k ∣ k ∣ 2 4 π e − σ 2 ∣ k ∣ 2 e i k ⋅ ( x − y ) = ν ∑ ∫ d 3 k f ν ( x ) f ν ( y ) . (202) where the existence of this kernel decomposition is guaranteed by the positivity of the kernel in the momentum space, and ν = 0 , 1 \nu=0,1 ν = 0 , 1 f ν f_\nu f ν M ( x ) α = m α δ ( x − x α ) M(\mathbf{x})_\alpha=m_\alpha\delta(\mathbf{x}-\mathbf{x}_\alpha) M ( x ) α = m α δ ( x − x α )
α ν ( k ) = G ℏ ∫ d 3 x f ν ( x ) M α ( x ) . (203) \alpha_\nu(\mathbf{k})=\sqrt{\frac{G}{\hbar}}\int\mathrm{d}^3x f_\nu(\mathbf{x})M_\alpha(\mathbf{x}). \tag*{(203)} α ν ( k ) = ℏ G ∫ d 3 x f ν ( x ) M α ( x ) . (203) Then the DP Lindbladian can be rewritten as
L D P ( ρ ) = ∑ ν ∫ d 3 k i 2 [ ( A ν ⊗ I B + I A ⊗ B ν ) 2 , ρ ] + D [ A ν ⊗ I B + I A ⊗ B ν ] ( ρ ) . (204) \mathcal{L}_{\mathrm{DP}}(\rho)=\sum_\nu\int\mathrm{d}^3k\frac{i}{2}[(A_\nu\otimes I_B+I_A\otimes B_\nu)^2,\rho]+\mathcal{D}[A_\nu\otimes I_B+I_A\otimes B_\nu](\rho). \tag*{(204)} L DP ( ρ ) = ν ∑ ∫ d 3 k 2 i [( A ν ⊗ I B + I A ⊗ B ν ) 2 , ρ ] + D [ A ν ⊗ I B + I A ⊗ B ν ] ( ρ ) . (204) For a general local code subspace defined by the projector Π A B = Π A ⊗ Π B \Pi_{AB}=\Pi_A\otimes\Pi_B Π A B = Π A ⊗ Π B Π A B ρ Π A B = ρ \Pi_{AB}\rho\Pi_{AB}=\rho Π A B ρ Π A B = ρ R ( ρ ) = ρ R(\rho)=\rho R ( ρ ) = ρ Π A B R ( L D P ( ρ ) ) Π A B = R ( L D P ( ρ ) ) \Pi_{AB}R(\mathcal{L}_{DP}(\rho))\Pi_{AB}=R(\mathcal{L}_{DP}(\rho)) Π A B R ( L D P ( ρ )) Π A B = R ( L D P ( ρ )) Prop. 2 , we can rewrite the Kraus operator of R ( ⋅ ) = ∑ i j R i A R j B R(\cdot)=\sum_{ij}R_i^A R_j^B R ( ⋅ ) = ∑ ij R i A R j B R i A † R j B † R_i^{A\dagger}R_j^{B\dagger} R i A † R j B † R i A Π A = δ i 0 Π A R_i^A\Pi_A=\delta_{i0}\Pi_A R i A Π A = δ i 0 Π A R j B Π B = δ j 0 Π B R_j^B\Pi_B=\delta_{j0}\Pi_B R j B Π B = δ j 0 Π B 4 ρ = ∣ ψ ⟩ ⟨ ψ ∣ \rho=|\psi\rangle\langle\psi| ρ = ∣ ψ ⟩ ⟨ ψ ∣
0 = ( I A B − Π A B ) R ( L D P ( ρ ) ) ( I A B − Π A B ) = ( I A B − Π A B ) ∑ i j R i A R j B ( ∑ ν ∫ d 3 k D [ A ν + B ν ] ( ρ ) ) R i A † R j B † ( I A B − Π A B ) = ∑ ν ∫ d 3 k ( I A B − Π A B ) ∑ i j R i A R j B ( A ν + B ν ) ∣ ψ ⟩ ⟨ ψ ∣ ( A ν † + B ν † ) R i A † R j B † ( I A B − Π A B ) (205) \begin{aligned}
0&=(I_{AB}-\Pi_{AB})R(\mathcal{L}_{DP}(\rho))(I_{AB}-\Pi_{AB})\\
&=(I_{AB}-\Pi_{AB})\sum_{ij}R_i^A R_j^B\left(\sum_\nu\int\mathrm{d}^3k\mathcal{D}[A_\nu+B_\nu](\rho)\right)R_i^{A\dagger}R_j^{B\dagger}(I_{AB}-\Pi_{AB})\\
&=\sum_\nu\int\mathrm{d}^3k(I_{AB}-\Pi_{AB})\sum_{ij}R_i^A R_j^B(A_\nu+B_\nu)|\psi\rangle\langle\psi|(A_\nu^\dagger+B_\nu^\dagger)R_i^{A\dagger}R_j^{B\dagger}(I_{AB}-\Pi_{AB}) \tag*{(205)}
\end{aligned} 0 = ( I A B − Π A B ) R ( L D P ( ρ )) ( I A B − Π A B ) = ( I A B − Π A B ) ij ∑ R i A R j B ( ν ∑ ∫ d 3 k D [ A ν + B ν ] ( ρ ) ) R i A † R j B † ( I A B − Π A B ) = ν ∑ ∫ d 3 k ( I A B − Π A B ) ij ∑ R i A R j B ( A ν + B ν ) ∣ ψ ⟩ ⟨ ψ ∣ ( A ν † + B ν † ) R i A † R j B † ( I A B − Π A B ) (205) Therefore every branch should be zero, i.e.
( I A B − Π A B ) R i A R j B ( A ν + B ν ) ∣ ψ ⟩ = 0 ∀ i , j , ν , k . (206) (I_{AB}-\Pi_{AB})R_i^A R_j^B(A_\nu+B_\nu)|\psi\rangle=0\qquad\forall i,j,\nu,k. \tag*{(206)} ( I A B − Π A B ) R i A R j B ( A ν + B ν ) ∣ ψ ⟩ = 0 ∀ i , j , ν , k . (206) Since it holds for every pure state, then we take i = 0 i=0 i = 0 j = 0 j=0 j = 0
( I A − Π A ) R i A A ν ( k ) Π A = ( I B − Π B ) R j B B ν ( k ) Π B = 0 , ∀ i , j , ν , k . (207) (I_A-\Pi_A)R_i^A A_\nu(k)\Pi_A=(I_B-\Pi_B)R_j^B B_\nu(k)\Pi_B=0,\ \forall i,j,\nu,k. \tag*{(207)} ( I A − Π A ) R i A A ν ( k ) Π A = ( I B − Π B ) R j B B ν ( k ) Π B = 0 , ∀ i , j , ν , k . (207) In other words, the recovery map turns every local error into a logical error operator
E i , ν , k α = R i α α ν ( k ) Π α , such that E i , ν , k α = Π α E i , ν , k α Π α . (208) E_{i,\nu,k}^{\alpha}=R_i^\alpha\alpha_\nu(k)\Pi_\alpha,\qquad\text{such that}\qquad E_{i,\nu,k}^{\alpha}=\Pi_\alpha E_{i,\nu,k}^{\alpha}\Pi_\alpha. \tag*{(208)} E i , ν , k α = R i α α ν ( k ) Π α , such that E i , ν , k α = Π α E i , ν , k α Π α . (208) For simplicity let us assume Tr E i , ν , k α = 0 \operatorname{Tr}E_{i,\nu,k}^{\alpha}=0 Tr E i , ν , k α = 0 Π α \Pi_\alpha Π α
In this case, the local error part of error-corrected channel can be rewritten as
∑ ν ∫ d 3 k R ( D [ α ν ] ( ρ ) ) = ∑ i , ν ∫ d 3 k D [ E i , ν , k α ] ( ρ ) . (209) \sum_\nu\int\mathrm{d}^3kR(\mathcal{D}[\alpha_\nu](\rho))=\sum_{i,\nu}\int\mathrm{d}^3k\mathcal{D}[E_{i,\nu,k}^{\alpha}](\rho). \tag*{(209)} ν ∑ ∫ d 3 k R ( D [ α ν ] ( ρ )) = i , ν ∑ ∫ d 3 k D [ E i , ν , k α ] ( ρ ) . (209) Specifically, note that Π α = ∑ i R i α † R i α Π α = R 0 α † R 0 α Π α = R 0 α † Π α \Pi_\alpha=\sum_iR_i^{\alpha\dagger}R_i^\alpha\Pi_\alpha=R_0^{\alpha\dagger}R_0^\alpha\Pi_\alpha=R_0^{\alpha\dagger}\Pi_\alpha Π α = ∑ i R i α † R i α Π α = R 0 α † R 0 α Π α = R 0 α † Π α E 0 , ν , k α = Π α R 0 α α ν Π α = Π α α ν Π α E_{0,\nu,k}^{\alpha}=\Pi_\alpha R_0^\alpha\alpha_\nu\Pi_\alpha=\Pi_\alpha\alpha_\nu\Pi_\alpha E 0 , ν , k α = Π α R 0 α α ν Π α = Π α α ν Π α
Let us now investigate the nonlocal effective interaction part, which can be rewritten as
∑ ν ∫ d 3 k i R ( [ A ν ⊗ B ν , ρ ] ) = ∑ ν ∫ d 3 k i R 0 A R 0 B ( [ A ν ⊗ B ν , ρ ] ) R 0 A † R 0 B † = ∑ ν ∫ d 3 k i [ Π A A ν Π A ⊗ Π B B ν Π B , ρ ] . (210) \sum_\nu\int\mathrm{d}^3k\,iR([A_\nu\otimes B_\nu,\rho])=\sum_\nu\int\mathrm{d}^3k\,iR_0^A R_0^B([A_\nu\otimes B_\nu,\rho])R_0^{A\dagger}R_0^{B\dagger}=\sum_\nu\int\mathrm{d}^3k\,i[\Pi_AA_\nu\Pi_A\otimes\Pi_BB_\nu\Pi_B,\rho]. \tag*{(210)} ν ∑ ∫ d 3 k i R ([ A ν ⊗ B ν , ρ ]) = ν ∑ ∫ d 3 k i R 0 A R 0 B ([ A ν ⊗ B ν , ρ ]) R 0 A † R 0 B † = ν ∑ ∫ d 3 k i [ Π A A ν Π A ⊗ Π B B ν Π B , ρ ] . (210) Let us denote Π A A ν ( k ) Π A = a ν , k 0 Π A + ∑ i = 1 a ν , k i O A i \Pi_AA_\nu(k)\Pi_A=a_{\nu,k}^0\Pi_A+\sum_{i=1}a_{\nu,k}^iO_A^i Π A A ν ( k ) Π A = a ν , k 0 Π A + ∑ i = 1 a ν , k i O A i { O A i } \{O_A^i\} { O A i } Π A \Pi_A Π A Tr O A i † O A j = δ i j \operatorname{Tr}O_A^{i\dagger}O_A^j=\delta_{ij} Tr O A i † O A j = δ ij a ν , k = ( a ν , k 1 , a ν , k 2 , ⋯ ) \mathbf{a}_{\nu,k}=(a_{\nu,k}^1,a_{\nu,k}^2,\cdots) a ν , k = ( a ν , k 1 , a ν , k 2 , ⋯ ) { O B j } \{O_B^j\} { O B j } b ν , k \mathbf{b}_{\nu,k} b ν , k
H A B = − ∑ i j J i j O A i ⊗ O B j , where J = ∑ ν ∫ d 3 k a ν , k b ν , k T . (211) H_{AB}=-\sum_{ij}J_{ij}O_A^i\otimes O_B^j,\qquad\text{where}\qquad J=\sum_\nu\int\mathrm{d}^3k\,\mathbf{a}_{\nu,k}\mathbf{b}_{\nu,k}^{T}. \tag*{(211)} H A B = − ij ∑ J ij O A i ⊗ O B j , where J = ν ∑ ∫ d 3 k a ν , k b ν , k T . (211) To account for the largest contribution to the interaction, let us make a singular value decomposition of J J J λ 1 \lambda_1 λ 1 u \mathbf{u} u v \mathbf{v} v
λ 1 ( u , v ) = u T J v = ∑ ν ∫ d 3 k ( u ⋅ a ν , k ) ( v ⋅ b ν , k ) (212) \lambda_1(\mathbf{u},\mathbf{v})=\mathbf{u}^{T}J\mathbf{v}=\sum_\nu\int\mathrm{d}^3k\,(\mathbf{u}\cdot\mathbf{a}_{\nu,k})(\mathbf{v}\cdot\mathbf{b}_{\nu,k}) \tag*{(212)} λ 1 ( u , v ) = u T J v = ν ∑ ∫ d 3 k ( u ⋅ a ν , k ) ( v ⋅ b ν , k ) (212) Then let us investigate local logical error rates along these two operator axes. Similarly, let us also write down the decomposition of error operators as E i , ν , k α = e i , ν , k α ⋅ O α E^{\alpha}_{i,\nu,k}=e^{\alpha}_{i,\nu,k}\cdot O_{\alpha} E i , ν , k α = e i , ν , k α ⋅ O α e 0 , ν , k 0 = a ν , k i e^{0}_{0,\nu,k}=a^{i}_{\nu,k} e 0 , ν , k 0 = a ν , k i w ⋅ O α w\cdot O_{\alpha} w ⋅ O α
Γ α ( w ) = ∑ i , ν ∫ d 3 k ∣ Tr ( w ⋅ O α ) † E i , ν , k α ∣ 2 . (213) \Gamma_{\alpha}(w)=\sum_{i,\nu}\int\mathrm{d}^{3}k\left|\operatorname{Tr}(w\cdot O_{\alpha})^{\dagger}E^{\alpha}_{i,\nu,k}\right|^{2}. \tag*{(213)} Γ α ( w ) = i , ν ∑ ∫ d 3 k Tr ( w ⋅ O α ) † E i , ν , k α 2 . (213) Specifically, along the axis u u u A A A v v v B B B
Γ A ( u ) ≥ ∑ ν ∫ d 3 k ( u ⋅ a ν , k ) 2 , Γ B ( v ) ≥ ∑ ν ∫ d 3 k ( v ⋅ b ν , k ) 2 . (214) \Gamma_{A}(u)\geq\sum_{\nu}\int\mathrm{d}^{3}k\,(u\cdot a_{\nu,k})^{2},\qquad\Gamma_{B}(v)\geq\sum_{\nu}\int\mathrm{d}^{3}k\,(v\cdot b_{\nu,k})^{2}. \tag*{(214)} Γ A ( u ) ≥ ν ∑ ∫ d 3 k ( u ⋅ a ν , k ) 2 , Γ B ( v ) ≥ ν ∑ ∫ d 3 k ( v ⋅ b ν , k ) 2 . (214) where we have taken only i = 0 i=0 i = 0
η u , v = λ 1 2 ( u , v ) + Γ A B 2 ( u , v ) Γ A ( u ) Γ B ( v ) ≤ 2 , (215) \eta_{u,v}=\frac{\sqrt{\lambda_{1}^{2}(u,v)+\Gamma_{AB}^{2}(u,v)}}{\sqrt{\Gamma_{A}(u)\Gamma_{B}(v)}}\leq\sqrt{2}, \tag*{(215)} η u , v = Γ A ( u ) Γ B ( v ) λ 1 2 ( u , v ) + Γ A B 2 ( u , v ) ≤ 2 , (215) where the nonlocal error rate ∣ Γ A B ( u , v ) ∣ = ∣ λ 1 ( u , v ) ∣ |\Gamma_{AB}(u,v)|=|\lambda_{1}(u,v)| ∣ Γ A B ( u , v ) ∣ = ∣ λ 1 ( u , v ) ∣ D [ A ν ⊗ I B + I A ⊗ B ν ] − D [ A ν ] − D [ B ν ] \mathcal{D}[A_{\nu}\otimes I_{B}+I_{A}\otimes B_{\nu}]-\mathcal{D}[A_{\nu}]-\mathcal{D}[B_{\nu}] D [ A ν ⊗ I B + I A ⊗ B ν ] − D [ A ν ] − D [ B ν ]
∣ ∑ ν ∫ d 3 k ( u ⋅ a ν , k ) ( v ⋅ b ν , k ) ∣ 2 ≤ ( ∑ ν ∫ d 3 k ( u ⋅ a ν , k ) 2 ) ( ∑ ν ∫ d 3 k ( v ⋅ b ν , k ) 2 ) . (216) \left|\sum_{\nu}\int\mathrm{d}^{3}k\,(u\cdot a_{\nu,k})(v\cdot b_{\nu,k})\right|^{2}\leq\left(\sum_{\nu}\int\mathrm{d}^{3}k\,(u\cdot a_{\nu,k})^{2}\right)\left(\sum_{\nu}\int\mathrm{d}^{3}k\,(v\cdot b_{\nu,k})^{2}\right). \tag*{(216)} ν ∑ ∫ d 3 k ( u ⋅ a ν , k ) ( v ⋅ b ν , k ) 2 ≤ ( ν ∑ ∫ d 3 k ( u ⋅ a ν , k ) 2 ) ( ν ∑ ∫ d 3 k ( v ⋅ b ν , k ) 2 ) . (216) And actually this 2 \sqrt{2} 2 u u u v v v 2 \sqrt{2} 2
Taken together, these results establish a qualitative distinction between the collapse models. For GRW and CSL, increasing the code distance permits an unbounded enhancement of the effective entangling scale relative to the logical errors while maintaining an exponentially small contact probability. For DP, by contrast, the common kernel underlying the coherent interaction and decoherence imposes a finite upper bound on the separability parameter. Thus the different recoverability behavior observed numerically is rooted in the analytical structure of the corresponding noise kernels.
Suppressing p 2 p^2 p 2
As discussed in App. A 1, gravity-induced decoherence may also arise from the interaction of matter with a quantized gravitational field, as described, for example, by the Anastopoulos–Hu (AH) model [38 ] . From the perspective of the present work, the AH model is distinguished by the fact that the Newtonian interaction and the gravitational decoherence are generated by distinct operators. In contrast to the collapse models discussed previously, where the effective interaction and error channels arise from closely related operators and therefore require a more detailed analysis, the AH model provides a particularly simple setting in which these channels can be more readily distinguished. As a result, the construction of a suitable error-correcting encoding can be illustrated in a comparatively transparent manner. In this appendix, we show that a stabilized Schrödinger-cat code can suppress the dominant decoherence channel while preserving a non-trivial interaction, and that the resulting asymptotic state becomes entangled. This provides a direct certification that the model is non-separable.
We first show that the Schrödinger-cat code suppresses the p 2 p^2 p 2 m A = m B = m m_A=m_B=m m A = m B = m d d d x A x_A x A x B x_B x B p A p_A p A p B p_B p B ω \omega ω
L t o t ( ρ ) = L N G ( ρ ) + L A H ( ρ ) = i ℏ [ G m A m B ∣ d + x A − x B ∣ , ρ ] + 8 π G Θ 9 ℏ m 2 c 5 ( D [ p A 2 ] + D [ p B 2 ] ) ( ρ ) ≈ − 2 i G m A m B ℏ d 3 [ x A x B , ρ ] + 8 π G Θ 9 ℏ m 2 c 5 ( D [ p A 2 ] + D [ p B 2 ] ) ( ρ ) , (217) \begin{aligned}
\mathcal{L}_{\mathrm{tot}}(\rho) &= \mathcal{L}_{\mathrm{NG}}(\rho) + \mathcal{L}_{\mathrm{AH}}(\rho) \\
&= \frac{i}{\hbar}\left[\frac{Gm_A m_B}{|d+x_A-x_B|},\rho\right] + \frac{8\pi G\Theta}{9\hbar m^2 c^5}\left(\mathcal{D}[p_A^2]+\mathcal{D}[p_B^2]\right)(\rho) \\
&\approx-\frac{2iGm_A m_B}{\hbar d^3}[x_Ax_B,\rho] + \frac{8\pi G\Theta}{9\hbar m^2 c^5}\left(\mathcal{D}[p_A^2]+\mathcal{D}[p_B^2]\right)(\rho), \tag*{(217)}
\end{aligned} L tot ( ρ ) = L NG ( ρ ) + L AH ( ρ ) = ℏ i [ ∣ d + x A − x B ∣ G m A m B , ρ ] + 9ℏ m 2 c 5 8 π G Θ ( D [ p A 2 ] + D [ p B 2 ] ) ( ρ ) ≈ − ℏ d 3 2 i G m A m B [ x A x B , ρ ] + 9ℏ m 2 c 5 8 π G Θ ( D [ p A 2 ] + D [ p B 2 ] ) ( ρ ) , (217) where we ignore the local free Hamiltonian for simplicity. Note that the nonlocal interaction is now based on displacement operators x A , B x_{A,B} x A , B p A 2 , p B 2 p_A^2,p_B^2 p A 2 , p B 2 x x x p 2 p^2 p 2 [60 ] , whose code states are defined as follows:
∣ 0 ˉ ⟩ = ∣ α ⟩ + ∣ − α ⟩ 2 ( 1 + e − 2 ∣ α ∣ 2 ) , ∣ 1 ˉ ⟩ = ∣ α ⟩ − ∣ − α ⟩ 2 ( 1 − e − 2 ∣ α ∣ 2 ) , ∣ α ⟩ = e − ∣ α ∣ 2 / 2 ∑ i = 0 ∞ α i i ! ∣ i ⟩ . (218) |\bar{0}\rangle=\frac{|\alpha\rangle+|-\alpha\rangle}{\sqrt{2(1+e^{-2|\alpha|^2})}},\qquad
|\bar{1}\rangle=\frac{|\alpha\rangle-|-\alpha\rangle}{\sqrt{2(1-e^{-2|\alpha|^2})}},\qquad
|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{i=0}^{\infty}\frac{\alpha^i}{\sqrt{i!}}|i\rangle. \tag*{(218)} ∣ 0 ˉ ⟩ = 2 ( 1 + e − 2∣ α ∣ 2 ) ∣ α ⟩ + ∣ − α ⟩ , ∣ 1 ˉ ⟩ = 2 ( 1 − e − 2∣ α ∣ 2 ) ∣ α ⟩ − ∣ − α ⟩ , ∣ α ⟩ = e − ∣ α ∣ 2 /2 i = 0 ∑ ∞ i ! α i ∣ i ⟩ . (218) Note that these two coherent states are not orthogonal, i.e. ⟨ − α ∣ α ⟩ = e − 2 ∣ α ∣ 2 \langle-\alpha|\alpha\rangle=e^{-2|\alpha|^2} ⟨ − α ∣ α ⟩ = e − 2∣ α ∣ 2 ∣ 0 ˉ ⟩ , ∣ 1 ˉ ⟩ |\bar{0}\rangle,|\bar{1}\rangle ∣ 0 ˉ ⟩ , ∣ 1 ˉ ⟩ α = α ∗ \alpha=\alpha^* α = α ∗ p 0 = m ℏ ω / 2 p_0=\sqrt{m\hbar\omega/2} p 0 = m ℏ ω /2 x 0 = ℏ / 2 m ω x_0=\sqrt{\hbar/2m\omega} x 0 = ℏ/2 mω σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ \sigma^z=|\bar{0}\rangle\langle\bar{0}|-|\bar{1}\rangle\langle\bar{1}| σ z = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ − ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ σ x = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ \sigma^x=|\bar{0}\rangle\langle\bar{1}|+|\bar{1}\rangle\langle\bar{0}| σ x = ∣ 0 ˉ ⟩ ⟨ 1 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 0 ˉ ∣ Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣ \Pi=|\bar{0}\rangle\langle\bar{0}|+|\bar{1}\rangle\langle\bar{1}| Π = ∣ 0 ˉ ⟩ ⟨ 0 ˉ ∣ + ∣ 1 ˉ ⟩ ⟨ 1 ˉ ∣
Π p 2 Π / p 0 2 = 1 + 4 ∣ α ∣ 2 e − 4 ∣ α ∣ 2 1 − e − 4 ∣ α ∣ 2 − 4 ∣ α ∣ 2 e − 2 ∣ α ∣ 2 1 − e − 4 ∣ α ∣ 2 σ z , (219) \Pi p^2\Pi/p_0^2=1+\frac{4|\alpha|^2e^{-4|\alpha|^2}}{1-e^{-4|\alpha|^2}}-\frac{4|\alpha|^2e^{-2|\alpha|^2}}{1-e^{-4|\alpha|^2}}\sigma^z, \tag*{(219)} Π p 2 Π/ p 0 2 = 1 + 1 − e − 4∣ α ∣ 2 4∣ α ∣ 2 e − 4∣ α ∣ 2 − 1 − e − 4∣ α ∣ 2 4∣ α ∣ 2 e − 2∣ α ∣ 2 σ z , (219) Π p 4 Π / p 0 4 = 3 − ( 16 α 4 − 24 α 2 ) e − 4 ∣ α ∣ 2 1 − e − 4 ∣ α ∣ 2 + ( 16 α 4 − 24 α 2 ) e − 2 ∣ α ∣ 2 1 − e − 4 ∣ α ∣ 2 σ z , (220) \Pi p^4\Pi/p_0^4=3-\frac{(16\alpha^4-24\alpha^2)e^{-4|\alpha|^2}}{1-e^{-4|\alpha|^2}}+\frac{(16\alpha^4-24\alpha^2)e^{-2|\alpha|^2}}{1-e^{-4|\alpha|^2}}\sigma^z, \tag*{(220)} Π p 4 Π/ p 0 4 = 3 − 1 − e − 4∣ α ∣ 2 ( 16 α 4 − 24 α 2 ) e − 4∣ α ∣ 2 + 1 − e − 4∣ α ∣ 2 ( 16 α 4 − 24 α 2 ) e − 2∣ α ∣ 2 σ z , (220) Π x Π / x 0 = 2 α 1 − e − 4 ∣ α ∣ 2 σ x . (221) \Pi x\Pi/x_0=\frac{2\alpha}{\sqrt{1-e^{-4|\alpha|^2}}}\sigma^x. \tag*{(221)} Π x Π/ x 0 = 1 − e − 4∣ α ∣ 2 2 α σ x . (221) Note that the interaction term is of order O ( α 2 ) O(\alpha^2) O ( α 2 ) σ z \sigma^z σ z O ( e − 2 ∣ α ∣ 2 ) O(e^{-2|\alpha|^2}) O ( e − 2∣ α ∣ 2 ) α → ∞ \alpha\to\infty α → ∞ (3) in the main text holds while the nonlocal interaction remains non-trivial in the code subspace, so entanglement generation is in principle possible.
We now consider an error-avoiding mechanism [60 ] . Note that ∣ α ⟩ |\alpha\rangle ∣ α ⟩ a − α a-\alpha a − α ∣ 0 ˉ ⟩ , ∣ 1 ˉ ⟩ |\bar{0}\rangle,|\bar{1}\rangle ∣ 0 ˉ ⟩ , ∣ 1 ˉ ⟩ ∣ α ⟩ |\alpha\rangle ∣ α ⟩ ∣ − α ⟩ |-\alpha\rangle ∣ − α ⟩ a 2 − α 2 a^2-\alpha^2 a 2 − α 2
d d t ρ = L N G ( ρ ) + L A H ( ρ ) + L e n c o d e ( ρ ) , L N G ( ρ ) = − 2 i G m A m B ℏ d 3 [ x A x B , ρ ] , L A H ( ρ ) = 8 π G Θ 9 ℏ m 2 c 5 ( D [ p A 2 ] + D [ p B 2 ] ) ( ρ ) , L e n c o d e ( ρ ) = Γ e n c o d e ( D [ a A 2 − α 2 ] + D [ a B 2 − α 2 ] ) ( ρ ) . (222) \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\rho&= \mathcal{L}_{\mathrm{NG}}(\rho)+\mathcal{L}_{\mathrm{AH}}(\rho)+\mathcal{L}_{\mathrm{encode}}(\rho),\\
\mathcal{L}_{\mathrm{NG}}(\rho) &= -\frac{2\mathrm{i}Gm_A m_B}{\hbar d^3}[x_Ax_B,\rho],\\
\mathcal{L}_{\mathrm{AH}}(\rho) &= \frac{8\pi G\Theta}{9\hbar m^2c^5}\left(\mathcal{D}[p_A^2]+\mathcal{D}[p_B^2]\right)(\rho),\\
\mathcal{L}_{\mathrm{encode}}(\rho) &= \Gamma_{\mathrm{encode}}\left(\mathcal{D}[a_A^2-\alpha^2]+\mathcal{D}[a_B^2-\alpha^2]\right)(\rho). \tag*{(222)}
\end{aligned} d t d ρ L NG ( ρ ) L AH ( ρ ) L encode ( ρ ) = L NG ( ρ ) + L AH ( ρ ) + L encode ( ρ ) , = − ℏ d 3 2 i G m A m B [ x A x B , ρ ] , = 9ℏ m 2 c 5 8 π G Θ ( D [ p A 2 ] + D [ p B 2 ] ) ( ρ ) , = Γ encode ( D [ a A 2 − α 2 ] + D [ a B 2 − α 2 ] ) ( ρ ) . (222) We consider the strong encoding limit Γ e n c o d e ≫ λ N G ≡ 2 G m 2 x 0 2 / ℏ d 3 \Gamma_{\mathrm{encode}}\gg\lambda_{\mathrm{NG}}\equiv2Gm^2x_0^2/\hbar d^3 Γ encode ≫ λ NG ≡ 2 G m 2 x 0 2 /ℏ d 3 Γ G D ≡ 8 π G Θ p 0 4 / 9 ℏ m 2 c 5 \Gamma_{\mathrm{GD}}\equiv8\pi G\Theta p_0^4/9\hbar m^2c^5 Γ GD ≡ 8 π G Θ p 0 4 /9ℏ m 2 c 5 L e n c o d e \mathcal{L}_{\mathrm{encode}} L encode C A B = span { ∣ i ˉ ⟩ A ⊗ ∣ j ˉ ⟩ B } i , j = 0 , 1 C_{AB}=\operatorname{span}\{|\bar{i}\rangle_A\otimes|\bar{j}\rangle_B\}_{i,j=0,1} C A B = span { ∣ i ˉ ⟩ A ⊗ ∣ j ˉ ⟩ B } i , j = 0 , 1 L N G \mathcal{L}_{\mathrm{NG}} L NG L A H \mathcal{L}_{\mathrm{AH}} L AH
In the following, we study the asymptotic state of this Lindbladian. For the numerical simulations, we choose α = 1 / 2 \alpha=1/2 α = 1/2
Π p 2 Π / p 0 2 = e e − 1 − e e − 1 σ z , (223) \Pi p^2\Pi/p_0^2=\frac{e}{e-1}-\frac{\sqrt{e}}{e-1}\sigma^z, \tag*{(223)} Π p 2 Π/ p 0 2 = e − 1 e − e − 1 e σ z , (223) Π p 4 Π / p 0 4 = 3 e + 2 e − 1 − 5 e e − 1 σ z , (224) \Pi p^4\Pi/p_0^4=\frac{3e+2}{e-1}-\frac{5\sqrt{e}}{e-1}\sigma^z, \tag*{(224)} Π p 4 Π/ p 0 4 = e − 1 3 e + 2 − e − 1 5 e σ z , (224) Π x Π / x 0 = 1 1 − e − 1 σ x . (225) \Pi x\Pi/x_0=\frac{1}{\sqrt{1-e^{-1}}}\sigma^x. \tag*{(225)} Π x Π/ x 0 = 1 − e − 1 1 σ x . (225) We initialize the system in ∣ 0 ˉ ⟩ A ⊗ ∣ 0 ˉ ⟩ B |\bar{0}\rangle_A\otimes|\bar{0}\rangle_B ∣ 0 ˉ ⟩ A ⊗ ∣ 0 ˉ ⟩ B { ∣ 0 ˉ 0 ˉ ⟩ , ∣ 1 ˉ 1 ˉ ⟩ } \{|\bar{0}\bar{0}\rangle,|\bar{1}\bar{1}\rangle\} { ∣ 0 ˉ 0 ˉ ⟩ , ∣ 1 ˉ 1 ˉ ⟩}
Π A ⊗ Π B ρ ( ∞ ) Π A ⊗ Π B = ( a 0 0 b i 0 0 0 0 0 0 0 0 − b i 0 0 c ) , (226) \Pi_A\otimes\Pi_B\rho(\infty)\Pi_A\otimes\Pi_B=
\begin{pmatrix}
a&0&0&b\mathrm{i}\\
0&0&0&0\\
0&0&0&0\\
-b\mathrm{i}&0&0&c
\end{pmatrix}, \tag*{(226)} Π A ⊗ Π B ρ ( ∞ ) Π A ⊗ Π B = a 0 0 − b i 0 0 0 0 0 0 0 0 b i 0 0 c , (226) where a , b , c a,b,c a , b , c b 2 ≤ a c b^2\leq ac b 2 ≤ a c
ρ p o s t ∞ = ( a a + c 0 0 b i a + c 0 0 0 0 0 0 0 0 − b i a + c 0 0 c a + c ) = a a + c ∣ 0 ˉ 0 ˉ ⟩ ⟨ 0 ˉ 0 ˉ ∣ + b i a + c ( ∣ 0 ˉ 0 ˉ ⟩ ⟨ 1 ˉ 1 ˉ ∣ − ∣ 1 ˉ 1 ˉ ⟩ ⟨ 0 ˉ 0 ˉ ∣ ) + c a + c ∣ 1 ˉ 1 ˉ ⟩ ⟨ 1 ˉ 1 ˉ ∣ . (227) \begin{aligned}
\rho^\infty_{\mathrm{post}}&=
\begin{pmatrix}
\frac{a}{a+c}&0&0&\frac{b\mathrm{i}}{a+c}\\
0&0&0&0\\
0&0&0&0\\
-\frac{b\mathrm{i}}{a+c}&0&0&\frac{c}{a+c}
\end{pmatrix}\\
&=\frac{a}{a+c}|\bar{0}\bar{0}\rangle\langle\bar{0}\bar{0}|+\frac{b\mathrm{i}}{a+c}\left(|\bar{0}\bar{0}\rangle\langle\bar{1}\bar{1}|-|\bar{1}\bar{1}\rangle\langle\bar{0}\bar{0}|\right)+\frac{c}{a+c}|\bar{1}\bar{1}\rangle\langle\bar{1}\bar{1}|. \tag*{(227)}
\end{aligned} ρ post ∞ = a + c a 0 0 − a + c b i 0 0 0 0 0 0 0 0 a + c b i 0 0 a + c c = a + c a ∣ 0 ˉ 0 ˉ ⟩ ⟨ 0 ˉ 0 ˉ ∣ + a + c b i ( ∣ 0 ˉ 0 ˉ ⟩ ⟨ 1 ˉ 1 ˉ ∣ − ∣ 1 ˉ 1 ˉ ⟩ ⟨ 0 ˉ 0 ˉ ∣ ) + a + c c ∣ 1 ˉ 1 ˉ ⟩ ⟨ 1 ˉ 1 ˉ ∣. (227) The amount of entanglement for this family of states can be characterized by the concurrence [61 ] . The post-selected asymptotic state has non-zero concurrence whenever b ≠ 0 b\ne0 b = 0
C ( ρ p o s t ∞ ) = 2 ∣ b ∣ a + c . (228) C(\rho^\infty_{\mathrm{post}})=\frac{2|b|}{a+c}. \tag*{(228)} C ( ρ post ∞ ) = a + c 2∣ b ∣ . (228) We now examine how a , b , c a,b,c a , b , c λ N G = 2 G m 2 x 0 2 / ℏ d 3 \lambda_{\mathrm{NG}}=2Gm^2x_0^2/\hbar d^3 λ NG = 2 G m 2 x 0 2 /ℏ d 3 Γ G D = 8 π G Θ p 0 4 / 9 ℏ m 2 c 5 \Gamma_{\mathrm{GD}}=8\pi G\Theta p_0^4/9\hbar m^2c^5 Γ GD = 8 π G Θ p 0 4 /9ℏ m 2 c 5 Γ e n c o d e \Gamma_{\mathrm{encode}} Γ encode Γ e n c o d e = 100 \Gamma_{\mathrm{encode}}=100 Γ encode = 100
Tr ( Π A Π B ρ ( ∞ ) Π A Π B ) = a + c , (229) \operatorname{Tr}\left(\Pi_A\Pi_B\rho(\infty)\Pi_A\Pi_B\right)=a+c, \tag*{(229)} Tr ( Π A Π B ρ ( ∞ ) Π A Π B ) = a + c , (229) and (ii) the concurrence of the post-selected state C ( ρ p o s t ∞ ) = 2 b / ( a + c ) C(\rho^\infty_{\mathrm{post}})=2b/(a+c) C ( ρ post ∞ ) = 2 b / ( a + c ) λ N G \lambda_{\mathrm{NG}} λ NG Γ G D \Gamma_{\mathrm{GD}} Γ GD Figure 11 . It decreases as λ N G \lambda_{\mathrm{NG}} λ NG Γ G D \Gamma_{\mathrm{GD}} Γ GD Γ e n c o d e \Gamma_{\mathrm{encode}} Γ encode λ N G = Γ G D = Γ e n c o d e \lambda_{\mathrm{NG}}=\Gamma_{\mathrm{GD}}=\Gamma_{\mathrm{encode}} λ NG = Γ GD = Γ encode a + c ≈ 0.31 a+c\approx0.31 a + c ≈ 0.31 C ≈ 0.52 C\approx0.52 C ≈ 0.52 C ( ρ p o s t ∞ ) C(\rho^\infty_{\mathrm{post}}) C ( ρ post ∞ ) Figure 12 . The maximum concurrence is achieved around λ N G / Γ e n c o d e ≈ 0.46 \lambda_{\mathrm{NG}}/\Gamma_{\mathrm{encode}}\approx0.46 λ NG / Γ encode ≈ 0.46 Γ G D = 0 \Gamma_{\mathrm{GD}}=0 Γ GD = 0 a + c ≈ 0.69 a+c\approx0.69 a + c ≈ 0.69 C ≈ 0.91 C\approx0.91 C ≈ 0.91 Figure 12 b displays the contour plot in the small λ N G \lambda_{\mathrm{NG}} λ NG λ N G \lambda_{\mathrm{NG}} λ NG Γ G D ≫ Γ e n c o d e \Gamma_{\mathrm{GD}}\gg\Gamma_{\mathrm{encode}} Γ GD ≫ Γ encode Figure 12 a. In the weak interaction limit, the contour can be well approximated by a linear relation; for instance, the fitted relation indicates that when
Figure 11. Probability for the post-selection into the code subspace { ∣ i ˉ ⟩ A ⊗ ∣ j ˉ ⟩ B } i , j = 0 , 1 \{\lvert\bar{i}\rangle_A \otimes\lvert\bar{j}\rangle_B\}_{i,j=0,1} {∣ i ˉ ⟩ A ⊗ ∣ j ˉ ⟩ B } i , j = 0 , 1 λ N G \lambda_{\mathrm{NG}} λ NG Γ G D \Gamma_{\mathrm{GD}} Γ GD Γ e n c o d e = 100 \Gamma_{\mathrm{encode}}=100 Γ encode = 100
Figure 12. Concurrence C ( ρ p o s t ∞ ) C(\rho_{\mathrm{post}}^\infty) C ( ρ post ∞ ) { ∣ i ˉ ⟩ A ⊗ ∣ j ˉ ⟩ B } i , j = 0 , 1 \{\lvert\bar{i}\rangle_A \otimes\lvert\bar{j}\rangle_B\}_{i,j=0,1} {∣ i ˉ ⟩ A ⊗ ∣ j ˉ ⟩ B } i , j = 0 , 1 λ N G \lambda_{\mathrm{NG}} λ NG Γ G D \Gamma_{\mathrm{GD}} Γ GD Γ e n c o d e = 100 \Gamma_{\mathrm{encode}}=100 Γ encode = 100 C ( ρ p o s t ∞ ) = 0.15 , 0.30 C(\rho_{\mathrm{post}}^\infty)=0.15, 0.30 C ( ρ post ∞ ) = 0.15 , 0.30 λ N G \lambda_{\mathrm{NG}} λ NG Γ G D \Gamma_{\mathrm{GD}} Γ GD Γ G D \Gamma_{\mathrm{GD}} Γ GD
Γ G D ≤ 20 λ N G − 7 5 Γ e n c o d e , (230) \Gamma_{\mathrm{GD}} \le20\lambda_{\mathrm{NG}}-\frac{7}{5}\Gamma_{\mathrm{encode}}, \tag*{(230)} Γ GD ≤ 20 λ NG − 5 7 Γ encode , (230) one can achieve an observable concurrence C ( ρ p o s t ∞ ) ≥ 0.15 C(\rho_{\mathrm{post}}^\infty)\ge0.15 C ( ρ post ∞ ) ≥ 0.15
λ N G ≥ Γ G D 20 + 7 Γ e n c o d e 100 , (231) \lambda_{\mathrm{NG}} \ge\frac{\Gamma_{\mathrm{GD}}}{20}+\frac{7\Gamma_{\mathrm{encode}}}{100}, \tag*{(231)} λ NG ≥ 20 Γ GD + 100 7 Γ encode , (231) which implies that entanglement can remain observable even in the regime λ N G ≪ Γ G D \lambda_{\mathrm{NG}}\ll\Gamma_{\mathrm{GD}} λ NG ≪ Γ GD
In summary, the AH model provides a particularly transparent example in which the dominant p 2 p^2 p 2
