Correlated Catalysis Removes the Strong Converse for Entanglement Distillation

Introduction

Quantum entanglement [1] is a fundamental type of correlation between quantum particles, enabling communication protocols that surpass classical limits. As a uniquely quantum phenomenon, entanglement acts as a vital resource for tasks such as quantum teleportation [2], superdense coding [3], and quantum key distribution [4, 5]. These applications motivate the development of methods for preparing high-quality entangled states.

Entanglement is often studied in the context of resource theory. In this setting, two parties, say Alice and Bob, can perform only local quantum channels, and communicate through a classical channel. These operations are called local operations and classical communication (LOCC) [6].

Entanglement distillation enables the extraction of nearly maximally entangled states from noisy entangled input using local operations and classical communication [7, 8, 9, 10, 11, 12, 13]. For a finite number of input copies, both deterministic protocols based on quantum error correction [8, 13] and probabilistic protocols based on postselection [7, 9] have been studied. In the asymptotic regime, the central objective is to determine the optimal rate at which maximally entangled pairs can be extracted as the number of input copies tends to infinity [8, 14, 15].

In the asymptotic regime, a distillation protocol $\mathcal{D}_n$ transforms $n$ copies of a bipartite state $\rho$ into an output state approximating $m_n$ maximally entangled qubit pairs. The distillation rate is the number of target pairs per input copy, $R = \lim_{n\to\infty} m_n/n$. The distillable entanglement $E_D(\rho)$ is the supremum of rates achievable such that the fidelity between the output state and $[Rn]$ maximally entangled states tends to unity [16]:

where the supremum is taken over rates $R$ for which there exists a sequence of LOCC channels $\{\mathcal{D}_n\}_{n\in\mathbb{N}}$, and $\Phi_2$ denotes the density operator of a maximally entangled qubit pair. Operationally, $E_D(\rho)$ quantifies the asymptotic number of Bell pairs that can be extracted per input copy with vanishing error [17].

The definition of distillable entanglement rules out asymptotically perfect distillation at rates $R > E_D(\rho)$, but does not determine how the fidelity behaves in this regime. A stronger requirement is the strong converse property: for every rate $R > E_D(\rho)$, the fidelity of the output with the target state $\Phi_2^{\otimes[Rn]}$ must tend to zero as $n\to\infty$, irrespective of the sequence of LOCC protocols employed [18]. Thus, when the strong converse holds, allowing a fixed nonzero error does not increase the asymptotically achievable distillation rate.

It is known that the strong converse holds for bipartite pure states [19] and for maximally correlated states [20]. More generally, the Rains relative entropy provides a strong-converse upper bound on the distillation rate: whenever the rate exceeds this bound, the fidelity vanishes exponentially with the number of input copies [21, 22]. Whether the strong converse holds for unassisted distillation from arbitrary bipartite states remains open. In this paper, we investigate whether the strong converse also holds for entanglement distillation protocols assisted by a catalyst.

Entanglement catalysis provides another way to extend the possibilities of quantum state manipulation under LOCC [23, 24, 25, 26, 27]. In its original formulation [23], Alice and Bob have access to an auxiliary entangled state $\omega_C$, called a catalyst, which assists a transformation while being returned unchanged and uncorrelated with the output. A catalytic transformation therefore takes the form

Remarkably, such a transformation may be possible even when $\rho$ cannot be converted into $\sigma$ by LOCC alone. This framework was subsequently extended to correlated catalysis [28, 29, 30], in which the catalyst is returned exactly in its marginal state but may become correlated with the output system [31, 32, 33]. The final state $\Omega_{SC}$ is such that

instead of $\Omega_{SC}=\sigma\otimes\omega_C$. Allowing such residual correlations can enlarge the set of achievable state transformations [31].

Entangled states with positive partial transpose remain undistillable under correlated catalytic local operations and classical communication, while retaining a positive entanglement cost [33]. Thus, correlated catalysis does not make entanglement theory asymptotically reversible.

In this work, we show that correlated catalysis removes the strong converse for every distillable bipartite state, even when the correlations between the output and the catalyst vanish asymptotically. More precisely, at any rate $R>E_D(\rho)$, the optimal asymptotic fidelity equals $E_D(\rho)/R$, whereas fidelity approaching unity is achievable for $R\le E_D(\rho)$. These limits remain achievable under any prescribed positive sequence of upper bounds on the output–catalyst mutual information tending to zero.

We also determine the catalytic resources needed to sustain non-vanishing fidelity above the reliable distillation rate. At rates where non-catalytic distillation obeys an exponential strong converse, the catalyst’s max-relative entropy of entanglement must grow at least linearly with the input blocklength $n$. Requiring the output–catalyst mutual information to satisfy $I(S:C)\le\delta_n$, for any prescribed positive sequence $\delta_n\to0$, strengthens this lower bound to $\Omega(n\log_2(1/\delta_n))$, and hence imposes the same lower bound on the logarithm of each local catalyst dimension. Conversely, for every distillable source, every $R>E_D(\rho)$, and every fixed $0<f_0<E_D(\rho)/R$, we construct correlated catalytic protocols achieving fidelity at least $f_0$ with $\log_2\dim C_A,\log_2\dim C_B=O(n\log_2(1/\delta_n))$. Thus, whenever the exponential-converse assumption holds, the optimal catalyst dimension satisfies $\log_2 D_n^{\mathrm{opt}}=\Theta(n\log_2(1/\delta_n))$. In particular, this conclusion applies to pure and maximally correlated source states at every rate above their distillable entanglement.

Definitions and problem statement

Let

be a maximally entangled state with local dimensions $d$. The $m$-fold tensor product of a maximally entangled qubit pair is therefore $\Phi_2^{\otimes m}=\Phi_{2^m}$.

As discussed in the introduction, catalysis can enable transformations that are otherwise impossible under LOCC alone. Here, we study entanglement distillation assisted by a bipartite catalyst. We define a distillation protocol with catalyst as follows [31, 32]:

Definition 1. Correlated catalytic distillation protocol

Fix a bipartite source state $\rho$ and a target of $m$ Bell pairs. A correlated catalytic distillation protocol is a deterministic LOCC channel $\mathcal{D}_{\mathrm{dist}}$ that takes $n$ copies of the state $\rho$ and a bipartite catalyst $\omega_C$ and outputs

We quantify the performance of a catalytic distillation protocol by the overlap of its output marginal with the target maximally entangled state:

The correlations between the output and the catalyst can be quantified by the quantum mutual information [34]

where $S(\rho)=-\operatorname{Tr}(\rho\log(\rho))$ denotes the von Neumann entropy [27]. Additionally, we can bound the trace distance to uncorrelated catalysis by the quantum Pinsker inequality with quantum mutual information [35],

In particular, $I(S:C)_\Omega=0$ implies $\Omega_{SC}=\Omega_S\otimes\omega_C$. Moreover, vanishing mutual information implies that the joint state approaches the product of its marginals in trace distance.

Throughout this work, we consider finite-dimensional catalysts with unrestricted dimension and entanglement, and investigate protocols in which the residual correlations between the system $\Omega_S$ and catalyst $\Omega_C$ can be made arbitrarily small while preserving the exact catalyst marginal.

Suppose that we have $n$ copies of a state $\rho$ and want to produce $m$ copies of $\Phi_2$ using correlated catalytic entanglement distillation. The maximum achievable fidelity with a target state $\Phi_{2^m}$ is:

where the supremum is taken over all finite-dimensional catalysts and deterministic LOCC channels satisfying $\Omega_C=\omega_C$ and $I(S:C)_\Omega\le\delta$. See Eq. (5) for the definition of $\Omega_{SC}$. We use $\delta=\infty$ when no restriction on the output–catalyst correlations is imposed.

For a fixed distillation rate $R>0$, let $m_n=\lfloor Rn\rfloor$. The reliable catalytic distillation rate is defined as

This quantity coincides with the correlated-catalytic distillable entanglement $E_{d,\mathrm{LOCCcc}}(\rho)$ defined in [33], since vanishing trace-distance error and fidelity tending to one are equivalent for the pure target state. The corresponding strong-converse threshold is

The strong converse holds if every rate exceeding the reliable distillation rate necessarily leads to vanishing global fidelity in the limit $n\to\infty$. In other words, for an input state $\rho$, the strong converse holds iff $D^{\mathrm{cc}}(\rho)=\widetilde{D}^{\mathrm{cc}}(\rho)$. In this paper we show that strong converse does not hold for catalytic entanglement distillation protocols.

Throughout this work, all logarithms are base two unless stated otherwise. We use standard asymptotic notation: for nonnegative sequences $a_n$ and $b_n$, we write $a_n=O(b_n)$ if there exists a constant $C>0$ such that $a_n\le Cb_n$ for all sufficiently large $n$. We write $a_n=\Omega(b_n)$ if there exists a constant $c>0$ such that $a_n\ge cb_n$ for all sufficiently large $n$. Finally, we write $a_n=\Theta(b_n)$ if there exist constants $c,C>0$ such that $cb_n\le a_n\le Cb_n$ for all sufficiently large $n$.

Throughout this work, fidelity refers to the squared fidelity; for a pure target $\Phi$, it is given by $\operatorname{Tr}(\Phi\rho)$.

Main results

We first show that exact catalysis with vanishing correlations removes the strong converse for every distillable source state. At any rate $R>E_D(\rho)$, the optimal asymptotic fidelity is $E_D(\rho)/R$, even when output–catalyst correlations vanish asymptotically.

The construction used in the proof of Theorem 2 achieves the optimal asymptotic fidelity $E_D(\rho)/R$ at rates $R>E_D(\rho)$, but requires a large catalyst memory.

In the following, we derive lower bounds on the catalyst resources needed to maintain a fixed positive fidelity at rates where non-catalytic distillation obeys an exponential decay of fidelity.

Let

denote the optimal fidelity without a catalyst. For a bipartite catalyst state $\omega_C$, define its max-relative entropy of entanglement [36]

The Rains relative entropy $R_{\mathrm{ent}}(\rho)$, defined as [37]

provides an exponential converse-bound for non-catalytic distillation of any finite-dimensional bipartite state [21, 22], i.e. every rate $R$ satisfying $R>R_{\mathrm{ent}}(\rho)$ leads to exponential decay of $F_{\rho\to\Phi_2}^{\mathrm{not}}$. In Eq. (19), $T_B$ denotes partial transposition on Bob’s subsystem [38, 39] and $D(\rho\mathbin\Vert\sigma)=\operatorname{Tr}[\rho(\log_2\rho-\log_2\sigma)]$ is the quantum relative entropy, with value $+\infty$ when $\operatorname{supp}\rho\not\subset\operatorname{supp}\sigma$.

For pure and maximally correlated states, $E_D(\rho)=R_{\mathrm{ent}}(\rho)$ [20], so for these states every rate above distillable entanglement leads to exponential decay of fidelity with target state $\Phi_2$. Our catalyst-resource lower bounds require an exponential fidelity bound at a threshold strictly below the target rate. We formulate this assumption below.

We next derive lower bounds on the catalyst entanglement required to maintain nonvanishing fidelity at rates where non-catalytic distillation obeys an exponential converse bound. The following theorem applies at every target rate $R>R_{\mathrm{ent}}(\rho)$, since its exponential-converse assumption is satisfied by choosing a threshold rate $r_{\mathrm{sc}}$ such that $R_{\mathrm{ent}}(\rho)<r_{\mathrm{sc}}<R$.

that it achieves the rate $R > r_{\mathrm{sc}}$ and the fidelity of this protocol with a target Bell state is greater than a constant $f_0 \in(0,1)$.

Then, for all sufficiently large $n$ and with no restriction on catalyst correlation, the catalyst’s max-relative entropy of entanglement satisfies

Furthermore, if the correlations of the joint output state $\Omega_{SC}$ are restricted by

then

The proof is given in § 1.4.

We next translate the lower bounds on $E_{\max}(\omega_C)$ into lower bounds on the local catalyst dimensions. Let

denote the smaller local catalyst dimension. We show the following corollary:

Proof. The max-relative entropy of entanglement satisfies the standard dimension bound [36, 40]

Consequently,

Combining this inequality with Theorem 4 gives

for protocols without a constraint on output–catalyst correlations. Under the additional correlation assumptions, Theorem 4 also yields

which proves the remaining claims. □

Although the above corollary provides a lower bound on the dimension of the catalyst, it does not provide a protocol that uses such a catalyst. The following theorem constructs a protocol which reaches the scaling specified by the lower bound on catalyst dimensions (26) of Corollary 5.

Now, let $D_n^{\mathrm{opt}}$ be the minimum value of $\min\{\dim C_A,\dim C_B\}$ over all deterministic correlated catalytic LOCC protocols with input $\rho^{\otimes n}$, target $\Phi_{2^{\lfloor Rn\rfloor}}$, fidelity at least $f_0$, and output–catalyst mutual information at most $\delta_n$. Using above results we show the following corollary.

The constants in these bounds may depend on $\rho$, $R$, $f_0$, and the fixed exponential-converse parameters, but not on $n$ or $\delta_n$. The strict inequality $f_0<E_D(\rho)/R$ provides the fixed positive gap required for our construction. Thus, under the assumptions of Corollary 7, the optimal catalyst dimension satisfies $\log_2 D_n^{\mathrm{opt}}=\Theta(n\log_2(1/\delta_n))$ for every fixed fidelity requirement strictly below the optimal asymptotic fidelity. This does not determine the optimal dimension scaling when the fidelity itself converges to $E_D(\rho)/R$.

Operationally, this corresponds to an optimal local catalyst memory cost of $\Theta(n\log_{2}(1/\delta_n))$ qubits: every admissible protocol requires $\Omega(n\log_{2}(1/\delta_n))$ qubits on each side, and our construction achieves the matching upper bound on both sides. This counts the memory needed to represent the catalyst, not the number of Bell pairs it contains.

Proof of Theorem 2

In this section, we detail the proof of theorem 2. Within the proof, we will use several lemmas whose full statements and proofs can be found in Section VII.

To begin, assume that we have a state $\rho$ such that $E_D(\rho)>0$. First, we find an upper bound on $\limsup_{n\to\infty}F_{\rho\to\Phi_2}^{\mathrm{cc}}(n,m_n;\infty)$.

Let a catalytic protocol take $\rho^{\otimes n}$ as input and produce an $m_n$-pair output of fidelity $f(\Omega_{SC},m_n)$. Lemmas 9 and 10 give, for $m_n\geq8$,

Hence

For $m_n=\lfloor Rn\rfloor$ this yields

The bound is independent of catalyst dimension, catalyst entanglement, and output–catalyst correlation.

We now construct a protocol that asymptotically attains this upper bound. The protocol consists of two stages. First, correlated catalytic amplification produces a mixture of $\widetilde{m}>n$ copies of the source state and a separable state. Second, ordinary LOCC distillation with a rate $r$ is applied to obtain $m_n$ Bell pairs. In the following paragraph we describe the first stage.

Fix a rate $r$ with $0<r<E_D(\rho)$ and a requested rate $R>r$. For all sufficiently large $n$ set

Since $r<E_D(\rho)$ and $\lfloor r\widetilde{m}\rfloor\geq m_n$, there are LOCC distillation channels on $\widetilde{m}$ source copies producing $m_n$ Bell pairs with fidelity tending to one, after discarding any excess output pairs.

Let $\tau$ be a separable bipartite state on the same Hilbert space as $\rho$. Choose an integer $N\geq2$ such that

Such a finite $N$ exists because $\log_2(N+1)/N\to0$ along the multiples of $\widetilde{m}/\gcd(n,\widetilde{m})$. By Proposition 12, there exists a correlated catalytic LOCC protocol with input $\rho^{\otimes n}$ and output marginal

satisfying

Now we proceed to the second stage. We apply the ordinary $\widetilde{m}$-copy distillation channel only to the system output. Let us denote it by $D_d$. This does not change the catalyst marginal and cannot increase $I(S:C)$. Additionally, we have

Since $r<E_D(\rho)$, ordinary distillation achieves

Moreover, since $\tau$ is separable and $D_d$ is LOCC, the state $D_d(\tau^{\otimes\widetilde{m}})$ is separable. Thus, by Lemma 8,

Consequently, the achieved fidelity satisfies

where we used Eq. (39).

For $R>E_D(\rho)$, this construction therefore gives

Since this holds for every fixed $0<r<E_D(\rho)$, letting $r\uparrow E_D(\rho)$ yields

For $R<E_D(\rho)$, ordinary unassisted distillation already achieves fidelity tending to one. At $R=E_D(\rho)$, the same catalytic construction gives an asymptotic fidelity of at least $r/E_D(\rho)$ for every fixed $r<E_D(\rho)$. Letting $r\uparrow E_D(\rho)$ therefore yields a lower bound of one. Combining all three cases, we obtain

For every $R<E_D(\rho)$, the limiting fidelity is one; for every $R>E_D(\rho)$, it is $E_D(\rho)/R<1$. Hence the reliable rate is $E_D(\rho)$. For every finite $R$, the limiting fidelity is positive because $E_D(\rho)>0$. Therefore the nonvanishing-fidelity threshold is infinite, proving (13).

Proof of Theorem 4

In this section, we detail the proof of theorem 4. Let $f_n$ denote the fidelity of the catalytic protocol with the target state $\Phi_{2^{m_n}}$. By assumption, we have $f_n > f_0$. Since $m_n/n \to R > r_{\mathrm{sc}}$, for all sufficiently large $n$, we have $n \ge N_0$ and $m_n \ge r_{\mathrm{sc}}n$. Lemma 13 and Eq. (20) therefore give

Taking logarithms proves Eq. (21).

We now repeat the protocol $k$ times, each time using a fresh input block $\rho^{\otimes n}$ and the same catalyst. Since the catalyst marginal is returned exactly after each use, the resulting process is a catalytic LOCC protocol with input $\rho^{\otimes kn}$, target $km_n$ Bell pairs, and catalyst $\omega_C$. Since

our exponential-converse assumption implies

Now, we impose the additional constraints that the correlations within the joint output state $\Omega_{SC}$ are restricted by

Then, by applying Lemma 14, we obtain

Set

Since $\delta_n\to0$, eventually $k_n\ge2$. Moreover,

Consequently,

where we used the fact that exponential decay dominates polynomial growth. Thus, for all sufficiently large $n$,

Substituting $k=k_n$ into Eq. (54) gives

Taking logarithms yields

For sufficiently large $n$, $k_n\ge\frac{\ell_n}{8b}$ and $n\ge\frac{2b}{\gamma}$. Consequently,

The threshold in $n$ and the constants in this bound depend only on the fixed parameters and the prescribed sequences $(m_n)$ and $(\delta_n)$, not on the protocol or catalyst. Thus, the bound holds uniformly over all admissible protocols. Eq. (23) follows.

Proof of Theorem 6

In this section, we detail the proof of theorem 6. As in the proof of Theorem 2, the protocol consists of two stages. First, correlated catalysis transforms $n$ copies of $\rho$ into a mixture of $\rho^{\otimes\tilde m}$, with $\tilde m>n$, and a separable state. Second, ordinary LOCC distillation is applied to this output to produce $m_n$ approximate Bell pairs. To reduce the catalyst memory cost, the first stage uses a truncated binomial distribution instead of the fixed-weight distribution used in Proposition 12.

Choose fixed constants $p,r$ such that

For all sufficiently large $n$, set

Then

Since $r<E_D(\rho)$ and $\lfloor r\tilde m\rfloor\ge m_n$, there exist deterministic LOCC distillation channels $\mathcal{D}_d$ such that

We now construct the first stage of the protocol. Let $\tau$ be a separable state on the same bipartite Hilbert space as $\rho$, and define

The additional copies of $\tau$ can be prepared freely by LOCC.

For an integer $N\ge2$, to be chosen below, set

Let $Q_p$ be the probability distribution on $\{0,1\}^N$ given by

where $|x| = \sum_{i=1}^{N} x_i$ is the Hamming weight of the binary string $x=(x_1,\ldots,x_N)$, i.e., the number of entries equal to one. Define

where

Thus, $P(x)$ is the probability of sampling the string $x$. Consider the permutation-invariant state

The transformation $a^{\otimes N}\to\Gamma_N$ is deterministic LOCC. Indeed, Alice samples $x$ according to $P$ and communicates it to Bob. Every string in the support of $P$ requires $|x|\widetilde{m}\le nN$ copies of $\rho$. The parties discard the unused copies of $\rho$, replace them with copies of $\tau$, and perform matching local permutations to prepare the corresponding tensor product.

Let $\Gamma_j$ denote the $j$-block marginal of $\Gamma_N$, with $\Gamma_0=1$ (it is a scalar, see the comment in Lemma 11) on the trivial system. By Lemma 11, the state

enables a correlated catalytic conversion of one fresh block $a$ into an output with marginal $\Gamma_1$. The complete catalyst marginal is returned exactly, and the output–catalyst mutual information at this stage equals

To bound this quantity, define

Let $q_p(0)=1-p$ and $q_p(1)=p$, so that $Q_p(x)=\prod_{i=1}^{N}q_p(x_i)$. Denote by $\mathcal{E}$ the classical-to-quantum channel that prepares $\theta_{x_1}\otimes\cdots\otimes\theta_{x_N}$ conditional on the classical string $x$. It maps the distribution $P$ to $\Gamma_N$ and $Q_p$ to $\sigma_p^{\otimes N}$. Indeed,

Data processing therefore gives

Furthermore, since every one-block marginal of $\Gamma_N$ equals $\Gamma_1$,

Consequently,

Choose a fixed $\eta>0$ such that, for all sufficiently large $n$,

For $X\sim\operatorname{Bin}(N,p)$, Hoeffding’s inequality yields [41]

Whenever $1-Z\le1/2$, we also have

Combining these inequalities with Eq. (78) gives

The condition $Z\ge1/2$ means that the admissible configurations carry at least half of the probability under the original distribution $Q_p$. Since $1-Z\le e^{-2\eta^2N}$, it is sufficient to take

As $\eta>0$ is fixed, this condition is automatically satisfied for sufficiently large $N$. It imposes no additional assumption on the source state. Note that $Z$ is a normalization factor, not the success probability of the catalytic protocol: the protocol samples directly from $P$, whose support contains only admissible configurations.

For all sufficiently large $n$, we have $\delta_n<1$. We may then choose

Here $\ln$ denotes the natural logarithm. This choice ensures

and $N \to\infty$.

By permutation invariance,

Since conditioning on $X \le K$ removes only an upper tail,

Thus $p_N \to p$.

Finally, apply $\mathcal{D}_d$ (recall that $\mathcal{D}_d$ is ordinary distillation protocol) only to the released output block. This preserves the catalyst marginal and cannot increase the output–catalyst mutual information. The final state therefore satisfies

Because $\mathcal{D}_d(\tau^{\otimes\tilde{m}})$ is separable,

Hence

which implies

In particular, $f_n \ge f_0$ for all sufficiently large $n$. No comparison between $\epsilon_n$ and $\delta_n$ is required.

It remains to bound the catalyst dimensions. Let $d_A,d_B$ be the local dimensions of $\rho$. Each catalyst block contains $\tilde{m}$ copies of the source Hilbert space, and the classical clock has $N$ values. Including the clock, the local dimensions can be chosen to satisfy

Since $\tilde{m}=O(n)$ and $N=O(\log_2(1/\delta_n))$, both quantities are $O(n\log_2(1/\delta_n))$, as claimed.

AUXILIARY LEMMAS AND PROPOSITIONS

Lemma 8 (Separable target overlap [22]). For every separable state $\sigma_{AB}$ with local dimension $D$,

Lemma 9 (Distillable entanglement of a finite catalytic output). Assume that $E_D(\rho)>0$. Suppose a finite-block exact-return catalytic LOCC protocol maps $\rho^{\otimes n}$ to an output marginal $\Omega_S$. Then

Proof. Fix $n$ and initial catalyst $\omega_C$ and assume instead that $E_D(\Omega_S)>nE_D(\rho)$. Consequently, there exists $r$ such that $nE_D(\rho)<r<E_D(\Omega_S)$. Run $t$ independent copies of the finite protocol on $\rho^{\otimes n}$, using independent copies of its finite catalyst. The system output is exactly $\Omega_S^{\otimes t}$ and the complete catalyst marginal is exactly $\omega_C^{\otimes t}$. Apply an ordinary distillation protocol only to $\Omega_S^{\otimes t}$. Such protocol does not change the catalyst marginal. Since $r<E_D(\Omega_S)$ there exist protocol which distill $\lfloor rt\rfloor$ copies of $\Phi_2$ state as $t\to\infty$. Thus, we constructed a protocol which consumes $nt$ copies of a state $\rho$ and distills $rt$ copies of $\Phi_2$. The rate of this protocol is $\frac{\lfloor rt\rfloor}{nt}=\frac{r}{n}>E_D(\rho)$ in the limit $t\to\infty$. This contradicts the catalytic rate-invariance theorem (Theorem 2 of Ref. [32]). Therefore, (95) holds.

Lemma 10 (Bell-product overlap lower-bounds $E_D$). Let $\sigma$ be a state on $m\ge8$ qubit pairs and let

Set

Then

Proof. Let $D=2^m$. Apply the isotropic twirl $U\otimes\overline{U}$, which is LOCC and preserves $f$. Since distillable entanglement cannot increase under LOCC, it is enough to lower-bound the distillable entanglement of

Decompose each local $D$-dimensional space into two subsystems of dimensions $d_1$ and $d_2$ such that:

The maximally entangled state factors as $\Phi_D=\Phi_{d_1}\otimes\Phi_{d_2}$.

Alice and Bob measure their $d_1$-dimensional subsystems in the computational basis. They keep the $d_2$-dimensional subsystems exactly when their outcomes agree and forget the common outcome. The ideal component $\Phi_D$ always passes and leaves $\Phi_{d_2}$. The identity operator has $d_1$ matching outcomes, so the unnormalized retained state is

Its success probability and unnormalized Bell overlap are

The normalized successful state is isotropic with local dimension $d_2$, with Bell fidelity

@references 久久精品 Its local marginals are maximally mixed, and therefore its coherent information is

where $h_2$ denotes the binary entropy. Here, $A$ and $B$ denote Alice’s and Bob’s retained subsystems, respectively, each of dimension $d_2$. The hashing inequality states that the positive part of coherent information is an achievable one-way distillation rate [15]. Applying the filter independently to many copies and hashing the successful outputs gives

The final inequality is valid even when the expression is negative, because $\max\{0,x\}\ge x$.

We now estimate the last expression. First, $h\ge f$, see Eq. (103). Second,

Plugging it into Eq. (102), we obtain that $s\le f+2^{-t}$. Since also $s\le1$,

where the second inequality uses the fact that $2^{-t}\ell\le m^{-2}m\le1$. Recalling (97) completes the proof of (98).

The following construction is a variant of the catalysts introduced in [29, 42].

Proof. We construct the LOCC protocol that implements the claimed transformation. Apply the following channel. For clock value $j<N-1$, regard one block of the $\Gamma_{N-1-j}$ part as the output, place the fresh $a$ into the bank, and increment the clock. The catalyst marginal in the new clock branch is

which is precisely the term of (112) corresponding to $j+1$. Correlations between the released block and the retained $\Gamma$ blocks are kept; only the marginal of the retained blocks is $\Gamma_{N-2-j}$.

For clock value $j=N-1$, the bank and fresh input contain $a^{\otimes N}$. Apply the assumed exact channel to obtain $\Gamma_N$, release one block, retain the remaining $N-1$ blocks, and reset the clock to zero. The retained marginal is $\Gamma_{N-1}$, again the required next catalyst branch. Since the clock is uniformly distributed and shifted cyclically, the complete catalyst marginal returns exactly. Every released block has marginal $\Gamma_1$ by permutation invariance.

It remains to compute the correlation. Put $S(\Gamma_0)=0$. Orthogonality of the classical clock labels and standard properties of von Neumann entropy give:

The second expression is indexed by the final clock value; at clock zero its conditional joint state is $\Gamma_N$, and at clock $j>0$ it is $a^{\otimes j}\otimes\Gamma_{N-j}$ with one $\Gamma$ block designated as $S$. Since the catalyst returns exactly and $\Omega_S=\Gamma_1$,

where the sums telescope. Finally, all one-block marginals of $\Gamma_N$ are $\Gamma_1$, so

Dividing by $N$ and comparing with Eq. (117) proves (113).

Proposition 12

The catalyst is returned exactly, $\Omega_C=\omega_C$, and the residual correlations satisfy

Proof. Set

Since $\tau$ is separable, the additional copies of $\tau$ can be prepared freely by LOCC. Define

and the permutation-invariant state

where $|x|=\sum_{i=1}^{N}x_i$. The state $a_1^{\otimes N}$ contains exactly $nN$ copies of $\rho$ and $(\widetilde{m}-n)N$ copies of $\tau$. Every tensor product term in Eq. (123) contains exactly $K\widetilde{m}=nN$ copies of $\rho$ and $(N-K)\widetilde{m}=(\widetilde{m}-n)N$ copies of $\tau$.

Alice chooses a binary string of weight $K$ uniformly at random and communicates it to Bob. Both parties then perform matching local permutations to arrange the copies of $\rho$ and $\tau$ into the corresponding tensor product term in Eq. (123). Consequently, $a_1^{\otimes N}\to\Gamma_N$ is a deterministic LOCC transformation.

Since $\Gamma_N$ is permutation invariant, every one-block marginal is identical. The probability that a given block contains $\rho^{\otimes\widetilde{m}}$ equals $K/N=n/\widetilde{m}$. Therefore,

By Lemma 11, the transformation $a_1^{\otimes N}\to\Gamma_N$ yields a correlated catalytic conversion of $a_1$ into an output with marginal $\Omega_S=\Gamma_1$. Since the additional copies of $\tau$ can be prepared freely, this gives the claimed protocol with input $\rho^{\otimes n}$ and proves Eq. (119).

By Lemma 11, the residual correlation is

Let $p=K/N=n/\widetilde{m}$, and let $P_{N,K}$ denote the uniform probability distribution over binary strings of length $N$ and weight $K$. Let $Q_p=\operatorname{Bern}(p)^{\otimes N}$ be the distribution of $N$ independent Bernoulli random variables. Consider the classical-to-quantum channel $\mathcal{E}$ that prepares $\theta_{x_1}\otimes\cdots\otimes\theta_{x_N}$ conditioned on a classical string $x=(x_1,\ldots,x_N)$. By construction,

The data-processing inequality for quantum relative entropy therefore gives

Finally, define

Since $K=Np$ is a mode of the binomial distribution, and the distribution has $N+1$ possible outcomes,

Consequently, by Eq. (127),

which proves Eq. (120).

Auxiliary-state bound

Lemma 13 (Auxiliary-state bound). Let $\Lambda$ be a deterministic LOCC channel acting on $\rho^{\otimes n}\otimes\omega_C$, where $\omega_C$ is a finite-dimensional bipartite auxiliary state. Write

Then

No return condition on the auxiliary state is required.

Proof. Fix $\eta>0$. By the definition of $E_{\max}$ (see Eq. (17)), there exists a normalized separable state $\tau_C$ such that [36]

Tensoring with $\rho^{\otimes n}$ and applying the positive map $\Lambda$ gives

Consequently,

Indeed, preparing the separable state $\tau_C$, applying $\Lambda$, and discarding $C$ defines a non-catalytic LOCC protocol. Taking $\eta\downarrow0$ and using $f\le1$ proves the claim. □

Lemma 14 (Repeated-use bound). Under the assumptions of Lemma 13, suppose additionally that

Then, for every integer $k\ge2$,

Proof. Apply the same protocol $k$ times, using fresh input blocks $\rho^{\otimes n}$ and the same catalyst, without acting again on previous outputs. Let $S_i$ denote the entire output block of use $i$, including both parties’ systems, and let $\Theta_{S_1\cdots S_k}$ denote the final joint state of all output blocks. Each use produces the same output marginal, so $\Theta_{S_i}=\Omega_S$. Note that the total output might be correlated and in general $\Theta_{S_1\cdots S_k}\ne\Omega_S^{\otimes k}$.

Before each use, the catalyst marginal is $\omega_C$, and the fresh input is independent of the catalyst and all previous outputs. The reduced input to each use is therefore $\rho^{\otimes n}\otimes\omega_C$. Consequently, immediately after each use, the joint marginal of the new output and the catalyst is

up to relabeling $S$ as $S_i$. In particular,

Here $\Omega_{S_iC}$ refers to the state immediately after use $i$. Since subsequent uses do not act on $S_i$, its final marginal also satisfies $\Theta_{S_i}=\Omega_S$.

Fix $i<k$. Starting immediately after use $i$, all future outputs are obtained by processing the catalyst together with fresh independent input blocks. Including these fixed input states in the channel definition, the remaining protocol followed by discarding the final catalyst is a quantum channel

Thus

Data processing gives

where each mutual information is evaluated on the corresponding joint state.

Using $\Theta_{S_i}=\Omega_S$ and the entropy chain identity, we obtain (recall that $D$ is quantum relative entropy.)

In the last step we used Eq. (142). Quantum Pinsker’s inequality therefore implies [35]

Eq. (131) gives $\operatorname{Tr}(\Phi_{2^m}^{\otimes k}\Omega_S^{\otimes k})=f^k$. Since $0\le\Phi_{2^m}^{\otimes k}\le I$, the trace-distance bound yields

Using the fact that $|a-b|\le c\Rightarrow a\le b+c$ we obtain:

The complete repeated-use process is a deterministic LOCC protocol acting on $\rho^{\otimes kn}\otimes\omega_C$, with the catalyst supplied only once. Since $\Phi_{2^m}^{\otimes k}$ is locally unitarily equivalent to $\Phi_{2^{mk}}$, the auxiliary-state bound gives

Combining Eqs. (147), (146) and (144), we obtain:

Conclusions

In this work, we have shown that correlated catalysis removes the strong converse for entanglement distillation from every finite-dimensional distillable bipartite state. At any target rate $R>0$, the optimal asymptotic fidelity is $\min\{1,E_D(\rho)/R\}$, even under an arbitrarily small positive bound on the output–catalyst mutual information tending to zero. The catalyst is returned exactly for every input blocklength. Consequently, catalysis preserves the reliable distillation rate while allowing nonvanishing fidelity at every finite rate.

An analogous catalytic enhancement occurs in coherence manipulation under time-translation covariant operations [43]. Under suitable conditions on the coherent modes, correlated catalysts can increase the reliable distillation rate from zero to infinity [43]. Our result identifies a different effect for entanglement: for every distillable source, the reliable rate remains $E_D(\rho)$, while the strong-converse threshold becomes infinite, even with asymptotically vanishing output–catalyst mutual information.

The construction used in the proof of Theorem 2 establishes the optimal asymptotic fidelity, but requires a larger catalyst memory. This motivates our second construction, which reduces memory cost while retaining any fixed fidelity requirement $0<f_0<E_D(\rho)/R$ at rates $R>E_D(\rho)$. For every distillable source, it requires only $O(n\log_2(1/\delta_n))$ qubits of local catalyst memory while ensuring that the fidelity $f_n$ with the target state satisfies $f_n\ge f_0$ and that $I(S:C)_\Omega\le\delta_n\to0$. Whenever noncatalytic distillation obeys an exponential strong converse at a threshold below $R$, a matching lower bound holds uniformly over all admissible protocols. Under this assumption, the optimal local memory cost for such fixed fidelity requirements is therefore $\Theta(n \log_2(1/\delta_n))$.

The fixed fidelity requirement can be chosen arbitrarily close to $E_D(\rho)/R$, but the constants in the memory bound and the required input blocklength may depend on this choice. Thus, although the construction of Theorem 2 establishes the optimal fidelity limit, our matching memory bounds do not determine the optimal memory cost when $f_n \to E_D(\rho)/R$. Characterizing the joint dependence of this cost on the fidelity gap and the allowed output–catalyst correlations remains an open question.

Acknowledgment

This work was supported by the National Science Centre Poland (Grant No. 2022/46/E/ST2/00115 and 2024/55/B/ST2/01590).

ChatGPT 6 Astra was used during the preparation of this manuscript. The authors checked and verified all the results.

References