Classical Extensions and Strongly Positive Realizations of Decoherent History Probabilities

Introduction

A decoherence functional assigns probabilities to events in each medium-decoherent family of histories. Two such families agree on the probabilities of their common events. Does a single probability measure on the Boolean algebra they generate reproduce both sets of probabilities? Given only the local probabilities, which strongly positive decoherence functionals reproduce them and make each family medium decoherent? The distinction already arises for two binary contexts on three atoms. If each context assigns probability one to a different singleton, the two marginals agree on their common events. No probability measure can give unit mass to both singletons, but a rank-one decoherence functional realizes the two marginals (§ 8.2).

The classical probability rules within a consistent set of histories are well established [1, 2]. Isham studied presheaves on the partially ordered set of consistent Boolean subalgebras [3]. Halliwell asked whether incompatible consistent sets admit a unifying probability, and used Bell and Leggett–Garg inequalities to test its existence [4]. A common decoherent refinement supplies such a probability. The converse can fail even for histories of a single qubit, as the class operators in § 8.5 show.

In the approach of Abramsky and Brandenburger, a noncontextual model is a probability distribution on global measurement assignments with the prescribed local marginals [5]. For subalgebras of a fixed history algebra, one must also specify which global assignments the algebra permits. A choice of one atom from each context can agree on every overlap while the chosen atoms have empty intersection. The two deterministic singleton outcomes above give an example. We relate this distinction to the amalgamation of Boolean subalgebras [6, 7] and show exactly when the two assignment spaces yield the same classical probability polytope (Proposition 2.4).

On the quantum side, Dowker, Henson, and Wallden studied joint quantum measures whose restrictions to measurement contexts are classical, and related strong positivity to bounds on quantum correlations [8]. Strong positivity also yields Tsirelson bounds [9]. Hilbert-space representations of decoherence functionals and the role of strong positivity in the histories formulation are discussed in [10, 11]. Here we fix the local probabilities and determine which atomic decoherence matrices realize them, how small their rank and trace can be, and with what probability they can be implemented by postselection.

Let $\mathcal{M}$ be the family of contexts and $X$ the set of atoms of the finite Boolean algebra they generate. For $C \in\mathcal{M}$, let $P_C$ sum coordinates on $X$ over each atom of $C$, and let $e_C$ be its probability vector. A signed extension is a real vector $q$ satisfying $P_Cq=e_C$ for all $C$. It is a classical extension if $q \ge0$. A strongly positive realization is a Hermitian matrix $H \succeq0$ satisfying

The diagonal entries give the context probabilities, and the off-diagonal equations express medium decoherence. Summing both indices gives $\mathbf{1}^{*}H\mathbf{1}=1$. Each entry of $P_C H P_C^{\mathsf{T}}$ sums the entries of $H$ over a pair of context atoms. Quantum coupling problems instead prescribe partial traces [12].

For any finite family of contexts, every realization has a unique decomposition

The equations for $z$ are $P_Cz=e_C$. Those for $K$ prescribe the local covariances $\operatorname{diag}(e_C)-e_Ce_C^{\mathsf{T}}$. Replacing $z$ by a real signed extension $q$ preserves the rank of $H$. Prescribing $H\mathbf{1}=q$ raises the minimum trace by $\lVert q-q_0\rVert^2$, where $q_0$ is the signed extension of least Euclidean norm (Theorems 3.1 and 3.3).

For two contexts, put an edge between two context atoms whenever they intersect. The edges of the resulting bipartite graph $G$ are exactly the atoms of the generated algebra. Overlap agreement suffices for a strongly positive realization, and the smallest rank is the larger of the two marginal support sizes (Theorem 4.2). A classical extension must also satisfy the supported transport inequalities in Proposition 4.5. For compatible marginals $p,q$, let $\eta_G(p,q)$ be the minimum negative mass of a signed extension, a quantity used in studies of contextuality [13]. We show that

where the cost of moving a unit of mass between vertices is their graph distance. Together with strong positivity, this identity gives bounds on negative real interference and trace that are attained on families of odd paths (Theorems 5.2 and 5.4 and Proposition 8.2).

Minimizing the trace uses the electrical minimum-energy principle [14] and a singular-value optimization of the type that occurs in the orthogonal Procrustes problem [15, 16]. For deterministic endpoints, the minimum trace is their effective resistance. For general compatible marginals, the minimizers are parametrized by a rectangular contraction ball, including when the graph is disconnected. Its extreme points attain minimum rank and minimum trace simultaneously. When every marginal entry is positive, the rank of the bipartite adjacency matrix determines the entire interval of minimizing ranks (Theorems 6.1 and 6.3 and Corollaries 6.2 and 6.4). For coordinate covers of full product spaces, the minimum trace is the fraction of Fourier characters visible in a context (Theorem 6.5).

We also consider pure-state histories with one intermediate projective decomposition and a selected final event. This is a finite instance of the histories formalism with initial and final conditions [17]. A Gram-matrix construction [18] gives the exact success criterion. For fixed $H$, the largest success probability is $1/\Gamma(H)$, where $\Gamma(H)$ is the least trace of a diagonal majorant of $H$. After trace normalization, this optimization is related to the robustness of coherence [19, 20]. Optimizing over all realizations gives success probability one exactly for classically extendable data (Theorems 7.1 and 7.2). The asymmetric three-atom example gives the exact gap in the bound from negative mass. A separate family shows how far the trace minimizer can fall short of optimal success even for classical data (Propositions 8.1 and 8.3).

Finite history algebras and their extensions

Decoherence functionals and contexts

Let $X = \operatorname{At}(\mathcal{G})$ be the atom set of a nontrivial finite Boolean algebra $\mathcal{G}$. Each event is the union of its atoms, so we identify $\mathcal{G}$ with $\mathcal{P}(X)$. Fix a nonempty finite family $\mathcal{M}$ of Boolean subalgebras, called contexts, that generates $\mathcal{G}$. For $C \in\mathcal{M}$, write $X_C = \operatorname{At}(C)$ and let

send each ambient atom to the unique atom of $C$ that contains it. The matrix that sums over the fibers of $r_C$ is

Here $b \in X_C$ and $a \in X$. Each column has exactly one nonzero entry.

Additivity determines $D$ from its atomic matrix $H$ by

Strong positivity is equivalent to $H \succeq0$. The list of all atoms gives necessity. For sufficiency, take $V$ with rows $\mathbf{1}_{A_i}^{\mathsf{T}}$ for any list of events. Their matrix is $VHV^{\mathsf{T}}$. The indicator vectors are real, and normalization reads $\mathbf{1}^{\mathsf{T}}H\mathbf{1} = 1$.

For $A,B \in C$, expand $D(A,B)$ over the atoms of $C$. Medium decoherence leaves only the diagonal terms, giving

Thus $p_C(A) = D(A,A)$ defines a probability measure on $C$, as in the history sum rule [1, 3]. Restriction preserves medium decoherence, and the measures induced by a common $D$ agree on intersections. The weaker condition $\operatorname{Re} D(b,b') = 0$ for $b \ne b'$ suffices for additivity of $D(A,A)$. We require medium decoherence to obtain (4) for every pair of events in $C$.

Classical extensions and strongly positive realizations

For each $C \in\mathcal{M}$, let $e_C$ be a probability vector on $X_C$. Assume that the induced measures agree on pairwise intersections. Stacking the matrices $P_C$ and vectors $e_C$ in the same order gives $M$ and $e$. Write

Both sets incorporate normalization. Summing $P_Cq = e_C$ over a single context gives $\mathbf{1}^{\mathsf{T}}q = 1$. Summing both indices of its matrix constraint gives $\mathbf{1}^{\mathsf{T}}H\mathbf{1} = 1$.

For an event $A$, the canonical extension is $q_H(A) = \operatorname{Re} D(A,1)$. For Heisenberg-picture class operators with $C_X = I$ and initial state $\rho$, this is the linear assignment $\operatorname{Re}\operatorname{tr}(C_A\rho)$ of Goldstein and Page [21]. Even if $q_H$ takes negative values, another signed extension of the same data may be nonnegative [22].

Ambient atoms and compatible local valuations

Let $\mathcal{C}$ consist of all intersections of one or more members of $\mathcal{M}$, together with the two-element algebra, ordered by inclusion. For $C' \subseteq C$, restriction sends each atom of $C$ to the atom of $C'$ containing it. These maps define an inverse system whose limit

consists of the compatible local atom choices. Each choice is determined by its components on $\mathcal{M}$. An ambient atom $a$ gives the choice $j(a) = (r_C(a))_{C\in\mathcal{M}}$.

Proof. The nonempty intersections obtained by choosing one atom from each generating context partition $X$. Every generator is a union of blocks of this partition. Since the contexts generate $\mathcal{G}$, the blocks are its atoms. This proves injectivity and the description of $j(X)$. Pushforward by $j$ identifies probabilities on $X$ with probabilities supported on $j(X)$, giving the inclusion and equality when $j$ is surjective. If $\ell\in L \setminus j(X)$, its deterministic marginal assigns probability one to $\ell_C$ in every context. A measure realizing all these marginals would assign probability one to their finite intersection, which is empty.

Surjectivity means that every compatible family of local ultrafilters extends to an ultrafilter on $\mathcal{G}$, the amalgamation condition for commuting Boolean subalgebras [6, 7]. With more than two contexts the condition applies jointly to the whole family. Under finite Stone duality, $j$ corresponds to the surjection from the Boolean colimit of the overlap diagram onto $\mathcal{G}$. Its kernel ideal consists of the subsets of $L$ disjoint from $j(X)$.