Operational separability of spacelike-separated local algebras in algebraic quantum field theory: its relation to operational independence, the W*-analogue, general spacetime regions, and complete characterization

In algebraic quantum field theory (AQFT), an operation is a completely positive unital map on a local observable algebra. Rédei and Valente (2010) called a pair of subalgebras operationally C*-separable if every operation on one subalgebra that can be extended to the larger algebra at all can be extended so as to act as the identity on the other, spacelike separated subalgebra. They proved that strictly spacelike separated double-cone algebras are operationally C*-separable in a larger double-cone algebra, using the split property and Connes' theorem that hyperfinite von Neumann algebras are injective, and posed four open questions that remain unanswered: (1) does operational C*-separability imply operational C*-independence (joint performability of arbitrary pairs of local operations), or is it strictly weaker? (2) are strictly spacelike separated double-cone von Neumann algebras operationally W*-separable in a larger double-cone algebra, given that injectivity guarantees completely positive but not normal extensions, and does operational W*-separability imply operational W*-independence? (3) does the double-cone separability theorem extend to arbitrary strictly spacelike separated open bounded regions? (4) what is the complete characterization of operational C*-separability among the standard subsystem-independence properties? The problem is to resolve all four parts. Part (4) is the inclusive complete characterization and subsumes parts (1) and (2)(b), so a resolution of (4) discharges those two implications; parts (2)(a) and (3) are posed independently of it.

Background

In algebraic quantum field theory (AQFT), a field theory on Minkowski space is described by an isotone net $V \mapsto \mathfrak{A}(V)$ assigning to each bounded open spacetime region $V$ a $C^*$-algebra of observables measurable in $V$ (with $\mathfrak{A}(V_1) \subseteq \mathfrak{A}(V_2)$ for $V_1 \subseteq V_2$; there is an analogous net $V \mapsto \mathfrak{N}(V)$ of von Neumann algebras). Microcausality requires the algebras of spacelike separated regions to commute elementwise. In standard models the local algebras of double cones (diamond-shaped regions: the domain of dependence of a spatial ball) are hyperfinite factors of type III — hyperfinite meaning approximable by an increasing sequence of finite-dimensional subalgebras, type III meaning that the factor carries no nonzero finite trace (the Murray-von Neumann class of factors typical of local relativistic algebras). Because the Reeh-Schlieder theorem (the vacuum vector is cyclic and separating for every local algebra, even for regions arbitrarily far away) makes spacelike separated local algebras highly entangled, independence of local algebras cannot mean a tensor-product decomposition, and a whole hierarchy of independence notions for pairs of subalgebras has been studied (commutation; statistical/C*-independence, meaning that any two states on the two subalgebras extend to a common state on the algebra they generate; W*-independence, the analogous requirement for normal states; the Schlieder property, meaning that the product of nonzero elements from the two subalgebras is nonzero; the split property; and operational variants; see Summers 1990, 2009).

An operation on a unital $C^*$-algebra $\mathfrak{B}$ is a completely positive (CP) unital map $T: \mathfrak{B} \to \mathfrak{B}$; in the von Neumann category, operations are additionally required to be normal (ultraweakly continuous). In finite dimensions every operation admits a Kraus (operator-sum) representation $T(A)=\sum_i K_i^* A K_i$ with bounded operators $K_i$, but this fails for the type III algebras typical of AQFT, so the behavior of local operations must be analyzed through extension properties of CP maps.

Rédei and Summers (2010) defined operational C*-independence: subalgebras $\mathfrak{A}_1, \mathfrak{A}_2 \subseteq \mathfrak{A}$ are operationally C*-independent in $\mathfrak{A}$ if any two operations $T_1: \mathfrak{A}_1 \to \mathfrak{A}_1$ and $T_2: \mathfrak{A}_2 \to \mathfrak{A}_2$ admit a joint extension to an operation $T: \mathfrak{A} \to \mathfrak{A}$. This expresses that any two actions in spacelike separated regions can be performed together. Summers (2009) proved that the split property (existence of a type I factor — an algebra of all bounded operators on some Hilbert space — interpolating between the two algebras) implies operational independence, with equivalence when either algebra is a factor or of type III; the split property holds for strictly spacelike separated double cones in standard nets (under phase-space conditions), fails at tangency, and is impossible for unbounded regions.

The complementary question — whether local actions can be performed without disturbing the spacelike separated region — is subtler, because microcausality alone does not prevent normal operations from acting nontrivially across the separation: the Accardi-Cecchini conditional expectation (a normal unital completely positive projection onto one of the local algebras) associated with the vacuum state is a normal operation on the generated algebra that changes expectations in the other region. Rédei and Valente (2010) therefore defined operational separability: $(\mathfrak{A}_1, \mathfrak{A}_2)$ is operationally C*-separable in $\mathfrak{A}$ if every operation on $\mathfrak{A}_1$ that has some extension to an operation on $\mathfrak{A}$ also has one acting as the identity on $\mathfrak{A}_2$ (and symmetrically with the subalgebras interchanged). The extendibility proviso is essential, since operations on subalgebras need not extend at all (not every $C^*$-algebra is injective, i.e., the target of a completely positive projection from every containing algebra, which is what would guarantee such extensions); the strong version drops the proviso.

Three delimiting results are known. (i) The strong version of operational separability is equivalent to operational independence (Rédei 2010). (ii) Under microcausality and the funnel property (for each pair of nested double cones with closure $\overline{O} \subset \tilde{O}$, a type I factor interpolates between $\mathfrak{N}(O)$ and $\mathfrak{N}(\tilde{O})$), strictly spacelike separated local von Neumann algebras $\mathfrak{N}(O_1), \mathfrak{N}(O_2)$ are operationally W*-separable in the generated algebra $\mathfrak{N}(O_1) \vee \mathfrak{N}(O_2)$, the smallest von Neumann algebra containing both (Summers 2009; Rédei 2010; Kitajima 2018, Prop. 11). (iii) Proposition 9 of Rédei-Valente (2010): if $D_1, D_2$ are strictly spacelike separated double cones contained in a double cone $D$, then $\mathfrak{A}(D_1), \mathfrak{A}(D_2)$ are operationally C*-separable in $\mathfrak{A}(D)$, the proof combining the split property (giving operational C*-independence in the generated algebra) with Connes' injectivity theorem for hyperfinite von Neumann algebras, which extends the CP maps to all of $\mathfrak{A}(D)$. The gap left open: injectivity yields CP extensions with no control of normality, and the logical relation between weak separability and independence is unknown.

Problem Statement

Let $\mathfrak{A}$ be a unital $C^*$-algebra containing unital subalgebras $\mathfrak{A}_1, \mathfrak{A}_2$. An operation on a $C^*$-algebra $\mathfrak{B}$ is a completely positive unital map $T: \mathfrak{B} \to \mathfrak{B}$; in the von Neumann category all algebras are von Neumann algebras and operations are additionally required to be normal (ultraweakly continuous).

Definitions (Rédei-Summers 2010; Rédei-Valente 2010):

1. $(\mathfrak{A}_1, \mathfrak{A}_2)$ is operationally C*-independent in $\mathfrak{A}$ if for any two operations $T_1: \mathfrak{A}_1 \to \mathfrak{A}_1$ and $T_2: \mathfrak{A}_2 \to \mathfrak{A}_2$ there is an operation $T: \mathfrak{A} \to \mathfrak{A}$ with $T(a_1) = T_1(a_1)$ for $a_1 \in \mathfrak{A}_1$ and $T(a_2) = T_2(a_2)$ for $a_2 \in \mathfrak{A}_2$.
2. $(\mathfrak{A}_1, \mathfrak{A}_2)$ is operationally C*-separable in $\mathfrak{A}$ if: (i) every operation $T_1: \mathfrak{A}_1 \to \mathfrak{A}_1$ that has at least one extension to an operation on $\mathfrak{A}$ has one such extension $T: \mathfrak{A} \to \mathfrak{A}$ with $T(a_2) = a_2$ for all $a_2 \in \mathfrak{A}_2$; and (ii) the symmetric condition with the subalgebras interchanged. (The strong version drops the extendibility proviso.)
3. Operational W*-independence and operational W*-separability are defined verbatim as above, with von Neumann algebras and normal operations.

In the setting of AQFT nets (and, where meaningful, abstractly for inclusions of operator algebras), determine: (1) Whether operational C*-separability of $(\mathfrak{A}_1, \mathfrak{A}_2)$ in $\mathfrak{A}$ implies operational C*-independence of $(\mathfrak{A}_1, \mathfrak{A}_2)$ in $\mathfrak{A}$; i.e., whether the weak version of separability is equivalent to, or strictly weaker than, independence (the strong version is already known to be equivalent to it). (2) (a) Whether strictly spacelike separated double-cone von Neumann algebras $\mathfrak{N}(D_1), \mathfrak{N}(D_2)$ are operationally W*-separable in $\mathfrak{N}(D)$ for a double cone $D \supseteq D_1 \cup D_2$ — the obstruction being that normal operations on hyperfinite algebras need not admit normal extensions, even though completely positive extensions exist. (b) Whether operational W*-separability (in an ambient von Neumann algebra) implies operational W*-independence. (3) Whether the double-cone theorem (Proposition 9 of Rédei-Valente 2010) extends from double cones to arbitrary strictly spacelike separated open bounded regions, and how the answer depends on the choice of ambient algebra. (4) A complete characterization of operational C*-separability in terms of the standard subsystem-independence properties (elementwise commutation, statistical C*-independence, W*-independence, the Schlieder property, the split property, and the operational versions thereof), covering the C*- versus W*-levels and the ambient-algebra distinctions arising in questions (2) and (3).

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