Convexity , Categorical Semantics and the Foundations of Physics

We consider symmetric monoidal categories of convex operational models, and adduce necessary and sufficient conditions for these to be compact-closed or daggercompact. Compact closure amounts to the condition that all processes be implementable by means of a “remote evaluation” protocol (generalizing standard conclusive quantum teleportation protocols), which amounts to a form of classical conditioning. This requires the existence, for each system, of a bipartite state involving a further system, whose corresponding conditioning map is an isomorphism, and an an effect whose corresponding map is the inverse of this isomorphism. Degenerate compact closure, in which systems act as their own duals in the compact structure, means that one may take this extension to be the system itself, so the isomorphism implies that systems are weakly self-dual as ordered vector spaces. Degenerate dagger compact categories emerge from a further restriction, namely, that the bipartite “isomorphism” state and effect be symmetric. It is natural to model physical theories as categories, with objects representing physical systems and morphisms, processes. It is reasonable to expect such a category to be symmetric monoidal, to allow for a treatment of composite systems. In the categorical approach to the foundations of quantum theory, associated with Abramsky and Coecke [1], Baez [2], Selinger [8, 9] and others, the focus is on compact closed, or, more narrowly, dagger-compact, categories. As a systematization of existing quantum protocols, this approach has been very successful. However, if our aim is more broadly foundational, then these strong structural constraints require independent physical motivation, or at any rate, characterization. It is useful to contrast the categorical approach with the older “convex operational” tradition (deriving from the work of Mackey [7], Davies and Lewis [4], Ludwig [6] and others in the 1950s and 60s). Here, state spaces of individual probabilistic systems are represented by essentially arbitrary compact convex sets, and physical processes by affine (convexlinear) mappings between these. This setting is conservative of classical probabilistic ideas, but entirely liberal as to the structures it embraces, so long as this content is respected. We 2 Barnum, Duncan, and Wilce show that almost any symmetric monoidal category can be represented as a category of concrete convex-operational models, with a “non-signaling” tensor product. This suggests that compact closure and dagger-compactness are rather special constraints. As we’ll show, compact closure amounts to the condition that all processes be implementable by means of a “remote evaluation” protocol (generalizing standard conclusive quantum teleportation protocols) — which amounts to a form of classical conditioning. Degenerate dagger compact categories emerge from a further restriction, namely, that a composite of two copies of a system in the theory admit a symmetric bipartite “isomorphism” state that can be used to teleport. 1 Some Categorical Preliminaries Let C be a symmetric monoidal category. Recall that a dual for an object A ∈ C consists of an object B and two morphisms, a unit, η : I → B ⊗ A and co-unit, : A ⊗ B → I , such that A idA⊗η // A ⊗ B ⊗ A ⊗idA // A = idA B η⊗idB // B ⊗ A ⊗ B idB⊗ // B = idB. (1) We shall be particularly interested in the case in which A = A (not just up to a canonical isomorphism, but on the nose). In this case, we shall say that C is degenerate. Adjoints In any compact closed category C, assignment A 7→ A extends to a canonical contravariant functor (−) : Cop → C taking φ ∈ C(A, B) to φ ∈ C(B, A) defined by