# Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models - Quantum Physics

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Abstract: In the categorical approach to the foundations of quantum theory, one beginswith a symmetric monoidal category, the objects of which represent physicalsystems, and the morphisms of which represent physical processes. Usually, thiscategory is taken to be at least compact closed, and more often, daggercompact, enforcing a certain self-duality, whereby preparation processesroughly, states are inter-convertible with processes of registrationroughly, measurement outcomes. This is in contrast to the more concrete-operational- approach, in which the states and measurement outcomes associatedwith a physical system are represented in terms of what we here call a -convexoperational model-: a certain dual pair of ordered linear spaces - generally,{\em not} isomorphic to one another. On the other hand, state spaces for whichthere is such an isomorphism, which we term {\em weakly self-dual}, play animportant role in reconstructions of various quantum-information theoreticprotocols, including teleportation and ensemble steering. In this paper, wecharacterize compact closure of symmetric monoidal categories of convexoperational models in two ways: as a statement about the existence ofteleportation protocols, and as the principle that every process allowed bythat theory can be realized as an instance of a remote evaluation protocol -hence, as a form of classical probabilistic conditioning. In a large class ofcases, which includes both the classical and quantum cases, the relevantcompact closed categories are degenerate, in the weak sense that every objectis its own dual. We characterize the dagger-compactness of such a categorywith respect to the natural adjoint in terms of the existence, for eachsystem, of a {\em symmetric} bipartite state, the associated conditioning mapof which is an isomorphism.

Author: ** Howard Barnum, Ross Duncan, Alexander Wilce**

Source: https://arxiv.org/