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

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## Abstract

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and measurement outcomes associated with a physical system are represented in terms of what we here call a *convex operational model*: a certain dual pair of ordered linear spaces–generally, *not* isomorphic to one another. On the other hand, state spaces for which there *is* such an isomorphism, which we term *weakly self-dual*, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a *symmetric* bipartite state, the associated conditioning map of which is an isomorphism.

## Keywords

Quantum foundations Convex operational theories Compact closed category Dagger-compact category## Notes

### Acknowledgments

HB and AW wish to thank Samson Abramsky and Bob Coecke for enabling them to visit the Oxford University Computing Laboratory in November 2009, where some of this work was done, and for helpful discussions during that time. The authors also wish to thank Peter Selinger for helpful discussions. RD was supported by EPSRC postdoctoral research fellowship EP/E04006/1. HB’s research was supported by Perimeter Institute for Theoretical Physics; work at Perimeter Institute is supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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