2010/04/16 by Howard Barnum, Ross Duncan, Barnum, Howard +3
Arts and Humanities · Computer Science · Physics and Astronomy · #Computability, Logic, AI Algorithms #FOS: Physical sciences #Philosophy and History of Science #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1004.2920
openalex publication_date 2010/04/16 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
In the categorical approach to the foundations of quantum theory, one begins\nwith a symmetric monoidal category, the objects of which represent physical\nsystems, and the morphisms of which represent physical processes. Usually, this\ncategory is taken to be at least compact closed, and more often, dagger\ncompact, enforcing a certain self-duality, whereby preparation processes\n(roughly, states) are inter-convertible with processes of registration\n(roughly, measurement outcomes). This is in contrast to the more concrete\n"operational" approach, in which the states and measurement outcomes associated\nwith a physical system are represented in terms of what we here call a "convex\noperational model": a certain dual pair of ordered linear spaces -- generally,\n em not isomorphic to one another. On the other hand, state spaces for which\nthere is such an isomorphism, which we term em weakly self-dual, play an\nimportant role in reconstructions of various quantum-information theoretic\nprotocols, including teleportation and ensemble steering. In this paper, we\ncharacterize compact closure of symmetric monoidal categories of convex\noperational models in two ways: as a statement about the existence of\nteleportation protocols, and as the principle that every process allowed by\nthat theory can be realized as an instance of a remote evaluation protocol ---\nhence, as a form of classical probabilistic conditioning. In a large class of\ncases, which includes both the classical and quantum cases, the relevant\ncompact closed categories are degenerate, in the weak sense that every object\nis its own dual. We characterize the dagger-compactness of such a category\n(with respect to the natural adjoint) in terms of the existence, for each\nsystem, of a em symmetric bipartite state, the associated conditioning map\nof which is an isomorphism.\n