Quantum Theory is a Quasi-stochastic Process Theory

Abstract

There is a long history of representing a quantum state using a quasi-probability distribution: a distribution allowing negative values. In this paper we extend such representations to deal with quantum channels. The result is a convex, strongly monoidal, functorial embedding of the category of trace preserving completely positive maps into the category of quasi-stochastic matrices. This establishes quantum theory as a subcategory of quasi-stochastic processes. Such an embedding is induced by a choice of minimal informationally complete POVM's. We show that any two such embeddings are naturally isomorphic. The embedding preserves the dagger structure of the categories if and only if the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of foundational interest in the QBism community. We also study general convex embeddings of quantum theory and prove a dichotomy that such an embedding is either trivial or faithful.

In Proceedings QPL 2017, arXiv:1802.09737

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• Bell non-locality using tensor networks and sparse recovery
• Ontological models for quantum theory as functors
• Simulating all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories
• Minimizing the negativity of quantum circuits in overcomplete quasiprobability representations
• Quantum Theory from Principles, Quantum Software from Diagrams