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A categorical approach to quantum Bayes' rule

Hitoshi Motoyama, Kohei Tanaka

Published Dec 18, 2014• 0 citations
OpenAlex

Abstract

We introduce a concept of conditional measures in quantum measure spaces from a view point of Gelfand duality between classical measure spaces and quantum measure spaces. Classical conditional measure spaces are based on subspaces of a space, hence we define quantum conditional measure spaces based on the dual notion, ideals and quotients of an algebra. Although the classical Bayes rule indicates the ratio of conditional measure (probability) and original one, the quantum Bayes rule describes the difference between them in quantum measure spaces. We also give categorical structures on classical and quantum measure spaces based on Bayes' rule, respectively. These extend the Riesz-Markov-Kakutani representation theorem to an equivalence of categories between the category of classical measure spaces and the category of commutative $C^{\ast}$-measure spaces.

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We introduce a concept of conditional measures in quantum measure spaces from a view point of Gelfand duality between classical measure spaces and quantum measure spaces. Classical conditional measure spaces are based on subspaces of a space, hence we define quantum conditional measure spaces based on the dual notion, ideals and quotients of an algebra. Although the classical Bayes rule indicates the ratio of conditional measure (probability) and original one, the quantum Bayes rule describes the difference between them in quantum measure spaces. We also give categorical structures on classical and quantum measure spaces based on Bayes' rule, respectively. These extend the Riesz-Markov-Kakutani representation theorem to an equivalence of categories between the category of classical measure spaces and the category of commutative C∗C^{\ast}C∗-measure spaces.
Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer-Spekkens acausal belief propagation is provided, illustrating some similarities and key differences.
The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.
Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. These are equivalent to Parzygnat's quantum Markov categories, but we argue that they offer a simpler and more practical approach. Our main examples of involutive Markov categories have pre-C*-algebras, including infinite-dimensional ones, as objects, together with completely positive unital maps as morphisms in the picture of interest. In this context, we prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms, and we develop a categorical description of such quantum de Finetti theorems which amounts to a universal property of state spaces.
The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.