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arXiv:2001.08375

Inverses, disintegrations, and Bayesian inversion in quantum Markov categories

Arthur J. Parzygnat

Published Jan 21, 2020• 5 citations
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Abstract

We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory. In this broader context, we analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inverses. We prove that each one is a strictly special instance of the latter for certain subcategories, providing a categorical foundation for Bayesian inversion as a generalization of reversing a process. We unify the categorical and $C^*$-algebraic notions of almost everywhere (a.e.) equivalence. As a consequence, we prove many results including a universal no-broadcasting theorem for S-positive categories, a generalized Fisher--Neyman factorization theorem for a.e. modular categories, a relationship between error correcting codes and disintegrations, and the relationship between Bayesian inversion and Umegaki's non-commutative sufficiency.

DOI
10.48550/arxiv.2001.08375

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We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory. In this broader context, we analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inverses. We prove that each one is a strictly special instance of the latter for certain subcategories, providing a categorical foundation for Bayesian inversion as a generalization of reversing a process. We unify the categorical and C∗C^*C∗-algebraic notions of almost everywhere (a.e.) equivalence. As a consequence, we prove many results including a universal no-broadcasting theorem for S-positive categories, a generalized Fisher--Neyman factorization theorem for a.e. modular categories, a relationship between error correcting codes and disintegrations, and the relationship between Bayesian inversion and Umegaki's non-commutative sufficiency.
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.
Reversibility lies at the heart of physics and information theory, linking entropy, efficiency and information recovery. Yet, definitions of a “reverse process” have been context-specific, relying on physical assumptions rather than a general framework. This thesis develops a universal recipe for reverse processes using Bayesian inference, formalized through Bayes’ rule and the Petz recovery map. We first show how these tools recover conventional notions of reversal while yielding new insights, and justify the Petz transpose as the quantum analog of Bayes’ rule via quasiprobability representations. We then compare Bayesian reversibility with dilation-based approaches, proving their equivalence through the role of correlations. This motivates the notion of tabletop time-reversible processes, special cases where correlations can be neglected with respect to reversal. Beyond the core results, we apply these methods to fluctuation relations, quantum thermal operations, and the Loschmidt paradox, and reflect on implications for the philosophy of time. In sum, Bayesian inference provides a regime-independent foundation for defining and analyzing (ir)reversibility across physics and information theory.
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.
Markov categories allow formalization of probabilistic and causal reasoning in a general setting that applies uniformly to many different kinds of classical probabilistic processes. It has so far been challenging, however, to generalize these techniques to reasoning about quantum processes, as the quantum no-cloning theorem forbids "copy" maps of the sort that have been used to axiomatize conditional independence, and the related notions of complete common causes and Markovianity, in classical Bayesian networks. Here, we introduce a new categorical notion of Markovian causal model, according to which a distinguished subcategory of "common cause" maps plays a similar role to that of "copy" maps in the categorical formulation of Bayesian networks. Moreover, defining causal models as second-order processes yields a clean and flexible formulation of interventions. Our formalism is both rich enough to handle "complete common cause" assumptions and general enough to encompass not only standard classical causal identification scenarios, but also quantum causal scenarios and new kinds of classical causal identification based on imperfect observations. Furthermore, we show that one can reason uniformly across all of these cases using string-diagrammatic techniques.
Markov categories allow formalization of probabilistic and causal reasoning in a general setting that applies uniformly to many different kinds of classical probabilistic processes. It has so far been challenging, however, to generalize these techniques to reasoning about quantum processes, as the quantum no-cloning theorem forbids "copy" maps of the sort that have been used to axiomatize conditional independence, and the related notions of complete common causes and Markovianity, in classical Bayesian networks. Here, we introduce a new categorical notion of Markovian causal model, according to which a distinguished subcategory of "common cause" maps plays a similar role to that of "copy" maps in the categorical formulation of Bayesian networks. Moreover, defining causal models as second-order processes yields a clean and flexible formulation of interventions. Our formalism is both rich enough to handle "complete common cause" assumptions and general enough to encompass not only standard classical causal identification scenarios, but also quantum causal scenarios and new kinds of classical causal identification based on imperfect observations. Furthermore, we show that one can reason uniformly across all of these cases using string-diagrammatic techniques.
Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.