A non-commutative Bayes' theorem

Journal Article · 28 February 2022 · Linear Algebra and Its Applications

DOI: https://doi.org/10.1016/j.laa.2022.02.030 · OSTI ID:1999097

Parzygnat, Arthur J.  ^[1]; Russo, Benjamin P.  ^[2]

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1. Institut des Hautes Études Scientifiques, Bures-sur-Yvette (France)
2. Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional C*-algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. In conclusion, we illustrate how the procedure works for several examples relevant to quantum information theory.

• Research Organization: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC05-00OR22725
• OSTI ID: 1999097
• Journal Information: Linear Algebra and Its Applications, Journal Name: Linear Algebra and Its Applications Journal Issue: 644 Vol. 644; ISSN 0024-3795
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English