A non-commutative Bayes' theorem

Parzygnat, Arthur J.; Russo, Benjamin P.

doi:10.1016/j.laa.2022.02.030

A non-commutative Bayes' theorem

Journal Article · Mon Feb 28 00:00:00 EST 2022 · Linear Algebra and Its Applications

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

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Institut des Hautes Études Scientifiques, Bures-sur-Yvette (France)
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.

View Accepted Manuscript (DOE)

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

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Related Subjects

97 MATHEMATICS AND COMPUTING
Bayesian inversion
Choi matrix
completely positive
positive semidefinite matrix completion
quantum information
quantum probability

A non-commutative Bayes' theorem

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