A non-commutative Bayes' theorem

Abstract

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 $C^{\ast }$-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. We illustrate how the procedure works for several examples relevant to quantum information theory.

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References in corpus (12)

• Introducing Quantum Discord
• Probabilistic theories with purification
• Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
• Reversing quantum dynamics with near-optimal quantum and classical fidelity
• A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
• A Characterization of Entropy in Terms of Information Loss
• Disintegration and Bayesian Inversion via String Diagrams
• Quantum Dynamics as an analog of Conditional Probability
• Can a quantum state over time resemble a quantum state at a single time?
• Picturing classical and quantum Bayesian inference
• A categorical foundation for Bayesian probability
• Conditional expectation and Bayes' rule for quantum random variables and positive operator valued measures

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• Bayesian inversion and the Tomita-Takesaki modular group
• A functorial characterization of von Neumann entropy
• Inverses, disintegrations, and Bayesian inversion in quantum Markov categories
• Quantum Operations on Conformal Nets
• Non-commutative disintegrations: existence and uniqueness in finite dimensions