Bayesian Generalized Probability Calculus for Density Matrices

TL;DR

This paper introduces a generalized probability calculus for density matrices in quantum physics, extending classical probability concepts like Bayes rules and total probability to quantum states, and providing a unified framework that includes traditional probability as a special case.

Contribution

It develops a novel Bayesian calculus for density matrices, incorporating analogs of classical probability rules and deriving new Bayes rules based on quantum relative entropy.

Findings

Unified framework for classical and quantum probability calculations.

Derived new Bayes rule for density matrices using quantum relative entropy.

Bounded negative log likelihood for quantum models similar to classical case.

Abstract

One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions can be seen as a special case when the density matrix is restricted to be diagonal. We develop a probability calculus based on these more general distributions that includes definitions of joints, conditionals and formulas that relate these, including analogs of the Theorem of Total Probability and various Bayes rules for the calculation of posterior density matrices. The resulting calculus parallels the familiar "conventional" probability calculus and always retains the latter as a special case when all matrices are diagonal. We motivate both the conventional and the generalized Bayes rule with a minimum relative entropy principle, where the Kullbach-Leibler version gives the conventional Bayes rule and Umegaki's quantum relative entropy the new Bayes rule for density matrices. Whereas the conventional Bayesian methods maintain uncertainty about which model has the highest data likelihood, the generalization maintains uncertainty about which unit direction has the largest variance. Surprisingly the bounds also generalize: as in the conventional setting we upper bound the negative log likelihood of the data by the negative log likelihood of the MAP estimator.