Quantum Bayesian Inference in Quasiprobability Representations

TL;DR

This paper extends quantum Bayesian inference by deriving the Petz recovery map within quasiprobability representations, highlighting the core difference from classical inference in the manipulation of the reference prior.

Contribution

It provides explicit formulas for the Petz recovery map in quasiprobability representations, including Discrete Wigner and SIC-POVM based representations.

Findings

Derived the Petz recovery map in quasiprobability frameworks

Explicit formulas for canonical quasiprobability representations

Identified the key difference in quantum vs. classical inference as prior manipulation

Abstract

Bayes' rule plays a crucial piece of logical inference in information and physical sciences alike. Its extension into the quantum regime has been the object of several recent works. These quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces. In this paper, we derive the expression of the Petz recovery map within any quasiprobability representation, with explicit formulas for the two canonical choices of normal quasiprobability representations (which include Discrete Wigner representations) and of representations based on symmetric, informationally complete positive operator-valued measures (SIC-POVMs). By using the same mathematical syntax of (quasi-)stochastic matrices acting on (quasi-)stochastic vectors, the core difference in logical inference between classical and quantum theory is found in the manipulation of the reference prior rather than in the representation of the channel.