Quantum Bayesian Inference in Quasiprobability Representations

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