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Quantum Bayesian Inference in Quasiprobability Representations
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DOI：10.1103/PRXQuantum.4.020352.png)
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Abstract
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Bayes' rule plays a crucial role in logical inference in information and physical sciences alike. Its extension into the quantum regime has been the subject 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 expres-sion for the Petz recovery map within any quasiprobability representation, with explicit formulas for the two canonical choices of normal quasiprobability representations (which include discrete-Wigner repre-sentations) 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.