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Quantum Conditional States, Bayes' Rule, and State Compatibility

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Quantum theory is a noncommutative generalization of classical probability theory. In the classical theory, conditional probability plays an important role in both theory and applications, but its quantum counterpart is conspicuous by its absence. In this talk, I will introduce the formalism of quantum conditional states, which is essentially just a change of notation that makes the equations of standard quantum theory look closer to their classical counterparts. This makes it easier to generalize classical concepts, and has the advantage that it unifies the treatment of quantum dynamics with the treatment of correlations between quantum systems. Conditional states allow for a quantum generalization of Bayes’ rule, which has appeared multiple times in the quantum information/foundations literature, albeit in a disguised form. Examples of the quantum Bayes’ rule include: the relationship between retrodictive states and predictive POV Ms in the retrodictive quantum formalism of Pegg et. al. (generalized to include biased sources), the rule for updating the state of a remote system after a measurement, the ``almost optimal error correction’’ of Knill and Barnum, and the ``pretty good’’ measurement of Hausladen and Wootters. As an application of the formalism, I present a novel justification for the Brun-Finklestein-Mermin criterion for state compatibility that would be acceptable to a Quantum Bayesian who thinks that quantum states represent subjective degrees of belief.

This talk is part of the CQIF Seminar series.

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