Graduate School of Informatics, Nagoya University, Chikusa-ku, 464-8601 Nagoya, Japan
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Abstract
We propose a category-theoretic definition of retrodiction and use it to exhibit a time-reversal symmetry for all quantum channels. We do this by introducing retrodiction families and functors, which capture many intuitive properties that retrodiction should satisfy and are general enough to encompass both classical and quantum theories alike. Classical Bayesian inversion and all rotated and averaged Petz recovery maps define retrodiction families in our sense. However, averaged rotated Petz recovery maps, including the universal recovery map of Junge-Renner-Sutter-Wilde-Winter, do not define retrodiction functors, since they fail to satisfy some compositionality properties. Among all the examples we found of retrodiction families, the original Petz recovery map is the only one that defines a retrodiction functor. In addition, retrodiction functors exhibit an inferential time-reversal symmetry consistent with the standard formulation of quantum theory. The existence of such a retrodiction functor seems to be in stark contrast to the many no-go results on time-reversal symmetry for quantum channels. One of the main reasons is because such works defined time-reversal symmetry on the category of quantum channels alone, whereas we define it on the category of quantum channels and quantum states. This fact further illustrates the importance of a prior in time-reversal symmetry.
Featured image: This figure shows sets and their inclusion structure depicting four families of channels. Standard time-reversal symmetry is obtained by taking the Hilbert–Schmidt adjoint of a reversible channel, or, more generally, a bistochastic channel. The states (drawn schematically as matrices with shaded entries) are irrelevant for reversible channels, but are implicitly the uniform states for bistochastic channels. For classical channels equipped with arbitrary probability distributions, standard Bayesian inversion provides a form of time-reversal symmetry that goes beyond the Hilbert–Schmidt adjoint for bistochastic channels. Finally, the Petz recovery map allows an extension of Bayesian inversion to all quantum channels and arbitrary states. In brief, this paper isolates axioms for retrodiction and inferential time-reversal symmetry that are simultaneously satisfied by all of these classes of channels and states.
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Popular summary
In the setting of classical mechanics, however, there is a well-known method of retrodicting using Bayes’ rule and the more general Jeffrey’s probability kinematics. Because of the ambiguities arising from extending Bayes’ rule to quantum systems, we instead isolate key properties of Jeffrey’s probability kinematics and classical retrodiction in order to provide precise logical axioms for quantum retrodiction. We then prove that, among a variety of proposals that could be used for quantum retrodiction, only one satisfies all of our proposed axioms.
Nevertheless, an important open problem remains, which we have been able to precisely state in mathematical terms. Namely, among all possible retrodiction algorithms that satisfy our axioms, is our solution truly the unique solution? Or are there other possible forms of retrodiction that satisfy our axioms? If so, what are they?
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[2] Arthur J. Parzygnat and James Fullwood, “From time-reversal symmetry to quantum Bayes’ rules”, arXiv:2212.08088, (2022).
[3] Ardra Kooderi Suresh, Markus Frembs, and Eric G. Cavalcanti, “A Semantics for Counterfactuals in Quantum Causal Models”, arXiv:2302.11783, (2023).
[4] Francesco Buscemi, Joseph Schindler, and Dominik Šafránek, “Observational entropy, coarse-grained states, and the Petz recovery map: information-theoretic properties and bounds”, New Journal of Physics 25 5, 053002 (2023).
[5] Mankei Tsang, “Operational meanings of a generalized conditional expectation in quantum metrology”, arXiv:2212.13162, (2022).
[6] Luca Giorgetti, Arthur J. Parzygnat, Alessio Ranallo, and Benjamin P. Russo, “Bayesian inversion and the Tomita-Takesaki modular group”, arXiv:2112.03129, (2021).
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