Axioms for retrodiction: achieving time-reversal symmetry with a prior

Arthur J. Parzygnat and Francesco Buscemi

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.

Among a variety of proposals, we prove that only one, namely Petz' transpose map, satisfies a list of natural desiderata, making it the canonical choice for inferring the past in quantum physics. Normally, quantum physics is used as a theory of prediction, where one predicts the outcomes of future measurements. However, defining retrodiction, which asks to infer about the past, is a nontrivial problem in quantum physics, with no agreed-upon solution. The only exceptions to this rule occur in two special cases, the first of which is reversible evolution, where one has a clear notion of time-reversal symmetry, and the second of which is the case of quantum channels that preserve the maximally mixed state. Understanding how to retrodict beyond these two special cases is the problem that we solve in our work.

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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[5] Arthur J. Parzygnat, James Fullwood, Francesco Buscemi, and Giulio Chiribella, "Virtual Quantum Broadcasting", Physical Review Letters 132 11, 110203 (2024).

[6] Clive Cenxin Aw, Lin Htoo Zaw, Maria Balanzó-Juandó, and Valerio Scarani, "Role of Dilations in Reversing Physical Processes: Tabletop Reversibility and Generalized Thermal Operations", PRX Quantum 5 1, 010332 (2024).

[7] Yu Guo, Zixuan Liu, Hao Tang, Xiao-Min Hu, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, and Giulio Chiribella, "Experimental Demonstration of Input-Output Indefiniteness in a Single Quantum Device", Physical Review Letters 132 16, 160201 (2024).

[8] Aw Clive Cenxin, Kelvin Onggadinata, Dagomir Kaszlikowski, and Valerio Scarani, "Quantum Bayesian Inference in Quasiprobability Representations", PRX Quantum 4 2, 020352 (2023).

[9] 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).

[10] James Fullwood and Arthur J. Parzygnat, "On dynamical measures of quantum information", arXiv:2306.01831, (2023).

[11] Arthur J. Parzygnat, "Reversing information flow: retrodiction in semicartesian categories", arXiv:2401.17447, (2024).

[12] Ardra Kooderi Suresh, Markus Frembs, and Eric G. Cavalcanti, "A Semantics for Counterfactuals in Quantum Causal Models", arXiv:2302.11783, (2023).

[13] Luca Giorgetti, Arthur J. Parzygnat, Alessio Ranallo, and Benjamin P. Russo, "Bayesian inversion and the Tomita-Takesaki modular group", arXiv:2112.03129, (2021).

[14] Zhian Jia and Dagomir Kaszlikowski, "The spatiotemporal doubled density operator: a unified framework for analyzing spatial and temporal quantum processes", arXiv:2305.15649, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 09:38:22) and SAO/NASA ADS (last updated successfully 2024-05-21 09:38:23). The list may be incomplete as not all publishers provide suitable and complete citation data.