State retrieval beyond Bayes’ retrodiction

Jacopo Surace and Matteo Scandi

ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain

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Abstract

In the context of irreversible dynamics, associating to a physical process its intuitive reverse can result to be a quite ambiguous task. It is a standard choice to define the reverse process using Bayes' theorem, but, in general, this choice is not optimal. In this work we explore whether it is possible to characterise an optimal reverse map building from the concept of state retrieval maps. In doing so, we propose a set of principles that state retrieval maps should satisfy. We find out that the Bayes inspired reverse is just one case in a whole class of possible choices, which can be optimised to give a map retrieving the initial state more precisely than the Bayes rule. Our analysis has the advantage of naturally extending to the quantum regime. In fact, we find a class of reverse transformations containing the Petz recovery map as a particular case, corroborating its interpretation as quantum analogue of the Bayes retrieval. Finally, we present numerical evidences that by adding a single extra axiom one can isolate the usual reverse process derived from Bayes' theorem.

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[1] 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).

[2] Arthur J. Parzygnat and James Fullwood, "From Time-Reversal Symmetry to Quantum Bayes’ Rules", PRX Quantum 4 2, 020334 (2023).

[3] Arthur J. Parzygnat and Francesco Buscemi, "Axioms for retrodiction: achieving time-reversal symmetry with a prior", Quantum 7, 1013 (2023).

[4] Paolo Abiuso, Matteo Scandi, Dario De Santis, and Jacopo Surace, "Characterizing (non-)Markovianity through Fisher information", SciPost Physics 15 1, 014 (2023).

[5] Akshaya Jayashankar and Prabha Mandayam, "Quantum Error Correction: Noise-Adapted Techniques and Applications", Journal of the Indian Institute of Science 103 2, 497 (2023).

[6] 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).

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

[8] Akshaya Jayashankar and Prabha Mandayam, "Quantum Error Correction: Noise-adapted Techniques and Applications", arXiv:2208.00365, (2022).

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