Observational entropy with general quantum priors

Ge Bai; Dominik Šafránek; Joseph Schindler; Francesco Buscemi; Valerio Scarani

doi:10.22331/q-2024-11-14-1524

Observational entropy with general quantum priors

Ge Bai¹, Dominik Šafránek^2,3, Joseph Schindler⁴, Francesco Buscemi⁵, and Valerio Scarani^1,6

¹Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
²Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea
³Basic Science Program, Korea University of Science and Technology (UST), Daejeon - 34113, Korea
⁴Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
⁵Department of Mathematical Informatics, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8601, Japan
⁶Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542

Published:	2024-11-14, volume 8, page 1524
Eprint:	arXiv:2308.08763v3
Doi:	https://doi.org/10.22331/q-2024-11-14-1524
Citation:	Quantum 8, 1524 (2024).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Observational entropy captures both the intrinsic uncertainty of a thermodynamic state and the lack of knowledge due to coarse-graining. We demonstrate two interpretations of observational entropy, one as the statistical deficiency resulting from a measurement, the other as the difficulty of inferring the input state from the measurement statistics by quantum Bayesian retrodiction. These interpretations show that the observational entropy implicitly includes a uniform reference prior. Since the uniform prior cannot be used when the system is infinite-dimensional or otherwise energy-constrained, we propose generalizations by replacing the uniform prior with arbitrary quantum states that may not even commute with the state of the system. We propose three candidates for this generalization, discuss their properties, and show that one of them gives a unified expression that relates both interpretations.

Featured image: A quantum system, like a cat, has intrinsic uncertainties. Observational entropy characterizes both the intrinsic uncertainty of the system and additional uncertianty introduced by the lens we use to observe it.

Popular summary

In his 1932 book, von Neumann not only introduced the now familiar von Neumann entropy, but also discussed another entropic quantity that he called "macroscopic". He argued that this macroscopic entropy, rather than the von Neumann entropy, is the key measure for understanding thermodynamic systems. Here we revisit and extend a concept derived from von Neumann's macroscopic entropy, observational entropy (OE) — an information-theoretic quantity that measures both the intrinsic uncertainty of a system and the additional uncertainty introduced by the measurement we use to observe it.

Typically, the definition of OE assumes a "uniform prior," i.e., it starts with an assumption of maximum uncertainty about the state of the system. However, this assumption is not always tenable, especially in more complex systems, such as those influenced by energy constraints or infinite dimensional systems, where instead other priors, such as the Gibbs distribution, would be preferable, both physically and mathematically.

To fill this gap and extend OE to arbitrary priors, we first show how OE can be interpreted in two ways: as a measure of how much a measurement scrambles the true state of a system (statistical deficiency), and as the difficulty of inferring the original state from the measurement results (irretrodictability). These two aspects provide complementary insights into how much we lose or gain in our knowledge of the original state of a system when we make observations on it.

This conceptual insight leads us to introduce three generalized versions of OE: two that capture either statistical deficiency or irretrodictability, but are inherently incompatible; and a third, based on Belavkin-Staszewski relative entropy, that instead is able to combine both perspectives and provide a unified view of commuting and non-commuting priors alike. We expect that our results will pave the way for a consistent treatment of the second law of thermodynamics and fluctuation relations in fully quantum scenarios.

► BibTeX data

@article{Bai2024observational,
  doi = {10.22331/q-2024-11-14-1524},
  url = {https://doi.org/10.22331/q-2024-11-14-1524},
  title = {Observational entropy with general quantum priors},
  author = {Bai, Ge and {\v{S}}afr{\'{a}}nek, Dominik and Schindler, Joseph and Buscemi, Francesco and Scarani, Valerio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1524},
  month = nov,
  year = {2024}
}

► References

[1] John von Neumann. ``Mathematical foundations of quantum mechanics''. Princeton university press. (1955).
https://doi.org/10.1515/9781400889921

[2] John von Neumann. ``Proof of the ergodic theorem and the H-theorem in quantum mechanics. Translation of: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik''. European Physical Journal H 35, 201–237 (2010).
https://doi.org/10.1140/epjh/e2010-00008-5

[3] Dominik Šafránek, J. M. Deutsch, and Anthony Aguirre. ``Quantum coarse-grained entropy and thermodynamics''. Phys. Rev. A 99, 010101 (2019). arXiv:1707.09722.
https://doi.org/10.1103/PhysRevA.99.010101
arXiv:1707.09722

[4] Dominik Šafránek, J. M. Deutsch, and Anthony Aguirre. ``Quantum coarse-grained entropy and thermalization in closed systems''. Phys. Rev. A 99, 012103 (2019). arXiv:1803.00665.
https://doi.org/10.1103/PhysRevA.99.012103
arXiv:1803.00665

[5] Dominik Šafránek, Anthony Aguirre, Joseph Schindler, and J. M. Deutsch. ``A Brief Introduction to Observational Entropy''. Foundations of Physics 51, 101 (2021). arXiv:2008.04409.
https://doi.org/10.1007/s10701-021-00498-x
arXiv:2008.04409

[6] Philipp Strasberg and Andreas Winter. ``First and second law of quantum thermodynamics: A consistent derivation based on a microscopic definition of entropy''. PRX Quantum 2, 030202 (2021).
https://doi.org/10.1103/PRXQuantum.2.030202

[7] Dominik Šafránek and Juzar Thingna. ``Quantifying information extraction using generalized quantum measurements''. Physical Review A 108, 032413 (2023). arXiv:2007.07246.
https://doi.org/10.1103/PhysRevA.108.032413
arXiv:2007.07246

[8] 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, 053002 (2023).
https://doi.org/10.1088/1367-2630/accd11

[9] Andreu Riera-Campeny, Anna Sanpera, and Philipp Strasberg. ``Quantum systems correlated with a finite bath: Nonequilibrium dynamics and thermodynamics''. PRX Quantum 2, 010340 (2021). arXiv:2008.02184.
https://doi.org/10.1103/PRXQuantum.2.010340
arXiv:2008.02184

[10] Dominik Šafránek, Anthony Aguirre, and J. M. Deutsch. ``Classical dynamical coarse-grained entropy and comparison with the quantum version''. Phys. Rev. E 102, 032106 (2020). arXiv:1905.03841.
https://doi.org/10.1103/PhysRevE.102.032106
arXiv:1905.03841

[11] Joshua M. Deutsch, Dominik Šafránek, and Anthony Aguirre. ``Probabilistic bound on extreme fluctuations in isolated quantum systems''. 101, 032112 (2020). arXiv:1806.08897.
https://doi.org/10.1103/PhysRevE.101.032112
arXiv:1806.08897

[12] Dana Faiez, Dominik Šafránek, J. M. Deutsch, and Anthony Aguirre. ``Typical and extreme entropies of long-lived isolated quantum systems''. 101, 052101 (2020). arXiv:1908.07083.
https://doi.org/10.1103/PhysRevA.101.052101
arXiv:1908.07083

[13] Charlie Nation and Diego Porras. ``Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories''. 102, 042115 (2020). arXiv:2003.08425.
https://doi.org/10.1103/PhysRevE.102.042115
arXiv:2003.08425

[14] Philipp Strasberg, María García Díaz, and Andreu Riera-Campeny. ``Clausius inequality for finite baths reveals universal efficiency improvements''. Phys. Rev. E 104, L022103 (2021). arXiv:2012.03262.
https://doi.org/10.1103/PhysRevE.104.L022103
arXiv:2012.03262

[15] Ryusuke Hamazaki. ``Speed Limits for Macroscopic Transitions''. PRX Quantum 3, 020319 (2022). arXiv:2110.09716.
https://doi.org/10.1103/PRXQuantum.3.020319
arXiv:2110.09716

[16] Ranjan Modak and S. Aravinda. ``Observational-entropic study of anderson localization''. Phys. Rev. A 106, 062217 (2022).
https://doi.org/10.1103/PhysRevA.106.062217

[17] Sreeram PG, Ranjan Modak, and S. Aravinda. ``Witnessing quantum chaos using observational entropy''. Phys. Rev. E 107, 064204 (2023).
https://doi.org/10.1103/PhysRevE.107.064204

[18] Joseph Schindler and Andreas Winter. ``Continuity bounds on observational entropy and measured relative entropies''. Journal of Mathematical Physics 64 (2023). arXiv:2302.00400.
https://doi.org/10.1063/5.0147294
arXiv:2302.00400

[19] Dominik Šafránek, Dario Rosa, and Felix C. Binder. ``Work extraction from unknown quantum sources''. Phys. Rev. Lett. 130, 210401 (2023).
https://doi.org/10.1103/PhysRevLett.130.210401

[20] Dominik Šafránek and Dario Rosa. ``Measuring energy by measuring any other observable''. Phys. Rev. A 108, 022208 (2023).
https://doi.org/10.1103/PhysRevA.108.022208

[21] Dominik Šafránek. ``Ergotropic interpretation of entanglement entropy'' (2023). arXiv:2306.08987.
arXiv:2306.08987

[22] Hisaharu Umegaki. ``On information in operator algebras''. Proc. Japan Acad. 37, 459–461 (1961).
https://doi.org/10.3792/pja/1195523632

[23] Hisaharu Umegaki. ``Conditional expectation in an operator algebra. iv. entropy and information''. Kodai Mathematical Journal 14 (1962).
https://doi.org/10.2996/kmj/1138844604

[24] S. Kullback and R. A. Leibler. ``On information and sufficiency''. Ann. Math. Statist. 22, 79–86 (1951).
https://doi.org/10.1214/aoms/1177729694

[25] Gavin E. Crooks. ``Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems''. Journal of Statistical Physics 90, 1481–1487 (1998).
https://doi.org/10.1023/A:1023208217925

[26] Massimiliano Esposito, Upendra Harbola, and Shaul Mukamel. ``Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems''. Rev. Mod. Phys. 81, 1665–1702 (2009).
https://doi.org/10.1103/RevModPhys.81.1665

[27] Gabriel T. Landi and Mauro Paternostro. ``Irreversible entropy production: From classical to quantum''. Rev. Mod. Phys. 93, 035008 (2021).
https://doi.org/10.1103/RevModPhys.93.035008

[28] Francesco Buscemi and Valerio Scarani. ``Fluctuation theorems from bayesian retrodiction''. Phys. Rev. E 103, 052111 (2021).
https://doi.org/10.1103/PhysRevE.103.052111

[29] Clive Cenxin Aw, Francesco Buscemi, and Valerio Scarani. ``Fluctuation theorems with retrodiction rather than reverse processes''. AVS Quantum Science 3, 045601 (2021). arXiv:https://doi.org/10.1116/5.0060893.
https://doi.org/10.1116/5.0060893
arXiv:https://doi.org/10.1116/5.0060893

[30] Richard Carl Jeffrey. ``The logic of decision''. McGraw-Hill. (1965).

[31] Judea Pearl. ``Probabilistic reasoning in intelligent systems: networks of plausible inference''. Elsevier. (1988).
https://doi.org/10.1016/C2009-0-27609-4

[32] Hei Chan and Adnan Darwiche. ``On the revision of probabilistic beliefs using uncertain evidence''. Artificial Intelligence 163, 67 – 90 (2005).
https://doi.org/10.1016/j.artint.2004.09.005

[33] Satosi Watanabe. ``Symmetry of physical laws. part iii. prediction and retrodiction''. Rev. Mod. Phys. 27, 179–186 (1955).
https://doi.org/10.1103/RevModPhys.27.179

[34] Satosi Watanabe. ``Conditional probabilities in physics''. Progr. Theor. Phys. Suppl. E65, 135–160 (1965).
https://doi.org/10.1143/PTPS.E65.135

[35] Arthur J. Parzygnat and Francesco Buscemi. ``Axioms for retrodiction: achieving time-reversal symmetry with a prior''. Quantum 7, 1013 (2023).
https://doi.org/10.22331/q-2023-05-23-1013

[36] Denes Petz. ``Sufficient subalgebras and the relative entropy of states of a von neumann algebra''. Comm. Math. Phys. 105, 123–131 (1986).
https://doi.org/10.1007/BF01212345

[37] Denes Petz. ``Sufficiency of channels over von Neumann algebras''. The Quarterly Journal of Mathematics 39, 97–108 (1988).
https://doi.org/10.1093/qmath/39.1.97

[38] Man-Duen Choi. ``Completely positive linear maps on complex matrices''. Linear Algebra and its Applications 10, 285–290 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0

[39] M. S. Leifer. ``Conditional density operators and the subjectivity of quantum operations''. In AIP Conference Proceedings. Volume 889, page 172–186. AIP (2007).
https://doi.org/10.1063/1.2713456

[40] M. S. Leifer and Robert W. Spekkens. ``Towards a formulation of quantum theory as a causally neutral theory of bayesian inference''. Physical Review A 88 (2013).
https://doi.org/10.1103/physreva.88.052130

[41] Viacheslav P. Belavkin and P. Staszewski. ``$C^{*}$-algebraic generalization of relative entropy and entropy''. In Annales de l'IHP Physique théorique. Volume 37 no. 1, pages 51–58. (1982). url: http://www.numdam.org/item/AIHPA_1982__37_1_51_0/.
http://www.numdam.org/item/AIHPA_1982__37_1_51_0/

[42] Sumeet Khatri and Mark M. Wilde. ``Principles of quantum communication theory: A modern approach'' (2020). arXiv:2011.04672.
arXiv:2011.04672

[43] Keiji Matsumoto. ``A new quantum version of f-divergence''. In Reality and Measurement in Algebraic Quantum Theory. Page 229–273. Springer Singapore (2018).
https://doi.org/10.1007/978-981-13-2487-1_10

[44] Fumio Hiai and Milán Mosonyi. ``Different quantum f-divergences and the reversibility of quantum operations''. Reviews in Mathematical Physics 29, 1750023 (2017).
https://doi.org/10.1142/s0129055x17500234

[45] Andreas Bluhm, Ángela Capel, Paul Gondolf, and Antonio Pérez-Hernández. ``Continuity of quantum entropic quantities via almost convexity''. IEEE Transactions on Information Theory 69, 5869–5901 (2023).
https://doi.org/10.1109/tit.2023.3277892

[46] Fumio Hiai, Milán Mosonyi, Dénes Petz, and Cédric Bény. ``Quantum f-divergences and error correction''. Reviews in Mathematical Physics 23, 691–747 (2011).
https://doi.org/10.1142/s0129055x11004412

[47] Andreas Bluhm and Ángela Capel. ``A strengthened data processing inequality for the belavkin–staszewski relative entropy''. Reviews in Mathematical Physics 32, 2050005 (2019).
https://doi.org/10.1142/s0129055x20500051

Cited by

[1] Tatsuaki Tsuruyama, "Thermodynamics reconstructed from information theory: an axiomatic framework via information-volume constraints and path-space KL divergence", The European Physical Journal Plus 141 2, 180 (2026).

[2] Teruaki Nagasawa, Eyuri Wakakuwa, Kohtaro Kato, and Francesco Buscemi, "Macroscopicity and observational deficit in states, operations, and correlations* ", Reports on Progress in Physics 88 11, 117601 (2025).

[3] Shintaro Minagawa, M. Hamed Mohammady, Kenta Sakai, Kohtaro Kato, and Francesco Buscemi, "Universal validity of the second law of information thermodynamics", npj Quantum Information 11 1, 18 (2025).

[4] Teruaki Nagasawa, Kohtaro Kato, Eyuri Wakakuwa, and Francesco Buscemi, "Generic increase of observational entropy in isolated systems", Physical Review Research 6 4, 043327 (2024).

[5] Ge Bai, Francesco Buscemi, and Valerio Scarani, "Quantum Bayes’ Rule and Petz Transpose Map from the Minimum Change Principle", Physical Review Letters 135 9, 090203 (2025).

[6] Joseph Schindler, Philipp Strasberg, Niklas Galke, Andreas Winter, and Michael G. Jabbour, "Unification of observational entropy with maximum entropy principles", arXiv:2503.15612, (2025).

[7] Leonardo Rossetti, Stefano Mancini, Andreas Winter, and Joseph Schindler, "Observational entropy of quantum correlations and entanglement", arXiv:2510.10058, (2025).

[8] Clive Cenxin Aw, "Classical and quantum reverse processes through Bayesian inference", International Journal of Quantum Information 24 2, 2530004-650 (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-07-15 11:22:25) and SAO/NASA ADS (last updated successfully 2026-07-14 19:40:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-07-15 11:22:25: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.