Observational entropy papers

Please write us at observationalentropy@gmail.com if you would like your research to be represented: write the citation in the standard form and add links to arXiv and the published version (if it exists). Then add a one- or two-sentence summary of your paper.  The research must be related to observational entropy.


Casey O. Barkan, "Convergence, and lack of convergence, of phase space distributions to microcanonical equilibrium", arXiv:2404.05123v1 (2024)

Shows that classical maximal coarse-grained entropy (without the volume term that makes it different from OE), consistent with observed observable averages on each macrostate, converges to the microcanonical entropy in the long time limit. Results not directly inlcuding OE, but possibly could be modified and the results generalized to include OE.

F Meier, T Rivlin, T Debarba, J Xuereb, M Huber, M P E Lock, "Emergence of a second law of thermodynamics in isolated quantum systems" arXiv:2406.01677, (2024)

Studies the difference between Shannon entropies of a state at time t, and a time averaged state, for an arbitrary but fixed coarse-graining. Shows that time average of the absolute value of this difference is smaller than  - Sqrt[N r / d_eff ] log r  in the leading order, where N is the size of the largest macrostate, r is the number of macrostates, and d_eff  an effective dimension. An equivalent result is obtained for OE, is the form of - Sqrt[r / d_eff ] log d  in the leading order, where d is the dimension. Further, a bound on the fluctuations of this difference is obtained for both Shannon and OE.

T Nagasawa, K Kato, E Wakakuwa, F Buscemi, "On the generic increase of observational entropy in isolated systems", arXiv:2404.11985 (2024)

Shows that OE generically increases in isolated systems based on statistical arguments: if the size of the smallest macrostate scales linearly with the size of the system, and evolution results in Haar measure random unitary, then the probability that OE will not be maximal is exponentially suppressed. More precise statemenent is given by introducing "asymptotic coarseness" of measurement, which guarantess that OE will be maximal as the dimension increases. Finally, the assumption of Haar random is weakened to 2-design.

P Strasberg, J Schindler, "Comparative Microscopic Study of Entropies and their Production", arXiv:2403.09403 (2024)

Performs numerical study of behaviors of various notion of entropies, including OE, in a paradigmatic example of heat exchange. Hamiltonian comes from random matrix theory. Three systems are considered: normal systems, systems with negative temperature, and systems with negative heat capacity . OE with local energy coarse-graining is denoted there as S_cgo, which is the same as "Factorized Observational entropy" in Physical Review A 99 (1), 012103 (2019), or "non-equilibrium dynamic entropy" in Foundations of Physics 51, 101 (2021).

P L Garrido, S Goldstein, D A Huse, and J L Lebowitz "Time evolution of the Boltzmann entropy for a nonequilibrium dilute gas" , arXiv:2403.07519 (2024)

Studies the Boltzmann entropy in the expansion of 200 classical hard disks when the partition is removed. Since OE is the sum of Shannon and Boltzmann entropies, and Shannon entropy is negligible in this case, it is almost the same as OE. Similar to the classical model simulated in Physical Review E 102 (3), 032106 (2020).

A Alonso-Serrano, S Schuster, M Visser, "Emergent Time and Time Travel in Quantum Physics", Universe10(2), 73 (2024)

Mentions that OE could show whether time travel is favored or disfavored on the thermodynamic grounds.


Adam Teixidó-Bonfill, Joseph Schindler, Dominik Šafránek, "Entropic partial orderings of quantum measurements", arXiv:2310.14086 (2023)

OE induces a partial ordering of coarse-grainings that differs from stochastic ordering.

Shivam Sinha, S. Aravinda, "Generalized α-Observational Entropy", arXiv:2312.03572 (2023)

Rényi OE is introduced and its properties studied.

Ge Bai, Dominik Šafránek, Joseph Schindler, Francesco Buscemi, Valerio Scarani, "Observational entropy with general quantum priors", arXiv:2308.08763 (2023)

Standard OE can be written as a quantum relative entropy with a maximally mixed state as a quantum prior. Three definitions for OE with general quantum priors are introduced, and their properties are discussed. Additionally, OE, and its generalizations, are interpreted as classical relative entropy between probabilities of forward and reversed processes.

Dominik Šafránek, "Ergotropic interpretation of entanglement entropy", arXiv:2306.08987 (2023)

Minimizing OE over local coarse-grainings gives entanglement entropy, and OE measures the amount of unitarily extracted work from unknown quantum sources. These results combined show that entanglement entropy measures the amount of the maximal unitarily extracted work from a source characterized only by local measurements. 

Joseph Schindler, Andreas Winter, "Continuity bounds on observational entropy and measured relative entropies", Journal of Mathematical Physics 64 (9) (2023) arXiv:2302.00400 

Shows that OE is a continuous function of the state (by a direct inequality), and also of the coarse-graining (for a fixed state, and without a direct inequality).

Dominik Šafránek, Dario Rosa, "Measuring energy by measuring any other observable", Phys. Rev. A 108, 022208 (2023), arXiv:2301.10428

OE is applied as a useful tool that determines which times of measurement are the most informative, to estimate an expectation value of a conserved observable (such as energy) of a quantum system. Intuitively, the most informative times of measurement are those of low OE.

Xiang Zhou, "Relations Between Observational Entropy and Other Measures Based on Tsallis-q Entropy", International Journal of Theoretical Physics 62(12) (2023) 

Some relations between OE and other measures derived, for small dimensional systems.

Xiang Zhou, "Dynamical Behavior of Quantum Correlation Entropy Under the Noisy Quantum Channel for Multiqubit Systems", International Journal of Theoretical Physics 62(2) (2023) 

Studies numerically quantum correlation entropy, which is OE minimized over local coarse-grainings, for bit-flip, phase-flip, and bit-phase flip channels.

D. Šafránek, D. Rosa, F. Binder, "Work extraction from unknown quantum sources", Phys. Rev. Lett. 130, 210401 (2023), arXiv:2209.11076

OE measures the amount of extracted work from a source of states about which nothing is known, apart from what can be learned from a single type of coarse-grained measurement.

Sreeram PG, R. Modak, S. Aravinda, "Witnessing quantum chaos using observational entropy", Phys. Rev. E 107, 064204 (2023), arxiv:2212.01585

Uses OE as a diagnostic tool for distinguishing regular and chaotic phases of the Quantum Kicked Top model. Compares it to OTOC approach. OE is as good as OTOC, but is more experimentally accessible.

F. Buscemi, J. Schindler, D. Šafránek, "Observational entropy, coarse-grained states, and the Petz recovery map: information-theoretic properties and bounds", New J. Phys. 25, 053002 (2023), arXiv:2209.03803

Proving identities between OE differences and the classical relative entropy. Showing that OE upper bounds the quantum relative entropy between the true and the recovered (coarse-grained) state of the system. Showing convexity properties of OE.


S. Gudder, "Entropy of quantum measurements", Entropy 2022, 24(11), 1686 (2022), arxiv:2210.15738

Defines entropy of an effect, which can be seen as the surprisal multiplied by the probability of an outcome. You can see this as a part of OE, which when summed gives the actual OE. Derives theorems about entropy of the sum of effects. Derives convexity properties, similar to those derived in "Observational entropy, coarse quantum states, and Petz recovery..."

A. Riera-Campeny, "Open quantum systems in and out of equilibrium theory and applications", PhD Thesis (2022)

A friendly overview of applications of OE to open quantum systems.

A. Stokes, "Nonconjugate quantum subsystems", Physical Review E 106 (3), 034111 (2022), arXiv:2106.11017

Derives that starting in a product state and a product coarse-graining, where the coarse-graining is time-dependent and observable A stops commuting with observable on B as time goes on (non-conjugate observables), the change in OE is positive.

R. Modak, S. Aravinda, "Observational entropic study of Anderson localization", Phys. Rev. A 106, 062217 (2022),  arXiv:2209.10273

OE can be used to detect Anderson localization.

P. Strasberg, A. Winter, J. Gemmer, J. Wang, "Classicality, Markovianity and local detailed balance from pure state dynamics", Phys. Rev. A 108, 012225 (2023), arXiv:2209.07977

It is shown from very basic principles when observational entropy increases monotonically (although this is only a corollary in the paper).

A. Teixidó-Bonfill, "Data processing makes POVMs coarser and observational entropy larger", arXiv:2209.04549

Describing OE in the context of data processing.

Xiang Zhou, Zhu-Jun Zheng  "Relations between the observational entropy and Rényi information measures", Quantum Information Processing 21, 228 (2022)

Derived several inequalities between OE and functions derived from Rényi entropies.


J. Schindler, E. Frangipane, and A. Aguirre. "Unitarity and the information problem in an explicit model of black hole evaporation", Class. Quantum Grav., 38:075025, (2021)

Points out as a remark that observational entropy (called coarse-grained entropy in this case) of a black hole is upper-bounded by the Bekenstein-Hawking entropy. See Eq. (22). 

R. Uzdin and S. Rahav. "Passivity deformation approach for the thermodynamics of isolated quantum setups", PRX Quantum, 2:010336, (2021).

Compares their approach, which provides changes in expectation values, to approach of Strasberg and Winter, which provides changes in OE in the open system dynamics.

A. Almheiri, T. Hartman, J. Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. "The entropy of Hawking radiation". Rev. Mod. Phys., 93:035002, (2021).

Discusses observational entropy (called coarse-grained entropy in this case), in relation to Hawking radiation, Eq. (4.2).

K. L. Ng, B. Opanchuk, M. Thenabadu, M. Reid, P. D. Drummond "Fate of the false vacuum: Finite temperature, entropy, and topological phase in quantum simulations of the early universe," PRX Quantum 2, 010350 (2021) arXiv:2010.08665

The OE with coarse-graining given by coarse-grained Q-function (i.e., imitating a coarse-grained Wehrl entropy) has been used to study whether the system of a bubbly Universe evolves into a true or false vacuum, and how the false vacuum decays. Number of simulations using this type of OE are shown.

P. Strasberg "Quantum Stochastic Thermodynamics: Foundations and Selected Applications," (Oxford Graduate texts, 2021)

Book integrating OE in the concept of entropy production for open quantum systems.

A. Riera-Campeny, A. Sanpera, P. Strasberg "Quantum Systems Correlated with a Finite Bath: Nonequilibrium Dynamics and Thermodynamics," PRX Quantum 2, 010340 (2021), arXiv:2008.02184

Starting from the microscopic description of the system and the bath, it is shown that OE production is positive.

D. Šafránek, A. Aguirre, J. Schindler, J. M. Deutsch, "A brief introduction to observational entropy," Foundations of Physics 51, 101 (2021), arXiv:2008.04409

Summary and overview of recent results for OE in isolated quantum systems, its interpretation and non-equilibrium thermodynamic entropy.


D. Šafránek, J. Thingna "Quantifying Information Extraction using Generalized Quantum Measurements", arXiv:2007.07246

OE for general measurements (quantum instruments), generalized properties, and illustrations on indirect measurement schemes.

P. Strasberg and A. Winter, "First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy", PRX Quantum 2, 030202 (2021), arXiv:2002.08817

Application of OE to open quantum systems and derivation of a hierarchy of second laws. Based on a microscopic definition of temperature valid out of equilibrium, the conventional Clausius inequality emerges as a part of that hierarchy.

J Schindler, D Šafránek, A Aguirre, "Quantum correlation entropy", Physical Review A 102 (5), 052407 (2020), arXiv:2005.05408

OE minimized over local coarse-grainings defines a measure of non-classical correlations called quantum correlation entropy, which is similar to quantum discord.

J. Schindler "Basics of observational entropy", arXiv:2010.00142

Short overview of OE.

S. Goldstein, J. L. Lebowitz, R. Tumulka, N. Zanghi "Gibbs and Boltzmann entropy in classical and quantum mechanics", In: V. Allori (ed.), Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature. Singapore: World Scientific (2020), arXiv:1903.11870

Comparison of OE to other types of entropies used in thermodynamics.

J.M. Deutsch, D Šafránek, A Aguirre "Probabilistic bound on extreme fluctuations in isolated quantum systems", Physical Review E 101 (3), 032112 (2020), arXiv:1806.08897

Showing that the probability of the Universe collapsing into a small box is 50%. This collapse corresponds to a sharp decrease in OE.

D. Faiez, D. Šafránek, J.M. Deutsch, A. Aguirre "Typical and extreme entropies of long-lived isolated quantum systems", Physical Review A 101 (5), 052101 (2020), arXiv:1908.07083

Comparing OE and entanglement entropy in isolated quantum systems, and differences between states that extremize these entropies.

OE can also be defined for classical systems, where phase-space density takes the role of the density matrix. Also, equivalent thermodynamic properties hold.


M. Brooks, "The universe tends towards disorder. But how come nobody knows why?" New Scientist (2019)

Popular science coverage of OE research and other approaches to entropy.

A. Aguirre, "Cosmological Koans: A Journey to the Heart of Physical Reality" (S.R. Tompson, 2019)

Popular science book, where OE is argued as the entropy that we have in mind when we say "Entropy of the Universe increases."

D Šafránek, J.M. Deutsch, A Aguirre "Quantum coarse-grained entropy and thermodynamics" Physical Review A 99 (1), 010101(R) (2019), arXiv:1707.09722

Short version of "Quantum coarse-grained entropy and thermalization in closed systems."

D Šafránek, J.M. Deutsch, A Aguirre "Quantum coarse-grained entropy and thermalization in closed systems" Physical Review A 99 (1), 012103 (2019), arXiv:1803.00665

The first paper introducing the term "observational entropy." It revived the research on OE after Alfred Wehrl in the 70s.  Plenty of definitions and theorems, and applications of OE in thermodynamics. The first definition of OE that includes multiple projective coarse-grainings.


C.M. Caves, “Resource Material for Promoting the Bayesian View of Everything”, online notes (2001)

A lot of promoting of the classical OE from the Bayesian perspective.


A. Wehrl, “General properties of entropy”, Rev. Mod. Phys. 50, 221–260 (1978).

Showing that the classical master equation leads to the increase in OE.


N.G. Van Kampen, “Quantum statistics of irreversible processes”, Physica, 20(1-6), 603–622. (1954)

Derives deviation of OE from the equilibrium value up to the second order.


J. von Neumann, "Mathematische Grundlagen der Quantenmechanik" (Julius Springer, 1932).

English translation: J. von Neumann, "Mathematical foundations of quantum mechanics (Princeton university press, 1955)

Book in which von Neumann introduced OE for general projective coarse-grainings.


J. 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). arXiv:1003.2133

The first paper that wrote down the definition of OE, and proved the quantum version of Boltzmann H-theorem for it.