Observational entropy papers

In the Supplemental Material, dynamical typicality of observational entropy production is derived. This shows that the difference between the mean value of observational entropy with coarse energy coarse-graining and the canonical entropy is very small, meaning that when measured at a random time in future, there will be almost never observed any difference.

Shows how many measurements are needed to estimate observational ergotropy up to an arbitrary precision. This fixes an issue with the original protocol,  Phys. Rev. Lett. 130, 210401 (2023), where an infinite amount of measurement is assumed.

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[N 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.

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).

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).

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

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.

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).

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

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.

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.

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..."

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.

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

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

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). 

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

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

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

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.

Short overview of OE.

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

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

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

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.

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

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

Observational entropy is mentioned as the relevant quantity most related to this work, since it also grows in isolated quantum systems and relates to the arrow of time. The paper itself however focuses on growth using autocorrelation functions.

Speculates that studying observational entropy could show whether in the case of time travel, going backwards in time is disfavored based on thermodynamics.