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1. A topological dynamical system with two different positive sofic entropies

   https://doi.org/10.1090/btran/101  

   Airey, Dylan; Bowen, Lewis; Lin, Yuqing (February 2022, Transactions of the American Mathematical Society, Series B)

   A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. This paper exhibits an explicit example of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-colorings of random hyper-graphs. 

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2. Entropy, products, and bounded orbit equivalence

   https://doi.org/10.1017/etds.2021.154  

   KERR, DAVID; LI, HANFENG (March 2023, Ergodic Theory and Dynamical Systems)

   Abstract We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities. 

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3. Locally compact sofic entropy theory

   https://doi.org/10.1090/btran/179  

   Bowen, Lewis (February 2025, Transactions of the American Mathematical Society, Series B)

   This paper generalizes sofic entropy theory, in both the topological and measure-theory settings, to actions of locally compact groups. We prove invariance under topological and measure conjugacy of these entropies and establish the variational principle. 

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4. Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings

   https://doi.org/10.1063/5.0066653  

   Junge, Marius; LaRacuente, Nicholas (December 2022, Journal of Mathematical Physics)

   Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states. 

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5. Relative State Counting for Semiclassical Black Holes

   https://doi.org/10.1103/PhysRevLett.133.201601  

   Akers, Chris; Sorce, Jonathan (November 2024, Physical Review Letters)

   It has been shown that entropy differences between certain states of perturbative quantum gravity can be computed without specifying an ultraviolet completion. This is analogous to the situation in classical statistical mechanics, where entropy differences are defined but absolute entropy is not. Unlike in classical statistical mechanics, however, the entropy differences computed in perturbative quantum gravity do not have a clear physical interpretation. Here we construct a family of perturbative black hole states for which the entropy difference can be interpreted as a relative counting of states. Conceptually, this Letter begins with the algebra of mass fluctuations around a fixed black hole background, and points out that while this is a type I algebra, it is not a factor and therefore has no canonical definition of entropy. As in previous work, coupling the mass fluctuations to quantum matter embeds the mass algebra within a type II factor, in which entropy differences (but not absolute entropies) are well defined. It is then shown that for microcanonical wave functions of mass fluctuation, the type II entropy difference equals the logarithm of the dimension of the extra Hilbert space that is needed to map one microcanonical window to another using gauge-invariant unitaries. The Letter closes with comments on type II entropy difference in a more general class of states, where the von Neumann entropy difference does not have a physical interpretation, but “one-shot” entropy differences do. Published by the American Physical Society2024 

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