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1. Sharp Maximum-Entropy Comparisons

   https://doi.org/10.1109/ISIT45174.2021.9517779  

   Aras, Efe; Courtade, Thomas A. (July 2021, 2021 IEEE International Symposium on Information Theory (ISIT))

   We establish a family of sharp entropy inequalities with Gaussian extremizers. These inequalities hold for certain dependent random variables, namely entropy-maximizing couplings subject to information constraints. Several well-known results, such as the Zamir–Feder and Brunn–Minkowski inequalities, follow as special cases. 

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2. Linear Complexity Entropy Regions

   https://doi.org/10.1109/ISIT45174.2021.9518030  

   Liu, Yirui; Walsh, John MacLaren (July 2021, 2021 IEEE International Symposium on Information Theory (ISIT))

   A great many problems in network information theory, both fundamental and applied, involve determining a minimal set of inequalities linking the Shannon entropies of a certain collection of subsets of random variables. In principle this minimal set of inequalities could be determined from the entropy region, whose dimensions are all the subsets of random variables, by projecting out dimensions corresponding to the irrelevant subsets. As a general solution technique, however, this method is plagued both by the incompletely known nature of the entropy region as well as the exponential complexity of its bounds. Even worse, for four or more random variables, it is known that the set of linear information inequalities necessary to completely describe the entropy region must be uncountably infinite. A natural question arises then, if there are certain nontrivial collections of subsets where the inequalities linking only these subsets is both completely known, and have inequality descriptions that are linear in the number of random variables. This paper answers this question in the affirmative. A decomposition expressing the collection of inequalities linking a larger collection of subsets from that of smaller collections of subsets is first proven. This decomposition is then used to provide systems of subsets for which it both exhaustively determines the complete list of inequalities, which is linear in the number of variables. 

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3. Quantum Extremal Surfaces and the Holographic Entropy Cone

   https://doi.org/10.1007/JHEP11(2021)177  

   Akers, Chris; Hernández-Cuenca, Sergio; Rath, Pratik (November 2021, Journal of High Energy Physics)

   A bstract Quantum states with geometric duals are known to satisfy a stricter set of entropy inequalities than those obeyed by general quantum systems. The set of allowed entropies derived using the Ryu-Takayanagi (RT) formula defines the Holographic Entropy Cone (HEC). These inequalities are no longer satisfied once general quantum corrections are included by employing the Quantum Extremal Surface (QES) prescription. Nevertheless, the structure of the QES formula allows for a controlled study of how quantum contributions from bulk entropies interplay with HEC inequalities. In this paper, we initiate an exploration of this problem by relating bulk entropy constraints to boundary entropy inequalities. In particular, we show that requiring the bulk entropies to satisfy the HEC implies that the boundary entropies also satisfy the HEC. Further, we also show that requiring the bulk entropies to obey monogamy of mutual information (MMI) implies the boundary entropies also obey MMI. 

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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. First-Order Greedy Invariant-Domain Preserving Approximation for Hyperbolic Problems: Scalar Conservation Laws, and p-System

   https://doi.org/10.1007/s10915-024-02592-4  

   Guermond, Jean-Luc; Maier, Matthias; Popov, Bojan; Saavedra, Laura; Tomas, Ignacio (August 2024, Journal of Scientific Computing)

   The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities. 

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