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1. 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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2. Quantification of electron correlation for approximate quantum calculations

   https://doi.org/10.1063/5.0119260  

   Yuan, Shunyue; Chang, Yueqing; Wagner, Lucas K. (November 2022, The Journal of Chemical Physics)

   State-of-the-art many-body wave function techniques rely on heuristics to achieve high accuracy at an attainable computational cost to solve the many-body Schrödinger equation. By far, the most common property used to assess accuracy has been the total energy; however, total energies do not give a complete picture of electron correlation. In this work, we assess the von Neumann entropy of the one-particle reduced density matrix (1-RDM) to compare selected configuration interaction (CI), coupled cluster, variational Monte Carlo, and fixed-node diffusion Monte Carlo for benchmark hydrogen chains. A new algorithm, the circle reject method, is presented, which improves the efficiency of evaluating the von Neumann entropy using quantum Monte Carlo by several orders of magnitude. The von Neumann entropy of the 1-RDM and the eigenvalues of the 1-RDM are shown to distinguish between the dynamic correlation introduced by the Jastrow and the static correlation introduced by determinants with large weights, confirming some of the lore in the field concerning the difference between the selected CI and Slater–Jastrow wave functions. 

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3. A Random Matrix Approach to Absorption in Free Products

   https://doi.org/10.1093/imrn/rnaa191  

   Hayes, Ben; Jekel, David; Nelson, Brent; Sinclair, Thomas (August 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $$1$$-bounded entropy. We show that if $${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$$, then $${\mathcal{P}}$$ absorbs any subalgebra of $${\mathcal{M}}$$ that intersects it diffusely and that has $$1$$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra $${\mathcal{P}} \leq{\mathcal{M}}$$ to have this absorption property, it suffices for $${\mathcal{M}}$$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto $${\mathcal{P}}$$. 

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4. Quantum entropy and central limit theorem

   https://doi.org/10.1073/pnas.2304589120  

   Bu, Kaifeng; Gu, Weichen; Jaffe, Arthur (June 2023, Proceedings of the National Academy of Sciences)

   We introduce a framework to study discrete-variable (DV) quantum systems based on qudits. It relies on notions of a mean state (MS), a minimal stabilizer-projection state (MSPS), and a new convolution. Some interesting consequences are: The MS is the closest MSPS to a given state with respect to the relative entropy; the MS is extremal with respect to the von Neumann entropy, demonstrating a “maximal entropy principle in DV systems.” We obtain a series of inequalities for quantum entropies and for Fisher information based on convolution, giving a “second law of thermodynamics for quantum convolutions.” We show that the convolution of two stabilizer states is a stabilizer state. We establish a central limit theorem, based on iterating the convolution of a zero-mean quantum state, and show this converges to its MS. The rate of convergence is characterized by the “magic gap,” which we define in terms of the support of the characteristic function of the state. We elaborate on two examples: the DV beam splitter and the DV amplifier. 

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5. An alternative method for extracting the von Neumann entropy from Rényi entropies

   https://doi.org/10.1007/JHEP01(2021)042  

   D’Hoker, Eric; Dong, Xi; Wu, Chih-Hung (January 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract An alternative method is presented for extracting the von Neumann entropy − Tr( ρ ln ρ ) from Tr( ρ n ) for integer n in a quantum system with density matrix ρ . Instead of relying on direct analytic continuation in n , the method uses a generating function − Tr{ ρ ln[(1 − zρ )/(1 − z )]} of an auxiliary complex variable z . The generating function has a Taylor series that is absolutely convergent within |z| < 1, and may be analytically continued in z to z = −∞ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in n . Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length. 

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