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This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps.…
Within the framework of linear vector Gaussian channels with arbitrary signaling, closed-form expressions for the Jacobian of the minimum mean square error and Fisher information matrices with respect to arbitrary parameters of the system…
Expressions for (EPI Shannon type) Divergence-Power Inequalities (DPI) in two cases (time-discrete and band-limited time-continuous) of stationary random processes are given. The new expressions connect the divergence rate of the sum of…
We establish a lower bound on the entropy of weighted sums of (possibly dependent) random variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of $(X_1, X_2, \dots,…
We extend quantum Stein's lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states…
In this appendix to our paper with the same title posted on arxiv we give a quick proof of an inequality that can be substituted to Hastings's result, quoted as Lemma 1.9 in our previous paper. Our inequality is less sharp but also appears…
This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei (2014) and Liu et al. (2015) are…
Relative to the Gaussian measure on $\mathbb{R}^d$, entropy and Fisher information are famously related via Gross' logarithmic Sobolev inequality (LSI). These same functionals also separately satisfy convolution inequalities, as proved by…
The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}|X_j|^{\alpha_j}]\geq\prod_{j=1}^{n}E[|X_j|^{\alpha_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and any non-negative real…
We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. Those inequalities can be considered as…
We introduce a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for the map…
We extend the result of Ball and Nguyen on the jump of entropy under convolution for logconcave random vectors. We show that the result holds for any pair of vectors (not necessarily identically distributed) and that a similar inequality…
We show that there is equality in Shannon's Entropy Power Inequality (EPI) if and only if the random variables involved are Gaussian, assuming nothing beyond the existence of differential entropies. This is done by justifying de Bruijn's…
We consider probability mass functions $V$ supported on the positive integers using arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry--\'{E}mery condition for a Markov birth and death operator with invariant measure $V$.…
In this article, we employ a standard convex argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.
When applying the quasi-Monte Carlo (QMC) method of numerical integration of univariate functions, Koksma's inequality provides a basic estimate of the error in terms of the discrepancy of the used evaluation points and the total variation…
The conditional entropy power inequality is a fundamental inequality in information theory, stating that the conditional entropy of the sum of two conditionally independent vector-valued random variables each with an assigned conditional…
We derive the monotonicity of the quantum relative entropy by an elementary operational argument based on Stein's lemma in quantum hypothesis testing. For the latter we present an elementary and short proof that requires the law of large…
The phenomenon of entropy concentration provides strong support for the maximum entropy method, MaxEnt, for inferring a probability vector from information in the form of constraints. Here we extend this phenomenon, in a discrete setting,…
In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal…