Generalized Shannon inequalities based on Tsallis relative operator entropy

Some Operator Inequalities via Convexity

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.

A Complementary Inequality to the Information Monotonicity for Tsallis Relative Operator Entropy

We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present converse of Ando's inequality, for each parameter. We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the Löwner-Heinz inequality under certain condition. Some interesting consequences of inner product spaces and norm inequalities are also presented.

A refinement of the Robertson-Schrödinger uncertainty principle and a Hirschman-Shannon inequality for Wigner distributions

We propose a refinement of the Robertson-Schrodinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product …

谱几何平均的若干不等式及其应用

Some inequalities for spectral geometric mean with applications

本文研究谱几何平均的不等式，推导了正可逆算子的Hölder型不等式、算子序关系及范数版本，证明矩阵情形的对数优超关系，得到量子Tsallis相对熵的序关系与新下界。

Inequalities for Quantum f-Divergence of Trace Class Operators in Hilbert Spaces

Some inequalities for quantum f-divergence of trace class operators in Hilbert spaces are obtained. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantum f-divergence in terms of variational and chi-distance are provided. Applications for some classes of divergence measures such as Umegaki and Tsallis relative entropies are also given.

New sharp inequalities for operator means

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.

Some remarks on Tsallis relative operator entropy

This paper intends to give some new estimates for Tsallis relative operator entropy ${{T}_{v}}\left( A|B \right)=\frac{A{{\natural}_{v}}B-A}{v}$. Let $A$ and $B$ be two positive invertible operators with the spectra contained in the interval $J \subset (0,\infty)$. We prove for any $v\in \left[ -1,0 \right)\cup \left( 0,1 \right]$, $$ (\ln_v t)A+\left( A{{\natural}_{v}}B+tA{{\natural}_{v-1}}B \right)\le {{T}_{v}}\left( A|B \right) \le (\ln_v s)A+{{s}^{v-1}}\left( B-sA \right) $$ where $s,t\in J$. Especially, the upper bound for Tsallis relative operator entropy is a non-trivial new result. Mea…

The geometry of Euclidean convolution inequalities and entropy

The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.