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arXiv · 2002.09649

A new matrix inequality involving partial traces

Yongtao Li·Weijun Liu·Yang Huang
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Abstract

Let $A$ be an $m\times m$ positive semidefinite block matrix with each block being $n$-square. We write $\mathrm{tr}_1$ and $\mathrm{tr}_2$ for the first and second partial trace, respectively. In this paper, we prove the following inequality \[ (\mathrm{tr} A)I_{mn} - (\mathrm{tr}_2 A) \otimes I_n \ge \pm \bigl( I_m\otimes (\mathrm{tr}_1 A) -A\bigr).\] This inequality is not only a generalization of Ando's result [ILAS Conference (2014)] and Lin [Canad. Math. Bull. 59 (2016) 585--591], but it also could be regarded as a complement of a recent result of Choi [Linear Multilinear Algebra 66 (2018) 1619--1625]. Additionally, some new partial traces inequalities for positive semidefinite block matrices are also included.

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Yongtao Li, Weijun Liu, Yang Huang. 2020-02-22. A new matrix inequality involving partial traces. https://doi.org/10.7153/oam-2021-15-75

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