A new matrix inequality involving partial traces

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.

11 pages. This is the final version. We added Corollary 2.6, which is an equivalent form of Theorem 2.5. In addition, we added an appendix, which provided an alternative proof of Theorem 2.2. The first author would like to express his hearty gratitude to Prof. Minghua Lin and Prof. Xiaohui Fu for detailed comments and constant encouragement

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