  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.

Functional Analysis