A new matrix inequality involving partial traces

Let $A$ be an $mtimes m$ 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 note, we prove the following inequality [ (mathrm{tr} A)I_{mn} - (mathrm{tr}_2 A) otimes I_n ge pm bigl( I_motimes (mathrm{tr}_1 A) -Abigr). ] This inequality is not only a generalization of Andos result [1], but it also could be regarded as a complement of a recent result of Choi [8]. Additionally, some new partial traces inequalities for positive semidefinite block matrices are also included.