A treatment of a determinant inequality of Fiedler and Markham

summary:Fiedler and Markham (1994) proved $$ \Big (\frac {\mathop {\rm det } \widehat {H}}{k}\Big )^{ k}\ge \mathop {\rm det } H, $$ where $H=(H_{ij})_{i,j=1}^n$ is a positive semidefinite matrix partitioned into $n\times n$ blocks with each block $k\times k$ and $\widehat {H}=(\mathop {\rm tr} H_{ij})_{i,j=1}^n$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $$ \mathop {\rm det }(I_n+\widehat {H}) \ge \mathop {\rm det }(I_{nk}+kH)^{{1}/{k}}.$