A treatment of a determinant inequality of Fiedler and Markham

Abstract

Fiedler and Markham (1994) proved

$${\left( {\frac{{\det \hat H}}{k}} \right)^k} \geqslant \det H,$$

where H = (H _ij ) _i,j=1 ⁿ is a positive semidefinite matrix partitioned into n × n blocks with each block k × k and $\hat H = \left( {tr{H_{ij}}} \right)_{i,j = 1}^n$ . We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove

$$\det \left( {{I_n} + \hat H} \right) \geqslant \det {\left( {{I_{nk}} + kH} \right)^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}}.$$