## A treatment of a determinant inequality of Fiedler and Markham.(English)Zbl 1413.15040

Summary: M. Fiedler and T. L. Markham [Math. Slovaca 44, No. 4, 441–444 (1994; Zbl 0828.15023)] proved $\Big(\frac{\det\widehat{H}}{k}\Big)^{k}\geq\det H,$ 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 $$\widehat{H}=(\operatorname{tr}H_{ij})_{i,j=1}^n$$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $\det(I_n+\widehat{H})\geq\det(I_{nk}+kH)^{1/k}.$

### MSC:

 15A45 Miscellaneous inequalities involving matrices 15A15 Determinants, permanents, traces, other special matrix functions

### Keywords:

determinant inequality; partial trace

Zbl 0828.15023
Full Text:

### References:

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