A treatment of a determinant inequality of Fiedler and Markham

Minghua Lin

Czechoslovak Mathematical Journal (2016)

• Volume: 66, Issue: 3, page 737-742
• ISSN: 0011-4642

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Abstract

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

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Lin, Minghua. "A treatment of a determinant inequality of Fiedler and Markham." Czechoslovak Mathematical Journal 66.3 (2016): 737-742. <http://eudml.org/doc/286831>.

@article{Lin2016,
abstract = {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 \[ \mathop \{\rm det \}(I\_n+\widehat\{H\}) \ge \mathop \{\rm det \}(I\_\{nk\}+kH)^\{\{1\}/\{k\}\}.\]},
author = {Lin, Minghua},
journal = {Czechoslovak Mathematical Journal},
keywords = {determinant inequality; partial trace},
language = {eng},
number = {3},
pages = {737-742},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A treatment of a determinant inequality of Fiedler and Markham},
url = {http://eudml.org/doc/286831},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Lin, Minghua
TI - A treatment of a determinant inequality of Fiedler and Markham
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 737
EP - 742
AB - 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 \[ \mathop {\rm det }(I_n+\widehat{H}) \ge \mathop {\rm det }(I_{nk}+kH)^{{1}/{k}}.\]
LA - eng
KW - determinant inequality; partial trace
UR - http://eudml.org/doc/286831
ER -

References

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9. Lin, M., Some applications of a majorization inequality due to Bapat and Sunder, Linear Algebra Appl. 469 (2015), 510-517. (2015) Zbl1310.15033MR3299075
10. Petz, D., Quantum Information Theory and Quantum Statistics, Theoretical and Mathematical Physics Springer, Berlin (2008). (2008) Zbl1145.81002MR2363070
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