Improvements on some partial trace inequalities for positive semidefinite block matrices

Abstract

We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove a celebrated result of Choi [Linear Algebra Appl. 516 (2017)]. The method also allows us to prove a generalization of another result of Choi [Linear Multilinear Algebra 66 (2018)]. Furthermore, we shall give an improvement on a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. In addition, we include with some majorization inequalities involving partial traces for two by two block matrices, and also provide inequalities related to the unitarily invariant norms as well as the singular values, which can be viewed as slight extensions of two results of Lin [Linear Algebra Appl. 459 (2014)] and [Electronic J Linear Algebra 31 (2016)].

KEYWORDS:

• Partial transpose
• partial traces
• Cauchy and Khinchin
• positive semidefinite

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A45
• 15A60
• 47B65

Acknowledgments

The paper is dedicated to Prof. Weijun Liu, my teacher on the occasion of his 60th birthday, October 22 of the lunar calendar in 2021. The author would like to thank Prof. Minghua Lin for bringing the topic on partial traces to his attention. Thanks also go to Prof. Fuzhen Zhang and Prof. Yuejian Peng for reading an earlier draft of the paper, Prof. Xiaohui Fu for the inspiring discussions over the years, and the anonymous referee for helpful suggestions on improving the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by NSFC [grant no. 11931002].