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arXiv · 1107.1047

Induced Metric And Matrix Inequalities On Unitary Matrices

H. F. Chau·C. -K. Li·Y. -T. Poon·N. -S. Sze
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Abstract

Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all $n\times n$ unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on ${\mathbb R}^n$ induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau [arXiv:1006.3614v1].

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H. F. Chau, C. -K. Li, Y. -T. Poon, N. -S. Sze. 2011-07-06. Induced Metric And Matrix Inequalities On Unitary Matrices. https://doi.org/10.1088/1751-8113/45/9/095201

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