Induced Metric And Matrix Inequalities On Unitary Matrices

TL;DR

This paper explores metrics on the unitary group derived from eigenvalue arguments, showing how symmetric norms induce such metrics and establishing a generalized eigenvalue inequality for matrix products, with implications for quantum information.

Contribution

It demonstrates that any symmetric norm induces a metric on U(n) and proves a new eigenvalue inequality for products of unitary matrices, extending previous results.

Findings

Symmetric norms induce metrics on U(n).

Established a generalized eigenvalue inequality for unitary matrix products.

Connected metrics on U(n) with quantum information theory.

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].