Induced Metric And Matrix Inequalities On Unitary Matrices

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].

6 pages, extensively rewritten with an earlier error fixed. It generalizes and simplifies the mathematical results concerning certain matrix inequalities originally reported in arXiv:1006.3614v1. To appear in J.Phys.A

References in corpus (4)

• Ultimate physical limits to computation
• Unambiguous discrimination among oracle operators
• Tight Upper Bound Of The Maximum Speed Of Evolution Of A Quantum State
• Metrics On Unitary Matrices And Their Application To Quantifying The Degree Of Non-Commutativity Between Unitary Matrices

Cited by in corpus (1)

• Relation Between Quantum Speed Limits And Metrics On U(n)