Metrics On Unitary Matrices And Their Application To Quantifying The Degree Of Non-Commutativity Between Unitary Matrices

TL;DR

This paper introduces metrics on unitary matrices linked to quantum speed limits, which serve as indicators of their non-commutativity, providing new tools for analyzing quantum transformations.

Contribution

The paper develops a family of metrics on unitary matrices that quantify non-commutativity and relate to quantum speed limits, offering novel analytical tools.

Findings

Metrics can be converted into indicators of non-commutativity

Metrics are related to quantum speed limits

Provides new insights into unitary transformations

Abstract

By studying the minimum resources required to perform a unitary transformation, families of metrics and pseudo-metrics on unitary matrices that are closely related to a recently reported quantum speed limit by the author are found. Interestingly, this family of metrics can be naturally converted into useful indicators of the degree of non-commutativity between two unitary matrices.