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

TL;DR

This paper generalizes the connection between metrics on unitary operators and quantum speed limits by extending from weighted -norms to weighted p-norms, exploring their quantum information-theoretic significance.

Contribution

It introduces a broader class of metrics on unitary operators derived from weighted p-norms and examines their relation to quantum speed limits.

Findings

Weighted p-norms induce metrics with quantum speed limit interpretations

The generalization from -norms to p-norms broadens the framework of quantum metrics

Exploration of the limits of the metric-speed limit correspondence

Abstract

Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] found a family of metrics and pseudo-metrics on $n$-dimensional unitary operators that can be interpreted as the minimum resources (given by certain tight quantum speed limit bounds) needed to transform one unitary operator to another. This result is closely related to the weighted $\ell^1$-norm on ${\mathbb R}^n$. Here we generalize this finding by showing that every weighted $\ell^p$-norm on ${\mathbb R}^n$ with $1\le p \le \limitingp$ induces a metric and a pseudo-metric on $n$-dimensional unitary operators with quantum information-theoretic meanings related to certain tight quantum speed limit bounds. Besides, we investigate how far the correspondence between the existence of metrics and pseudo-metrics of this type and the quantum speed limits can go.