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

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