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

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 ${\mathrm{\ell }}^1$-norm on ${\mathbb{R}}^n$. Here we generalize this finding by showing that every weighted ${\mathrm{\ell }}^p$-norm on ${\mathbb{R}}^n$ with $1\le p\le \text{limitingp}$ 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.