  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.
