Contractivity of the Hilbert-Schmidt distance under open-system dynamics

It is shown that the Hilbert-Schmidt (HS) norm and distance, unlike the trace norm and distance, are generally not contractive for open quantum systems under Lindblad dynamics. Necessary and sufficient conditions for contractivity of the HS norm and distance are given, and explicit criteria in terms of the Lindblad operators are derived. It is also shown that the requirements for contractivity of the HS distance are strictly weaker than those for the HS norm, although simulations suggest that noncontractivity is a typical case, i.e., that systems for which the HS distance between quantum states is monotonically decreasing are exceptional for $N>2$, in contrast to the $N=2$ case where it is always monotonically decreasing.