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

TL;DR

This paper investigates the behavior of the Hilbert-Schmidt distance in open quantum systems, revealing it is generally non-monotonic under Lindblad dynamics, with specific conditions for contractivity and implications for quantum state distinguishability.

Contribution

It provides sufficient conditions for the contractivity of the Hilbert-Schmidt norm and demonstrates that non-contractivity is typical in higher-dimensional systems.

Findings

Hilbert-Schmidt distance is not generally monotonic in open quantum systems.

Contractivity conditions depend on dissipation generators.

Non-contractivity is prevalent in systems with dimension greater than two.

Abstract

We show that the Hilbert-Schmidt distance, unlike the trace distance, between quantum states is generally not monotonic for open quantum systems subject to Lindblad semigroup dynamics. Sufficient conditions for contractivity of the Hilbert-Schmidt norm in terms of the dissipation generators are given. Although these conditions are not necessary, simulations suggest that non-contractivity is the typical case, i.e., that systems for which the Hilbert-Schmidt distance between quantum states is monotonically decreasing form only a small set of all possible dissipative systems for N>2, in contrast to the case N=2 where the Hilbert-Schmidt distance is always monotonically decreasing.