Contractivity of the Hilbert-Schmidt-Speed in Unital Channels: Non-Markovianity and Non-Unital Markovian Dynamics

Abstract

We investigate the Hilbert--Schmidt speed (HSS), a measure of parameter sensitivity based on the tangent vector of a parametrized family of quantum states, for finite-dimensional open quantum systems. The parameter is assumed to be encoded only in the initial state, while the dynamics and the intermediate propagators are taken to be independent of the parameter. We prove that a positive trace-preserving linear map, acting on a fixed matrix algebra, contracts the Hilbert--Schmidt norm of every Hermitian operator if and only if it is unital. Therefore, the HSS is non-increasing for any evolution whose parameter-independent intermediate propagators are unital, positive, and trace preserving, namely for unital P-divisible dynamics. This class also includes unital CP-divisible dynamics. For time-local GKSL equations with nonnegative rates, we show that, when the generator is unital, the time derivative of the squared HSS is given by a nonpositive weighted sum of squared commutator norms. We also clarify that HSS growth, by itself, rules out unital P-divisibility, but not CP-divisibility in general; the latter conclusion follows only when the relevant intermediate maps are known to be unital. For qubit systems, we prove that every positive trace-preserving map contracts all physical HSS tangents. Hence, $\HSS_\varphi(t)\le \HSS_\varphi(0)$, and every P-divisible qubit evolution is HSS-monotone. In contrast, we provide an explicit non-unital CP-divisible qutrit semigroup for which the HSS strictly increases with time. This shows that non-unital CP-divisible dynamics can lead to HSS growth in all dimensions $d\ge3$. Our results identify unitality as a simple sufficient condition for HSS monotonicity and determine the precise range of validity of HSS-based witnesses of quantum non-Markovianity.