Contractivity of the Hilbert--Schmidt Speed in Unital Quantum Channels: Foundation for Witnessing Non-Markovianity and Discriminating Unital from Non-Unital Markovian Dynamics

Abstract

We investigate the Hilbert--Schmidt speed (HSS), a geometric indicator defined through the Hilbert--Schmidt norm of the tangent vector to a parametrized family of quantum states, under general open-system dynamics. Working in the framework of finite-dimensional, parameter-independent completely positive trace-preserving (CPTP) evolution where the parameter is encoded solely in the initial state, we prove that the HSS is contractive under every unital CPTP map. Consequently, for any CP-divisible evolution whose intermediate propagators are unital, the HSS is monotonically non-increasing in time. We then establish the generator-level counterpart for Markovian dynamics governed by a Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) master equation with Hermitian Lindblad operators, deriving an explicit non-positive expression for the time derivative of the squared HSS. These results provide a rigorous foundation for using HSS backflow as a sufficient witness of non-Markovianity in physical settings where the relevant CP-divisible Markovian dynamics is known \emph{a priori} to be unital. Conversely, we show by an explicit qutrit counterexample that HSS can increase even in perfectly Markovian but non-unital dynamics, demonstrating that HSS non-monotonicity is not, in general, a faithful indicator of memory effects unless unitality is guaranteed. Our findings clarify the exact scope of HSS-based diagnostics and identify unitality as the crucial structural ingredient underlying their validity.