Contractivity of the Hilbert--Schmidt Speed and Unitality--Divisibility-Based Witnesses in Finite-Dimensional Quantum Dynamics

We investigate the Hilbert–Schmidt speed as a witness of non-Markovianity in finite-dimensional quantum dynamics. For a differentiable one-parameter family of quantum states, the Hilbert–Schmidt speed is defined, up to a conventional factor, as the Hilbert–Schmidt norm of the corresponding Hermitian traceless tangent operator. We show that, for unital dynamics in arbitrary finite dimension, P-divisibility implies the monotonic decrease of the Hilbert–Schmidt speed. Hence, any increase in this quantity signals the breakdown of P-divisibility, and therefore also excludes CP-divisibility. For qubit systems, we establish a stronger result: every P-divisible evolution decreases the Hilbert–Schmidt speed, independently of unitality. Thus, in dimension two, HSS growth is a valid witness of non-Markovian behaviour for arbitrary positive divisible dynamics. This conclusion is dimension dependent. In dimensions $d\ge 3$, non-unitality can generate HSS growth even under CP-divisible dynamics. We demonstrate this by constructing an explicit non-unital CP-divisible qutrit semigroup for which the Hilbert–Schmidt speed strictly increases. Finally, for unital GKSL dynamics, we derive a generator-level dissipation identity explaining the monotonic decay of the Hilbert–Schmidt speed. These results specify the regimes in which HSS growth can be interpreted as evidence of the failure of divisibility and clarify its limitation for higher-dimensional non-unital evolutions.