(A)Symmetric complexity and the Quantum Mpemba Effect

A bstract The Quantum Mpemba Effect (QME) — the counter-intuitive phenomenon where states further from equilibrium can relax faster than those closer to it — challenges standard expectations of quantum thermalization. In this work, we introduce Krylov complexity as a sensitive diagnostic for the QME. We show that Krylov spread complexity encodes the asymmetry essential to the effect, and we define a new class of projective (a)symmetric complexities that sharpen this connection. Strikingly, the structure of these projective complexities at the initial moment (t=0) already carries predictive power for the onset of Mpemba-like inversions, obviating the need for explicit time evolution. Our results suggest that the geometry of states in Krylov space captures deep information about non-monotonic relaxation and provides a powerful framework for diagnosing and anticipating anomalous thermalization phenomena in quantum systems.