The quantum Mpemba effect — a hotter quantum system relaxing to equilibrium faster than a cooler one — depends on which distance metric you use to define “closer.” In trace distance, the effect appears reliably. In quantum relative entropy, it vanishes at low temperatures. Same system, same dynamics, different measurement, different answer.
The resolution is geometric. On the Bloch sphere, initial states related by rotation around a specific axis generically display inverted relaxation ordering. The axis is determined by the system-bath coupling and the bath spectral density, not by temperature. The Mpemba effect is not a thermal anomaly. It is a geometric one: certain directions in state space relax faster than others, and “hotter” states can happen to align with the fast-relaxation direction while “cooler” states align with the slow one.
The metric dependence follows from this geometry. Trace distance and relative entropy weight different directions on the Bloch sphere differently, so they disagree about which state is “closer” to equilibrium when the states lie along different relaxation axes. The effect doesn't appear and disappear — it's always there in the geometric structure. The metrics just project that structure differently.
More surprising: increasing the system-bath coupling strength restores the Mpemba effect even where weak coupling destroys it. Strong coupling rotates the Bloch sphere's relaxation axes, bringing previously hidden geometric relationships into alignment. The bath isn't just a source of dissipation — its coupling strength reshapes the geometry of approach to equilibrium.
Whether a quantum system remembers its initial temperature is determined not by the temperature itself but by where the initial state sits relative to the relaxation geometry. Hotter cooling faster is a statement about direction, not about heat.