Probability-Phase Mutual Information

Quantum coherence is an exquisitely quantum phenomenon that depends on both probability amplitudes and relative phases. Standard coherence measures quantify superposition within density matrices but cannot distinguish ensembles that produce the same mixed state through different distributions of pure states. Building on the geometric formulation of quantum mechanics, we introduce the probability-phase mutual information $I(P;\Phi)$. We show that it characterizes quantum coherence at the ensemble level and that ensemble coherence systematically exceeds density-matrix coherence, thus quantifying the structure lost when averaging over pure states. Eventually, its relevance for quantum thermodynamics, quantum information theory, and deep thermalization is highlighted by explicit examples: canonical ensembles reveal temperature-dependent probability-phase correlations absent from thermal density matrices; we show that the probability of converting an ensemble into another one is bound by the ratio of their $I(P;\Phi)$; and, that a non-vanishing $I(P;\Phi)$ signals the breakdown of deep thermalization.