Probability-Phase Mutual Information

Building on the geometric formulation of quantum mechanics, we develop a coherence theory for ensembles that exploits the probability-phase structure of the quantum state space. Standard coherence measures quantify superposition within density matrices but cannot distinguish ensembles that produce the same mixed state through different distributions of pure states. First, we introduce the probability-phase mutual information $I(P;\Phi)$, which measures statistical correlations between measurement-accessible probabilities and measurement-inaccessible phases across an ensemble. Then, we prove this satisfies the axioms of a coherence monotone, establishing it as a bona-fide measure of ensemble-level coherence. Eventually, through the definition of the \emph{coherence surplus} $\delta_{\mathcal{C}} \geq 0$, we show how ensemble coherence relates to, but exceeds, density-matrix coherence, thus quantifying structure lost in statistical averaging.