Probability-Phase Mutual Information

Quantum coherence is usually quantified through density-matrix measures, which cannot distinguish different ensembles that yield the same mixed state.  Using the probability–phase coordinates of the complex projective state space, we introduce the probability–phase mutual information I(P;Φ), quantifying statistical correlations between measurement-accessible probabilities and inaccessible phases across an ensemble. We show that I(P;Φ) satisfies all coherence monotone axioms, establishing a coherence theory native to ensembles rather than density matrices. Comparing I(P;Φ) with the relative entropy of coherence C(ρ), we derive a non-negative coherence surplus δ_C ≥ 0, bounding C(ρ) and revealing structure lost under statistical averaging.  The coherence surplus establishes a direct connection between the ensemble-level and density-matrix descriptions of quantum systems, providing a new understanding of how coherence emerges from the underlying ensembles which generate a given mixed state.