Quantum association entropy for characterizing coherence and entanglement-related quantum correlations

Quantum entropies based on the eigenspectrum characterize spectral mixedness, whereas diagonal entropies describe basis-dependent measurement uncertainty but neglect off-diagonal structure. Here, we introduce quantum association entropy (QAE), which combines the Shannon entropy of the diagonal probability distribution with the quantum Tanimoto coefficient fidelity between a quantum state and its dephased counterpart. QAE reduces to the Shannon, diagonal, and von Neumann entropies for incoherent states, remains nonzero for coherent pure states, and distinguishes states with identical diagonal statistics but different off-diagonal structures. The complementary fidelity deviation defines a QAE-based coherence-related quantity, which equals the QAE difference between the dephased and original states and quantifies the off-diagonal contribution weighted by measurement uncertainty. The same state–reference construction is extended to entanglement-related correlations using a marginal-constrained separable reference state obtained through relative-entropy optimization. For bipartite pure states in the Schmidt product basis, the resulting QAE difference is non-increasing under deterministic LOCC transformations. These results provide a common state–reference entropy-difference framework for characterizing coherence-related structure and entanglement-related correlations.