Coherence measure of ensembles with nonlocality without entanglement

Irreversibility between preparation and discrimination processes is manifested in the indistinguishability of orthogonal product states via local operations and classical communication (LOCC). Characterizing quantum properties for sets of states according to their local distinguishing property is one of the avenues to explain the surprising results obtained in the LOCC indistinguishability domain. We propose a measure based on the $l_1$ norm of coherence to quantitatively assess the quantumness of ensembles composed of orthogonal product states. Furthermore, to establish a hierarchy among different product ensembles, we establish a relationship between the coherence-based measure of an ensemble and the optimal success probability of distinguishing states within the ensemble using LOCC, constrained by a limited amount of classical communication and projective measurements, in the framework of minimum error state discrimination.