Characterizing bipartite states with vanishing basis-dependent correlations

Since both coherence and quantum correlations arise from the superposition principle and can be regarded as resources in quantum information tasks, it is of significance to investigate the interplay between them from different perspectives. In this work we focus on the basis-dependent correlations in a bipartite state defined by the coherence difference between global state and local state relative to a local basis and characterize bipartite states with vanishing basis-dependent correlations. Using the relative entropy of coherence, the structure of such states has been determined by Yadin et al. [Phys. Rev. X 6, 041028 (2016)], which we call block-diagonal product states here. The first result of this work is to demonstrate that the set of block-diagonal product states can also be characterized by the property of possessing vanishing basis-dependent correlations using the coherence measure based on skew information. As a by-product of this result, we describe the structure of quantum ensembles saturating the convexity inequality in the resource theory of coherence using the coherence measure based on skew information, which may be of independent interest. Next, we characterize the set of bipartite states with vanishing basis-dependent correlations using the ${l}_{1}$ norm of coherence, and show that the set of block-diagonal product states is a subset of it. Furthermore, we provide an operational interpretation of block-diagonal product states in an interference model. Finally, we compare the amount of basis-dependent correlations using the three mentioned coherence measures through several examples such as Werner states, isotropic states, Bell-diagonal states, and a family of classical-quantum states.