Two-qubit correlations revisited: average mutual information, relevant (and useful) observables and an application to remote state preparation

Understanding how correlations can be used for quantum communication protocols is a central goal of quantum information science. While many authors have linked global measures of correlations such as entanglement or discord to the performance of specific protocols, in general the protocols may require only on correlations between specific observables. Here, we introduce a pairwise measure of correlations between local observables, based on the classical mutual information, and show how it can be used to classify the quantum states and assess the role of correlations in specific protocols in a standard way. We exemplify our approach in a two-qubit scenario. We first focus on states with maximally mixed marginals (MMMS) and show that pairwise correlations between local observables are complementary to the coherence of the product bases they define, and that the purity and the symmetry of the states determines the average correlations. Next, we address the role of correlations in the remote state preparation protocol (RSP). One can introduce a proper figure-of-merit for RSP that depends only on a subset of relevant observables. While for MMMS the corresponding relevant correlations are always useful, i.e., they can be fuitfully used for RSP, for non-MMMS only a subset of the relevant correlations has this feature. In both cases, one can define a gain function that quantifies the benefit of using the useful correlations, which turns out to coincide with the average correlations between the useful observables. The gain provides a consistent measure of correlations tied to the specific protocol and one can show that again purity and symmetry are the resources that determine the overall performance. It turns out that optimizing the protocol means maximizing the useful correlations and thus minimizing the corresponding coherence. The scheme we propose can be straightforwardly extended to other communication protocols.