Algebraic probability-theoretic characterization of quantum correlations

Quantum entanglement and nonlocality are inequivalent notions: There exist entangled states that nevertheless admit local-realistic interpretations. This paper studies a special class of local-hidden-variable theories, in which the linear structure of quantum measurement operators is preserved. It has been proven that a quantum state has such linear hidden-variable representations if, and only if, it is not entangled. Separable states are known to admit nonclassical correlations as well, which are captured by quantum discord and related measures. In the unified framework presented in this paper, zero-discordant states are characterized as the only states that admit fully consistent classical probability representations. Possible generalization of this framework to the quasiprobability representation of multipartite quantum states is also discussed.