Partial correlations in multipartite quantum systems

Quantum correlations are more general than entanglement. A problem on how to characterize and quantify quantum correlations has attracted substantial attention. Quantum correlations in bipartite systems have been researched extensively. In this paper, partial correlations in a multipartite quantum system a(1) a(2)...a(n) are discussed in detail. In order to reveal correlations of a given state with respect to some subsystems, for a nonempty subset Delta of the index set {1, 2,..., n}, Delta-classical correlations (Delta-CC), Delta-quantum correlations (Delta-QC), partially classical correlations (PCC) and genuinely quantum correlations (GQC) are introduced in a multipartite system. Subsequently, characterizations of Delta-CC are obtained. Secondly, a measure function G(Delta)(rho) of partial correlations of a state rho is defined and called the Delta-quantum discord. It is proved that G(Delta) (rho) is nonnegative, independent of the choice of basis and invariant under local unitary transformations. It is also proved that a state rho is Delta-CC if and only if G(Delta)(rho) = 0, and then it is Delta-QC if and only if G(Delta)(rho) > 0. Moreover, some relationships among partial correlations with respect to different Delta are discussed in light of the measure function. Finally, an illustrative example with 6-qubit GHZ state is given. (C) 2014 Elsevier Inc. All rights reserved.