Partial correlations in multipartite quantum systems

Abstract

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 a1a2…an are discussed in detail. In order to reveal correlations of a given state with respect to some subsystems, for a nonempty subset Δ of the index set {1,2,…,n}, Δ-classical correlations (Δ-CC), Δ-quantum correlations (Δ-QC), partially classical correlations (PCC) and genuinely quantum correlations (GQC) are introduced in a multipartite system. Subsequently, characterizations of Δ-CC are obtained. Secondly, a measure function GΔ(ρ) of partial correlations of a state ρ is defined and called the Δ-quantum discord. It is proved that GΔ(ρ) is nonnegative, independent of the choice of basis and invariant under local unitary transformations. It is also proved that a state ρ is Δ-CC if and only if GΔ(ρ)=0, and then it is Δ-QC if and only if GΔ(ρ)>0. Moreover, some relationships among partial correlations with respect to different Δ are discussed in light of the measure function. Finally, an illustrative example with 6-qubit GHZ state is given.