Multipartite entanglement

In the case of systems composed of m > 2 {displaystyle m>2} subsystems, the classification of quantum-entangled states is richer than in the bipartite case. Indeed, in multipartite entanglement apart from fully separable states and fully entangled states, there also exists the notion of partially separable states. The definitions of fully separable and fully entangled multipartite states naturally generalizes that of separable and entangled states in the bipartite case, as follows. Definition [Full m {displaystyle ;m} -partite separability ( m {displaystyle ;m} -separability) of m {displaystyle ;m} systems]: The state ϱ A 1 … A m {displaystyle ;varrho _{A_{1}ldots A_{m}}} of m {displaystyle ;m} subsystems A 1 , … , A m {displaystyle ;A_{1},ldots ,A_{m}} with Hilbert space H A 1 … A m = H A 1 ⊗ … ⊗ H A m {displaystyle ;{mathcal {H}}_{A_{1}ldots A_{m}}={mathcal {H}}_{A_{1}}otimes ldots otimes {mathcal {H}}_{A_{m}}} is fully separable if and only if it can be written in the form Correspondingly, the state ϱ A 1 … A m {displaystyle ;varrho _{A_{1}ldots A_{m}}} is fully entangled if it cannot be written in the above form. As in the bipartite case, the set of m {displaystyle ;m} -separable states is convex and closed with respect to trace norm, and separability is preserved under m {displaystyle ;m} -separable operations ∑ i Ω i 1 ⊗ … ⊗ Ω i n {displaystyle ;sum _{i}Omega _{i}^{1}otimes ldots otimes Omega _{i}^{n}} which are a straightforward generalization of the bipartite ones: As mentioned above, though, in the multipartite setting we also have different notions of partial separability. Definition : The state ϱ A 1 … A m {displaystyle ;varrho _{A_{1}ldots A_{m}}} of m {displaystyle ;m} subsystems A 1 , … , A m {displaystyle ;A_{1},ldots ,A_{m}} is separable with respect to a given partition { I 1 , … , I k } {displaystyle ;{I_{1},ldots ,I_{k}}} , where I i {displaystyle ;I_{i}} are disjoint subsets of the indices I = { 1 , … , m } , ∪ j = 1 k I j = I {displaystyle ;I={1,ldots ,m},cup _{j=1}^{k}I_{j}=I} , if and only if it can be written