Multipartite entanglement

In the case of systems composed of ${\displaystyle m>2}$ subsystems the classification of 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 ${\displaystyle \;m}$-partite separability (${\displaystyle \;m}$-separability) of ${\displaystyle \;m}$ systems]: The state ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ of ${\displaystyle \;m}$ subsystems ${\displaystyle \;A_{1},\ldots ,A_{m}}$ with Hilbert space ${\displaystyle \;{\mathcal {H}}_{A_{1}\ldots A_{m}}={\mathcal {H}}_{A_{1}}\otimes \ldots \otimes {\mathcal {H}}_{A_{m}}}$ is fully separable if and only if it can be written in the form

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