# Multipartite Entanglement - Full and Partial Separability

Full and Partial Separability

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 : The state of subsystems with Hilbert space is fully separable if and only if it can be written in the form

Correspondingly, the state is fully entangled if it cannot be written in the above form.

As in the bipartite case, the set of -separable states is convex and closed with respect to trace norm, and separability is preserved under -separable operations which are a straightforward generalization of the bipartite ones:

$; varrho_{A_1ldots A_m}to frac{sum_iOmega_i^1otimesldotsotimesOmega_i^nvarrho_{A_1ldots A_m} (Omega_i^1otimesldotsotimesOmega_i^n)^dagger}{Tr[sum_i Omega_i^1otimesldotsotimesOmega_i^nvarrho_{A_1ldots A_m} (Omega_i^1otimesldotsotimesOmega_i^n)^dagger]} .$

As mentioned above, though, in the multipartite setting we also have different notions of partial separability.

Definition : The state of subsystems is separable with respect to a given partition, where are disjoint subsets of the indices, if and only if it can be written

Definition : The state is semiseparable if and only if it is separable under all - partitions, .

Definition : An -particle system can have at most -particle entanglement if it is a mixture of all states such that each of them is separable with respect to some partition, where all sets of indices have cardinality .