**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:

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 .

Read more about this topic: Multipartite Entanglement

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