subsystems, the classification of quantum-entangled states is richer than in the bipartite case.
[1] The definitions of fully separable and fully entangled multipartite states naturally generalizes that of separable and entangled states in the bipartite case, as follows.
is fully separable if and only if it can be written in the form Correspondingly, the state
-separable states is convex and closed with respect to trace norm, and separability is preserved under
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.
Such experiments are often referred to as detecting the entanglement depth of the quantum state.
[3][4][5] The definition can be extended to mixed states in the usual manner.
One can define further properties based on the partitioning of particles into groups, which have extensively been studied.
[6] An equivalent definition to Full m-partite separability is given as follows: The pure state
-partite separable if and only if it can be written In order to check this, it is enough to compute reduced density matrices of elementary subsystems and see whether they are pure.
-partite entangled if and only if all bipartite partitions produce mixed reduced density matrices.
[1] In the multipartite case there is no simple necessary and sufficient condition for separability like the one given by the PPT criterion for the
However, many separability criteria used in the bipartite setting can be generalized to the multipartite case.
[1] The definition of entanglement witnesses and the Choi–Jamiołkowski isomorphism that links PnCP maps to entanglement witnesses in the bipartite case can also be generalized to the multipartite setting.
[1] The "range criterion" can also be immediately generalized from the bipartite to the multipartite case.
[1] Finally, the contraction criterion generalizes immediately from the bipartite to the multipartite case.
[1] The relative entropy of entanglement, for example, can be generalized to the multipartite case by taking a suitable set in place of the set of bipartite separable states.
In order to analyze truly multipartite entanglement one has to consider the set of states containing no more than
[1] In the case of squashed entanglement, its multipartite version can be obtained by simply replacing the mutual information of the bipartite system with its generalization for multipartite systems, i.e.
[1] However, in the multipartite setting many more parameters are needed to describe the entanglement of the states, and therefore many new entanglement measures have been constructed, especially for pure multipartite states.
[1] The tangle measure is permutationally invariant; it vanishes on all states that are separable under any cut; it is nonzero, for example, on the GHZ-state; it can be thought to be zero for states that are 3-entangled (i.e. that are not product with respect to any cut) as, for instance, the W-state.
Moreover, there might be the possibility to obtain a good generalization of the tangle for multiqubit systems by means of hyperdeterminant.
[1] This was one of the first entanglement measures constructed specifically for multipartite states.
is the number of terms in an expansion of the state in product basis.
[1] This is an interesting class of multipartite entanglement measures obtained in the context of classification of states.
Namely, one considers any homogeneous function of the state: if it is invariant under SLOCC (stochastic LOCC) operations with determinant equal to 1, then it is an entanglement monotone in the strong sense, i.e. it satisfies the condition of strong monotonicity.
Indeed, for two qubits concurrence is simply the modulus of the determinant, which is the hyperdeterminant of first order; whereas the tangle is the hyperdeterminant of second order, i.e. a function of tensors with three indices.
is the minimum of with respect to all the separable states This approach works for distinguishable particles or the spin systems.
For identical or indistinguishable fermions or bosons, the full Hilbert space is not the tensor product of those of each individual particle.
Namely, one chooses two spins and performs LOCC operations that aim at obtaining the largest possible bipartite entanglement between them (measured according to a chosen entanglement measure for two bipartite states).