Multipartite entanglement
In the case of systems composed of formula_1 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. 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. Full "m"-partite separability ("m"-separability) of "m" systems. The state formula_2 of formula_3 subsystems formula_4 with Hilbert space formula_5 is fully separable if and only if it  can be written in the form Correspondingly, the state formula_2 is fully entangled if it cannot be written in the above form. As in the bipartite case, the set of formula_3-separable states is "convex" and "closed" with respect to trace norm, and separability is preserved under formula_3-separable operations formula_10 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. Separability with respect to partitions. The state formula_2 of formula_3 subsystems formula_4 is separable with respect to a given partition formula_15, where formula_16  are disjoint subsets of the indices formula_17, if and only if it can be written Semiseparability. The state formula_2 is semiseparable if and only if it is separable under all formula_20-formula_21 partitions, formula_22. Section::::"s"-particle entanglement. An formula_3-particle system can have at most formula_24-particle entanglement if it is a mixture of all states such that each of them is separable with respect to some partition formula_25, where all sets of indices formula_26 have cardinality formula_27. Separability characterization and criteria. Pure states. An equivalent definition to Full m-partite separability is given as follows: The pure state formula_28 of  formula_3 subsystems formula_4 is fully formula_3-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. However, this cannot be done so easily in the multipartite case, as only rarely multipartite pure states admit the generalized Schmidt decomposition formula_33. A multipartite state admits generalized Schmidt decomposition if, tracing out any subsystem, the rest is in a fully separable state. Thus, in general the entanglement of a pure state is described by the spectra of the reduced density  matrices of all bipartite partitions: the state is genuinely formula_3-partite entangled if and only if all bipartite partitions produce mixed reduced density matrices. Mixed states. In the multipartite case there is no simple necessary and sufficient condition for separability like the one given by the PPT criterion for the formula_35 and formula_36 cases. However, many separability criteria used in the bipartite setting can be generalized to the multipartite case. Positive but not completely positive (PnCP) maps and entanglement witnesses. The characterization of separability in terms of positive but not completely positive maps can be naturally generalized from the bipartite  case, as follows. Any positive but not completely positive (PnCP) map formula_37 provides a nontrivial necessary separability criterion in the form: where formula_39 is the identity acting on the first subsystem formula_40. The state formula_2 is "separable" if and only if the above condition is satisfied for all PnCP maps formula_37. 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. We therefore get a separability condition from entanglement witnesses for multipartite states: the state formula_2 is separable if it has  non-negative mean value formula_44 for all entanglement witnesses formula_45. Correspondingly, the entanglement of formula_2 is detected by the witness formula_47 if and only if formula_48. The above description provides a full characterization of formula_3-separability of formula_3-partite systems. Range criterion. The "range criterion" can also be immediately generalized from the bipartite to the multipartite case. In the latter case the range of formula_2 must be spanned by the vectors formula_52, while the range of formula_53 partially transposed with respect to the subset formula_54 must be spanned by the products of these vectors where those with indices formula_55 are  complex conjugated. If the state formula_2 is "separable", then all such partial transposes must lead to matrices with non-negative spectrum, i.e. all the matrices formula_53 should be states themselves. Realignment criteria. The "realignment criteria" from the bipartite case are generalized to permutational criteria in the multipartite setting: if the state formula_58 is separable, then the matrix formula_59, obtained from the original state via permutation formula_60 of matrix indices in product basis, satisfies formula_61. Contraction criterion. Finally, the contraction criterion generalizes immediately from the bipartite to the multipartite case. Multipartite entanglement measures. Many of the axiomatic entanglement measures for bipartite  states, such as relative entropy of entanglement, robustness of entanglement and squashed entanglement can be generalized to the multipartite setting. 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. One can take the set of fully separable states, even though with this choice the measure will not distinguish between truly multipartite entanglement and several instances of bipartite entanglement, such as formula_62. In order to analyze truly multipartite entanglement one has to consider the set of states containing no more than formula_63 -particle entanglement. 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. formula_64. 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. Multipartite entanglement measures for pure states. In the multipartite setting there are entanglement measures that simply are functions of sums of bipartite entanglement measures, as, for instance, the global entanglement, which is given by the sum of  concurrences between one qubit and all others. For these multipartite entanglement measures the 'monotonicity under LOCC is simply inherited from the bipartite measures. But there are also entanglement measures that were constructed specifically for multipartite states, as the following: Tangle. The first multipartite entanglement measure that is neither a direct generalization nor an easy combination of bipartite measures was introduced by Coffman "et al." and called tangle. Definition: where the formula_66-tangles on the right-hand-side are the squares of "concurrence". 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. Schmidt measure. This was one of the first entanglement measures constructed specifically for multipartite states. Definition: The minimum of formula_67, where formula_68 is the number of terms in an expansion of the state in product basis. This measure is zero if and  only if the state is fully product; therefore, it cannot distinguish between truly multipartite entanglement and bipartite entanglement, but it may nevertheless be useful in many contexts. Measures based on normal forms. 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. Measures based on hyperdeterminant. It was proved by Miyake  that hyperdeterminants are entanglement monotones and they describe truly multipartite entanglement in the sense that states such as products of formula_69's have zero entanglement. In particular concurrence and tangle are special cases of hyperdeterminant. 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. Geometric entanglement. The geometric measure of entanglement of formula_70 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. Therefore, a simple modification is necessary. For example, for identical fermions, since the full wave function formula_70 is now completely anti-symmetric, so is required for formula_74. This means, the formula_74 taken to approximate formula_70 should be a Slater determinant wave function. Localisable entanglement. This entanglement measure is a generalization of the entanglement of assistance and was constructed in the context of spin chains. 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).