Multipartite entanglement

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

${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}=\sum _{i=1}^{k}p_{i}\varrho _{A_{1}}^{i}\otimes \ldots \otimes \varrho _{A_{m}}^{i}.}$

Correspondingly, the state ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ is fully entangled if it cannot be written in the above form.

As in the bipartite case, the set of ${\displaystyle \;m}$-separable states is convex and closed with respect to trace norm, and separability is preserved under ${\displaystyle \;m}$-separable operations ${\displaystyle \;\sum _{i}\Omega _{i}^{1}\otimes \ldots \otimes \Omega _{i}^{n}}$ which are a straightforward generalization of the bipartite ones:

${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}\to {\frac {\sum _{i}\Omega _{i}^{1}\otimes \ldots \otimes \Omega _{i}^{n}\varrho _{A_{1}\ldots A_{m}}(\Omega _{i}^{1}\otimes \ldots \otimes \Omega _{i}^{n})^{\dagger }}{\operatorname {Tr} [\sum _{i}\Omega _{i}^{1}\otimes \ldots \otimes \Omega _{i}^{n}\varrho _{A_{1}\ldots A_{m}}(\Omega _{i}^{1}\otimes \ldots \otimes \Omega _{i}^{n})^{\dagger }]}}.}$[1]

As mentioned above, though, in the multipartite setting we also have different notions of partial separability.[1]

The state ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ of ${\displaystyle \;m}$ subsystems ${\displaystyle \;A_{1},\ldots ,A_{m}}$ is separable with respect to a given partition ${\displaystyle \;\{I_{1},\ldots ,I_{k}\}}$, where ${\displaystyle \;I_{i}}$ are disjoint subsets of the indices ${\displaystyle \;I=\{1,\ldots ,m\},\cup _{j=1}^{k}I_{j}=I}$, if and only if it can be written

${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}=\sum _{i=1}^{N}p_{i}\varrho _{1}^{i}\otimes \ldots \otimes \varrho _{k}^{i}.}$[1]

The state ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ is semiseparable if and only if it is separable under all ${\displaystyle \;1}$-${\displaystyle \;(m-1)}$ partitions, ${\displaystyle \;{\big \{}I_{1}=\{k\},I_{2}=\{1,\ldots ,k-1,k+1,\ldots ,m\}{\big \}},1\leq k\leq m}$.[1]

An ${\displaystyle \;m}$-particle system can have at most ${\displaystyle \;s}$-particle entanglement if it is a mixture of all states such that each of them is separable with respect to some partition ${\displaystyle \;\{I_{1},\ldots ,I_{k}\}}$, where all sets of indices ${\displaystyle \;I_{k}}$ have cardinality ${\displaystyle \;N\leq s}$.[1]

An equivalent definition to Full m-partite separability is given as follows: The pure state ${\displaystyle |\Psi _{A_{1}\ldots A_{m}}\rangle }$ of ${\displaystyle \;m}$ subsystems ${\displaystyle \;A_{1},\ldots ,A_{m}}$ is fully ${\displaystyle \;m}$-partite separable if and only if it can be written

${\displaystyle \;|\Psi _{A_{1}\ldots A_{m}}\rangle =|\psi _{A_{1}}\rangle \otimes \ldots \otimes |\psi _{A_{m}}\rangle .}$[1]

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 ${\displaystyle \;|\Psi _{A_{1}\ldots A_{m}}\rangle =\sum _{i=1}^{\min\{d_{A_{1}},\ldots ,d_{A_{m}}\}}a_{i}|e_{A_{1}}^{i}\rangle \otimes \ldots \otimes |e_{A_{m}}^{i}\rangle }$. 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 ${\displaystyle \;m}$-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 ${\displaystyle 2\otimes 2}$ and ${\displaystyle 2\otimes 3}$ cases. However, many separability criteria used in the bipartite setting can be generalized to the multipartite case.[1]

The characterization of separability in terms of positive but not completely positive maps can be naturally generalized from the bipartite case, as follows.[1]

Any positive but not completely positive (PnCP) map ${\displaystyle \;\Lambda _{A_{2}\ldots A_{m}}:{\mathcal {B}}({\mathcal {H}}_{A_{2}\ldots A_{m}})\to {\mathcal {B}}({\mathcal {H}}_{A_{1}})}$ provides a nontrivial necessary separability criterion in the form:

${\displaystyle \;(I_{A_{1}}\otimes \Lambda _{A_{2}\ldots A_{m}})[\varrho _{A_{1}\ldots A_{m}}]\geq 0,}$

where ${\displaystyle \;I_{A_{1}}}$ is the identity acting on the first subsystem ${\displaystyle \;{\mathcal {H}}_{A_{1}}}$. The state ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ is separable if and only if the above condition is satisfied for all PnCP maps ${\displaystyle \;\Lambda _{A_{2}\ldots A_{m}}:{\mathcal {B}}({\mathcal {H}}_{A_{2}\ldots A_{m}})\to {\mathcal {B}}({\mathcal {H}}_{A_{1}})}$.[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. We therefore get a separability condition from entanglement witnesses for multipartite states: the state ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ is separable if it has non-negative mean value ${\displaystyle \;\operatorname {Tr} (W\varrho _{A_{1}\ldots A_{m}})\geq 0}$ for all entanglement witnesses ${\displaystyle W}$. Correspondingly, the entanglement of ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ is detected by the witness ${\displaystyle \;W}$ if and only if ${\displaystyle \;\operatorname {Tr} (W\varrho _{A_{1}\ldots A_{m}})<0}$.[1]

The above description provides a full characterization of ${\displaystyle \;m}$-separability of ${\displaystyle \;m}$-partite systems.[1]

The "range criterion" can also be immediately generalized from the bipartite to the multipartite case. In the latter case the range of ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ must be spanned by the vectors ${\displaystyle \;\{|\phi _{A_{1}}\rangle ,\ldots ,|\phi _{A_{m}}\rangle \}}$, while the range of ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}^{T_{A_{k_{1}}\ldots A_{k_{l}}}}}$ partially transposed with respect to the subset ${\displaystyle \;\{A_{k_{1}}\ldots A_{k_{l}}\}\subset \{A_{1}\ldots A_{m}\}}$ must be spanned by the products of these vectors where those with indices ${\displaystyle \;k_{1},\ldots ,k_{l}}$ are complex conjugated. If the state ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}}$ is separable, then all such partial transposes must lead to matrices with non-negative spectrum, i.e. all the matrices ${\displaystyle \;\varrho _{A_{1}\ldots A_{m}}^{T_{A_{k_{1}}\ldots A_{k_{l}}}}}$ should be states themselves.[1]

The "realignment criteria" from the bipartite case are generalized to permutational criteria in the multipartite setting: if the state ${\displaystyle \varrho _{A_{1}\ldots A_{m}}}$ is separable, then the matrix ${\displaystyle \;[R_{\pi }(\varrho _{A_{1}\ldots A_{m}})]_{i_{1}j_{1},i_{2}j_{2},\ldots ,i_{n}j_{n}}\equiv \varrho _{\pi (i_{1}j_{1},i_{2}j_{2},\ldots ,i_{n}j_{n})}}$, obtained from the original state via permutation ${\displaystyle \;\pi }$ of matrix indices in product basis, satisfies ${\displaystyle \;||R_{\pi }(\varrho _{A_{1}\ldots A_{m}})]||_{\operatorname {Tr} }\leq 1}$.[1]

Finally, the contraction criterion generalizes immediately from the bipartite to the multipartite case.[1]

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:[1]

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.[1]

Definition:

${\displaystyle \;\tau (A:B:C)=\tau (A:BC)-\tau (AB)-\tau (AC),}$

where the ${\displaystyle \;2}$-tangles on the right-hand-side are the squares of concurrence.[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.[1]

Definition:

The minimum of ${\displaystyle \;\log r}$, where ${\displaystyle \;r}$ is the number of terms in an expansion of the state in product basis.[1]

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.[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.[1]

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 ${\displaystyle EPR}$'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.[1]

The geometric measure of entanglement[2] of ${\displaystyle \psi }$ is the minimum of

${\displaystyle \|\psi -\phi \|_{2}}$

with respect to all the separable states

${\displaystyle \phi =\bigotimes _{i=1}^{N}\phi _{i}.}$

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 ${\displaystyle \psi }$ is now completely anti-symmetric, so is required for ${\displaystyle \phi }$. This means, the ${\displaystyle \phi }$ taken to approximate ${\displaystyle \psi }$ should be a Slater determinant wave function.[3]

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).[1]