Separability criteria

Although there exists a clear definition of what separable and entangled states are, in general it is difficult to determine whether a given state is entangled or separable. Linear maps which are positive but not completely positive (PnCP) are a useful tool to investigate the entanglement of given states via separability criteria.


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PnCP maps and separability criteria

Every linear map \; \Lambda which describes a physical transformation must preserve the positivity of every state \varrho: if this were not true, the transformed system could have negative eigenvalues, which would be in contradiction with the statistical interpretation of the eigenvalues as probabilities. In order to preserve the positivity of every state \varrho,\; \Lambda must be a positive map. But the system \; S_d could be statistically coupled to another system \; S_n, called "ancilla". If we perform a physical transformation, represented by the positive map \; \Lambda, on the system \; S_d statistically coupled to the system \;S_n, we must consider the action of the tensor product of the maps  id_n \otimes \Lambda on the compound system \; S_n \otimes S_d , where \; id_n is the identity on the state space of the system \; S_n. If we want \; \Lambda to be a fully consistent physical transformation it isn't sufficient for \; \Lambda to be positive: the tensor product  id_n \otimes \Lambda must be positive for every n, i.e. the map \; \Lambda must be completely positive. Complete positivity is necessary because of entangled states of the bipartite system \; S_n \otimes S_d . If all the physical states of a bipartite system were separable, then positivity of the map \; \Lambda would be sufficient. Indeed we know that if  \varrho \geq 0 is separable, then \varrho \equiv \varrho_{nd} = \sum_i p_i \varrho_i^n \otimes \varrho_i^d , and therefore:

 (id_n \otimes \Lambda)[\varrho] = \sum_i p_i \Big(id_n[\varrho_i^n] \otimes \Lambda[\varrho_i^d]\Big) = 
\sum_i p_i \Big(\varrho_i^n \otimes \Lambda[\varrho_i^d]\Big) \geq 0 \, .

If instead the state \varrho of the bipartite system is entangled (\varrho \equiv \varrho^{ent}), it cannot be written as a convex combination of product states as above, and therefore, in order to have (id_n \otimes \Lambda)[\varrho^{ent}] \geq 0 , the tensor product id_n \otimes \Lambda must be positive for every n, i.e. the map \; \Lambda must be completely positive.

Therefore positive but not completely positive (PnCP) maps move entangled states out of the space of physical states and thus are a useful tool in the identification of separable or entangled states via separability criteria, such as the following.


Theorem [Separability criterion via PnCP maps]: A state  \varrho \in \mathcal{S}_{d \times d} is separable if and only if (id_d \otimes \Lambda)[\varrho] \geq 0 for all PnCP maps  \Lambda : M_d \rightarrow M_d .


The following theorem provides an operationally useful separability criterion:

A state  \varrho \in \mathcal{S}_{d \times d} is entangled if and only if there exists a PnCP map \;\Lambda such that

 Tr[(id_d \otimes \Lambda)[P_d^+]\varrho] < 0 .

P_d^+ is the projector onto the totally symmetric state |\Psi_d^+\rangle = 1/\sqrt{d}\sum_{i=1}^d |i \rangle \otimes |i\rangle . The operator \;(id_d \otimes \Lambda)[P_d^+] is called entanglement witness and is uniquely associated to the positive map \; \Lambda via the Choi-Jamiolkowski isomorphism.


The most simple example of PnCP map is transposition, from which we get the PPT criterion.

But there are also two other PnCP maps that provide important separability criteria.


Reduction criterion

Since it is based on a decomposable map, this criterion is not very strong; however, it is interesting because it plays an important role in entanglement distillation and it leads to the extended reduction criterion, which we will analyze in the following subsection.

Definition: The linear map \; \Lambda_r: \mathcal{S}_{d_A \times d_B} \to  \mathcal{S}_{d_A \times d_B} such that

\; \Lambda_r[\varrho] = \mathbf{I}(Tr\varrho) - \varrho,

with \; \varrho_{AB} \in \mathcal{S}_{d_A \times d_B} and \; \mathbf{I} the identity operator, is called reduction map.

It can be easily proved that the reduction map is positive but not completely positive (PnCP) and decomposable.

Theorem [Reduction criterion]: If the state \; \varrho_{AB} \in \mathcal{S}_{d_A \times d_B} is separable, then \; (\mathbf{I} \otimes \Lambda_r)[\varrho_{AB} ] \geq 0, i.e. the following two conditions hold:

\; \varrho_A \otimes \mathbf{I}_B - \varrho_{AB} \geq 0 \qquad   
\mathbf{I}_A \otimes \varrho_B - \varrho_{AB} \geq 0 ,

where \; \varrho_A and \; \varrho_B are the reduced density matrices of the subsystems \; S_A and \; S_B respectively.


Extended reduction criterion

This criterion is based on a PnCP non-decomposable map, found independently by Breuer and Hall, which is an extension of the reduction map on even-dimensional Hilbert spaces with \; d=2k. On these subspaces there exist antisymmetric unitary operations \; U^T = -U. The corresponding antiunitary map \; U[\cdot]^T U^\dagger maps any pure state to some state that is orthogonal to it. Therefore we can define the positive map \; \Lambda_{er} as follows.

Definition: The linear map \; \Lambda_{er}: \mathcal{S}_d \to  \mathcal{S}_d such that

\; \Lambda_{er}[\varrho] = \Lambda[\varrho] - U[\varrho]^T U^\dagger

is called extended reduction map.

This map is positive but not completely positive and non-decomposable; moreover, the entanglement witness corresponding to \; \Lambda_{er} can be proved to be optimal.

From \; \Lambda_{er} we get the following separability condition.

Theorem [Extended reduction criterion]: If the state \; \varrho \in \mathcal{S}_{d \times d } is separable, then \; (\mathbf{I}_d \otimes \Lambda_{er})[\varrho] \geq 0.

Notice that, since \; \Lambda_{er} is indecomposable, it can detect the entanglement of PPT entangled states and thus turns out to be useful for the characterization of the entanglement properties of various classes of quantum states.


Other separability criteria

There are also separability criteria which are not based on PnCP maps, such as the range criterion and the matrix realignment criterion.


Range criterion

Let us consider a state \; \varrho_{AB} where the dimensions of the two subsystem are \; d_A respectively \; d_B. If \; d_A \cdot d_B >6 then there exist states which are entangled but nevertheless PPT. Therefore, a separability criterion independent of the PPT criterion is needed in order to detect the entanglement of these states. This can be done with separability criteria based on PnCP maps where the chosen PnCP map is not decomposable. However, in (M., P., R. Horodecki, Phys. Rev. Lett. 78, 574, 1997) another criterion was especially formulated to detect the entanglement of some PPT states: the range criterion.

Range criterion: If the state \; \varrho_{AB} is separable, then there exists a set of product vectors \; \{\psi_A^i \otimes \phi_B^i\} that spans the range of \; \varrho_{AB}, while \; \{\psi_A^i \otimes (\phi_B^i)^*\} spans the range of the partial transpose \; \varrho_{AB}^{T_B}, where the complex conjugation \; (\phi_B^i)^* is taken in the same basis in which the partial transposition operation on \; \varrho_{AB} is performed.


An interesting application of the range criterion in detecting PPT entangled states is the unextendible product basis methods.

Definition: An unextendible product basis is a set \; \mathcal{S}_{UPB} of orthonormal product vectors in \; \mathcal{H}_{AB}= \mathcal{H}_A \otimes \mathcal{H}_B such that there is no product vector that is orthogonal to all of them.

Thus, from the definition it directly follows that any vector belonging to the orthogonal subspace \; \mathcal{H}_{UPB}^\bot is entangled and, by the range criterion, any mixed state with support contained in \; \mathcal{H}_{UPB}^\bot is entangled.


Matrix realignment criterion and linear contractions criteria

Another strong class of separability criteria which are independent of the separability criteria based on PnCP maps and, in particular, of the PPT criterion, is those based on linear contractions on product states.

Matrix realignment criterion or computable cross norm (CCN) criterion: If the state \; \varrho_{AB} is separable, then the matrix \; \mathcal{R}(\varrho_{AB}) with elements

\; \langle m|\langle \mu| \mathcal{R}(\varrho_{AB})|n\rangle |\nu\rangle \equiv 
\langle m|\langle n| \varrho_{AB}|\nu \rangle |\mu\rangle

has trace norm not greater than 1.


The above condition can be generalized as follows.

Linear contraction criterion: If the map \; \Lambda satisfies the condition

\; ||\Lambda[ |\phi_A\rangle \langle \phi_A| \otimes |\phi_B\rangle \langle \phi_B|]||_{Tr} \leq 1

for all pure product states \; |\phi_A\rangle \langle \phi_A| \otimes |\phi_B\rangle \langle \phi_B|, then for any separable state \; \varrho_{AB} one has

\; ||\Lambda[\varrho_{AB}]||_{Tr} \leq 1.


The matrix realignment criterion is just a particular case of the above criterion where the matrix realignment map \; \mathcal{R}, which permutes matrix elements, satisfies the above contraction condition on product states. Moreover, this criterion has been found to be useful for the detection of some PPT entanglement.



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