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.
PnCP maps and separability criteria
Every linear map which describes a physical transformation must preserve the positivity of every state : 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 must be a positive map. But the system could be statistically coupled to another system , called "ancilla". If we perform a physical transformation, represented by the positive map , on the system statistically coupled to the system , we must consider the action of the tensor product of the maps on the compound system , where is the identity on the state space of the system . If we want to be a fully consistent physical transformation it isn't sufficient for to be positive: the tensor product must be positive for every n, i.e. the map must be completely positive. Complete positivity is necessary because of entangled states of the bipartite system If all the physical states of a bipartite system were separable, then positivity of the map would be sufficient. Indeed we know that if is separable, then , and therefore:
If instead the state of the bipartite system is entangled (), it cannot be written as a convex combination of product states as above, and therefore, in order to have , the tensor product must be positive for every n, i.e. the map 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 is separable if and only if for all PnCP maps .
The following theorem provides an operationally useful separability criterion:
A state is entangled if and only if there exists a PnCP map such that
is the projector onto the totally symmetric state . The operator is called entanglement witness and is uniquely associated to the positive map 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.
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 such that
with and 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 is separable, then , i.e. the following two conditions hold:
where and are the reduced density matrices of the subsystems and 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 . On these subspaces there exist antisymmetric unitary operations . The corresponding antiunitary map maps any pure state to some state that is orthogonal to it. Therefore we can define the positive map as follows.
Definition: The linear map such that
is called extended reduction map.
This map is positive but not completely positive and non-decomposable; moreover, the entanglement witness corresponding to can be proved to be optimal.
From we get the following separability condition.
Theorem [Extended reduction criterion]: If the state is separable, then .
Notice that, since 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.
Let us consider a state where the dimensions of the two subsystem are respectively . If 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 is separable, then there exists a set of product vectors that spans the range of , while spans the range of the partial transpose , where the complex conjugation is taken in the same basis in which the partial transposition operation on 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 of orthonormal product vectors in 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 is entangled and, by the range criterion, any mixed state with support contained in 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 is separable, then the matrix with elements
has trace norm not greater than 1.
The above condition can be generalized as follows.
Linear contraction criterion: If the map satisfies the condition
for all pure product states , then for any separable state one has
The matrix realignment criterion is just a particular case of the above criterion where the matrix realignment map , 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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