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 $\backslash ;\; \backslash Lambda$ which describes a physical transformation must preserve the positivity of every state $\backslash varrho$: 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 $\backslash varrho,\backslash ;\; \backslash Lambda$ must be a positive map. But the system $\backslash ;\; S\_d$ could be statistically coupled to another system $\backslash ;\; S\_n$, called "ancilla". If we perform a physical transformation, represented by the positive map $\backslash ;\; \backslash Lambda$,
on the system $\backslash ;\; S\_d$ statistically coupled to the system $\backslash ;S\_n$, we must consider the action of the tensor product of the maps
$id\_n\; \backslash otimes\; \backslash Lambda$ on the compound system $\backslash ;\; S\_n\; \backslash otimes\; S\_d$, where $\backslash ;\; id\_n$ is the identity on the state space of the system $\backslash ;\; S\_n$. If we want $\backslash ;\; \backslash Lambda$ to be a fully consistent physical transformation it isn't sufficient for $\backslash ;\; \backslash Lambda$ to be positive: the tensor product $id\_n\; \backslash otimes\; \backslash Lambda$ must be positive
for every $n$, i.e. the map $\backslash ;\; \backslash Lambda$ must be completely positive. Complete positivity is necessary because of entangled states of the bipartite system $\backslash ;\; S\_n\; \backslash otimes\; S\_d\; .$
If all the physical states of a bipartite system were separable, then positivity of the
map $\backslash ;\; \backslash Lambda$ would be sufficient. Indeed we know that if $\backslash varrho\; \backslash geq\; 0$ is separable, then
$\backslash varrho\; \backslash equiv\; \backslash varrho\_\{nd\}\; =\; \backslash sum\_i\; p\_i\; \backslash varrho\_i^n\; \backslash otimes\; \backslash varrho\_i^d$, and therefore:
:$(id\_n\; \backslash otimes\; \backslash Lambda)[\backslash varrho]\; =\; \backslash sum\_i\; p\_i\; \backslash Big(id\_n[\backslash varrho\_i^n]\; \backslash otimes\; \backslash Lambda[\backslash varrho\_i^d]\backslash Big)\; =\; \backslash sum\_i\; p\_i\; \backslash Big(\backslash varrho\_i^n\; \backslash otimes\; \backslash Lambda[\backslash varrho\_i^d]\backslash Big)\; \backslash geq\; 0\; \backslash ,\; .$
If instead the state $\backslash varrho$ of the bipartite system is entangled ($\backslash varrho\; \backslash equiv\; \backslash varrho^\{ent\}$), it cannot be written as a convex combination of product states as above, and therefore, in order to have $(id\_n\; \backslash otimes\; \backslash Lambda)[\backslash varrho^\{ent\}]\; \backslash geq\; 0$, the tensor product $id\_n\; \backslash otimes\; \backslash Lambda$ must be positive for every $n$,
i.e. the map $\backslash ;\; \backslash Lambda$ 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 $\backslash varrho\; \backslash in\; \backslash mathcal\{S\}\_\{d\; \backslash times\; d\}$ is separable if and only if $(id\_d\; \backslash otimes\; \backslash Lambda)[\backslash varrho]\; \backslash geq\; 0$ for all PnCP maps
$\backslash Lambda\; :\; M\_d\; \backslash rightarrow\; M\_d$.'''
The following theorem provides an operationally useful separability criterion:
'''A state $\backslash varrho\; \backslash in\; \backslash mathcal\{S\}\_\{d\; \backslash times\; d\}$ is ''entangled'' if and only if there exists a PnCP map $\backslash ;\backslash Lambda$ such that
:$Tr[(id\_d\; \backslash otimes\; \backslash Lambda)[P\_d^+]\backslash varrho]\; <\; 0\; .$ '''
$P\_d^+$ is the projector onto the totally symmetric state $|\backslash Psi\_d^+\backslash rangle\; =\; 1/\backslash sqrt\{d\}\backslash sum\_\{i=1\}^d\; |i\; \backslash rangle\; \backslash otimes\; |i\backslash rangle$. The operator $\backslash ;(id\_d\; \backslash otimes\; \backslash Lambda)[P\_d^+]$ is called ''entanglement witness'' and is uniquely associated to the positive map $\backslash ;\; \backslash Lambda$ 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 $\backslash ;\; \backslash Lambda\_r:\; \backslash mathcal\{S\}\_\{d\_A\; \backslash times\; d\_B\}\; \backslash to\; \backslash mathcal\{S\}\_\{d\_A\; \backslash times\; d\_B\}$ such that
:$\backslash ;\; \backslash Lambda\_r[\backslash varrho]\; =\; \backslash mathbf\{I\}(Tr\backslash varrho)\; -\; \backslash varrho,$
with $\backslash ;\; \backslash varrho\_\{AB\}\; \backslash in\; \backslash mathcal\{S\}\_\{d\_A\; \backslash times\; d\_B\}$ and $\backslash ;\; \backslash mathbf\{I\}$ 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 $\backslash ;\; \backslash varrho\_\{AB\}\; \backslash in\; \backslash mathcal\{S\}\_\{d\_A\; \backslash times\; d\_B\}$ is ''separable'', then $\backslash ;\; (\backslash mathbf\{I\}\; \backslash otimes\; \backslash Lambda\_r)[\backslash varrho\_\{AB\}\; ]\; \backslash geq\; 0$, i.e. the following two conditions hold:
:$\backslash ;\; \backslash varrho\_A\; \backslash otimes\; \backslash mathbf\{I\}\_B\; -\; \backslash varrho\_\{AB\}\; \backslash geq\; 0\; \backslash qquad\; \backslash mathbf\{I\}\_A\; \backslash otimes\; \backslash varrho\_B\; -\; \backslash varrho\_\{AB\}\; \backslash geq\; 0\; ,$
where $\backslash ;\; \backslash varrho\_A$ and $\backslash ;\; \backslash varrho\_B$ are the reduced density matrices of the subsystems $\backslash ;\; S\_A$ and $\backslash ;\; S\_B$ 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 $\backslash ;\; d=2k$. On these subspaces there exist antisymmetric unitary operations $\backslash ;\; U^T\; =\; -U$. The corresponding antiunitary map $\backslash ;\; U[\backslash cdot]^T\; U^\backslash dagger$ maps any pure state to some state that is orthogonal to it. Therefore we can define the positive map $\backslash ;\; \backslash Lambda\_\{er\}$ as follows.
'''Definition:''' The linear map $\backslash ;\; \backslash Lambda\_\{er\}:\; \backslash mathcal\{S\}\_d\; \backslash to\; \backslash mathcal\{S\}\_d$ such that
:$\backslash ;\; \backslash Lambda\_\{er\}[\backslash varrho]\; =\; \backslash Lambda[\backslash varrho]\; -\; U[\backslash varrho]^T\; U^\backslash dagger$
is called '''extended reduction map'''.
This map is positive but not completely positive and non-decomposable; moreover, the ''entanglement witness'' corresponding to $\backslash ;\; \backslash Lambda\_\{er\}$ can be proved to be optimal.
From $\backslash ;\; \backslash Lambda\_\{er\}$ we get the following separability condition.
'''Theorem [Extended reduction criterion]:''' If the state $\backslash ;\; \backslash varrho\; \backslash in\; \backslash mathcal\{S\}\_\{d\; \backslash times\; d\; \}$ is ''separable'', then $\backslash ;\; (\backslash mathbf\{I\}\_d\; \backslash otimes\; \backslash Lambda\_\{er\})[\backslash varrho]\; \backslash geq\; 0$.
Notice that, since $\backslash ;\; \backslash Lambda\_\{er\}$ 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 $\backslash ;\; \backslash varrho\_\{AB\}$ where the dimensions of the two subsystem are $\backslash ;\; d\_A$ respectively $\backslash ;\; d\_B$. If $\backslash ;\; d\_A\; \backslash cdot\; d\_B\; >6$ 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 (P. Horodecki, ''Phys. Lett. A'' 232, 1997) another criterion was especially formulated to detect the entanglement of some PPT states: the ''range criterion''.
'''Range criterion:''' If the state $\backslash ;\; \backslash varrho\_\{AB\}$ is separable, then there exists a set of product vectors $\backslash ;\; \backslash \{\backslash psi\_A^i\; \backslash otimes\; \backslash phi\_B^i\backslash \}$ that spans the range of $\backslash ;\; \backslash varrho\_\{AB\}$, while $\backslash ;\; \backslash \{\backslash psi\_A^i\; \backslash otimes\; (\backslash phi\_B^i)^*\backslash \}$ spans the range of the partial transpose $\backslash ;\; \backslash varrho\_\{AB\}^\{T\_B\}$, where the complex conjugation $\backslash ;\; (\backslash phi\_B^i)^*$ is taken in the same basis in which the partial transposition operation on $\backslash ;\; \backslash varrho\_\{AB\}$ 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 $\backslash ;\; \backslash mathcal\{S\}\_\{UPB\}$ of orthonormal product vectors in $\backslash ;\; \backslash mathcal\{H\}\_\{AB\}=\; \backslash mathcal\{H\}\_A\; \backslash otimes\; \backslash mathcal\{H\}\_B$ 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 $\backslash ;\; \backslash mathcal\{H\}\_\{UPB\}^\backslash bot$ is entangled and, by the '''range criterion''', any mixed state with support contained in $\backslash ;\; \backslash mathcal\{H\}\_\{UPB\}^\backslash bot$ 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 $\backslash ;\; \backslash varrho\_\{AB\}$ is separable, then the matrix $\backslash ;\; \backslash mathcal\{R\}(\backslash varrho\_\{AB\})$ with elements
: $\backslash ;\; \backslash langle\; m|\backslash langle\; \backslash mu|\; \backslash mathcal\{R\}(\backslash varrho\_\{AB\})|n\backslash rangle\; |\backslash nu\backslash rangle\; \backslash equiv\; \backslash langle\; m|\backslash langle\; n|\; \backslash varrho\_\{AB\}|\backslash nu\; \backslash rangle\; |\backslash mu\backslash rangle$
has trace norm not greater than 1.
The above condition can be generalized as follows.
'''Linear contraction criterion:''' If the map $\backslash ;\; \backslash Lambda$ satisfies the condition
: $\backslash ;\; ||\backslash Lambda[\; |\backslash phi\_A\backslash rangle\; \backslash langle\; \backslash phi\_A|\; \backslash otimes\; |\backslash phi\_B\backslash rangle\; \backslash langle\; \backslash phi\_B|]||\_\{Tr\}\; \backslash leq\; 1$
for all pure product states $\backslash ;\; |\backslash phi\_A\backslash rangle\; \backslash langle\; \backslash phi\_A|\; \backslash otimes\; |\backslash phi\_B\backslash rangle\; \backslash langle\; \backslash phi\_B|$, then for any separable state $\backslash ;\; \backslash varrho\_\{AB\}$ one has
: $\backslash ;\; ||\backslash Lambda[\backslash varrho\_\{AB\}]||\_\{Tr\}\; \backslash leq\; 1$.
The '''matrix realignment criterion''' is just a particular case of the above criterion where the ''matrix realignment map'' $\backslash ;\; \backslash mathcal\{R\}$, 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.
== Related papers ==
* M. Keyl, ''Phys. Rep.'' '''369''', no.5, 431-548 (2002).
* C. H. Bennett ''et al.'', ''Phys. Rev. Lett.'' '''76''', 722 (1996).
* M. Horodecki, P. Horodecki, R. Horodecki, ''Phys. Lett. A'' '''223''', 1 (1996).
* G. Lindblad, ''Commun. Math. Phys.'' '''40''', 147-151 (1975).
* M. D. Choi, ''Linear Alg. Appl.'' '''10''', 285 (1975).
* A. Jamiolkowski, ''Rep. Math. Phys.'' '''3''', 275 (1972).
*N. Cerf, R. Adami, ''Phys. Rev. A'' '''60''', 898-909 (1999).
*H.-P. Breuer, ''Phys. Rev. Lett.'' '''97''', 080501 (2006).
*W. Hall, ''Construction of indecomposable positive maps based on a new criterion for indecomposability'', e-print quant-ph/0607035.
*M. Horodecki, P. Horodecki, R. Horodecki, ''Phys. Rev. Lett.'' '''78''', 574 (1997)
*C. H. Bennet ''et al.'', ''Phys. Rev. Lett.'' '''82''', 5385 (1999)
*D. DiVincenzo ''et al.'', ''Comm. Math. Phys.'' '''238''', 379 (2003)
*O. Rudolph, "Lett. Math. Phys." '''70''', 57 (2004)
*K. Chen, L.-A. Wu, ''Quantum Inf. Comp.'' '''3''', 193 (2003)
Category:Entanglement
Category:Handbook of Quantum Information

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