The Peres–Horodecki criterion is a necessary condition, for the joint density matrix
It is also called the PPT criterion, for positive partial transpose.
It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply.
The theorem was discovered in 1996 by Asher Peres[1] and the Horodecki family (Michał, Paweł, and Ryszard)[2] In higher dimensions, the test is inconclusive, and one should supplement it with more advanced tests, such as those based on entanglement witnesses.
Its partial transpose (with respect to the B party) is defined as Note that the partial in the name implies that only part of the state is transposed.
This definition can be seen more clearly if we write the state as a block matrix: Where
The converse of these statements is true if and only if the dimension of the product space is
Consider this 2-qubit family of Werner states: It can be regarded as the convex combination of
Its density matrix is and the partial transpose Its least eigenvalue is
If ρ is separable, it can be written as In this case, the effect of the partial transposition is trivial: As the transposition map preserves eigenvalues, the spectrum of
Showing that being PPT is also sufficient for the 2 X 2 and 3 X 2 (equivalently 2 X 3) cases is more involved.
This is a result of geometric nature and invokes the Hahn–Banach theorem (see reference below).
being positive for all positive maps Λ is a necessary and sufficient condition for the separability of ρ, where Λ maps
Loosely speaking, the transposition map is therefore the only one that can generate negative eigenvalues in these dimensions.
In higher dimensions, however, there exist maps that can't be decomposed in this fashion, and the criterion is no longer sufficient.
Consequently, there are entangled states which have a positive partial transpose.
Such states have the interesting property that they are bound entangled, i.e. they can not be distilled for quantum communication purposes.
The Peres–Horodecki criterion has been extended to continuous variable systems.
Rajiah Simon[3] formulated a particular version of the PPT criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for
Simon's condition can be generalized by taking into account the higher order moments of canonical operators [6][7] or by using entropic measures.
[8][9] For symmetric states of bipartite systems, the positivity of the partial transpose of the density matrix is related to the sign of certain two-body correlations.
is the flip or swap operator exchanging the two parties
A full basis of the symmetric subspace is of the form
has a positive partial transpose if and only if [10] holds for all operators
Moreover, a bipartite symmetric PPT state can be written as where
are not necessarily physical pure density matrices since they can have negative eigenvalues.
are physical density matrices, which is consistent with the fact that for two qubits all PPT states are separable.
The concept of such pseudomixtures has been extended to non-symmetric states and to the multipartite case, by the definition of pseudoseparable states[11] where
are just states that live on the higher dimensional equivalent of the Bloch sphere even for systems that are larger than a qubit.
For systems larger than qubits, such quantum states can be entangled, and in this case they can have PPT or non-PPT bipartitions.