- Peres-Horodecki criterion
The Peres-Horodecki criterion is a necessary condition, for the joint
density matrix of two systems and , to be separable. It is also called the PPT criterion, for "positive partial transpose". In the 2x2 and 2x3 dimensional cases the condition is also sufficient.The criterion reads:
If is separable, then the
partial transpose : of , taken in some basis , has non negativeeigenvalues .In matrix notation, if we write a "N" X "M" mixed state ρ as block matrix:
:
,where each is "M" X "M" and "n" runs from 1 to "N". The partial transpose of ρ is then given by
:
So ρ is PPT if is positive, where "T" is the transposition map on matrices.
That necessity of PPT for separability follows immediately from the fact that if ρ is separable, then must be positive for all positive maps Φ. The transposition map is clearly a positive map.
Showing that being PPT is also sufficient for the 2 X 2 and 2 X 3 (therefore 3 X 2) cases is more involved. It was shown by the Horodecki's that for every entangled state there exists an
entanglement witness . This is a result of geometric nature and invokes theHahn-Banach theorem (see reference below). From the existence of entanglement witnesses, one can show that being positive for all positive maps Φ is not only necessary but also sufficient for separability of ρ. Furthermore, every positive map from theC*-algebra of matrices to or matrices can be decomposed into a sum of completely positive and completely copositive maps. In other words, every such map Λ can be written as:
,where and are completely positive and "T" is the transposition map. Combining the above two facts, we can conclude PPT is also sufficient for separability in the 2 X 2 and 2 X 3 cases.
Due to the existence of non-decomposable positive maps in higher dimensions, PPT is no longer sufficient in higher dimensions. In higher dimension, there are entangled states which are PPT. Such states have some interesting properties including the fact that they are bound entangled, i.e. they can not be distilled for quantum communication purposes.
References
*
Asher Peres , [http://prola.aps.org/abstract/PRL/v77/i8/p1413_1 "Separability Criterion for Density Matrices"] , Phys. Rev. Lett. 77, 1413–1415 (1996)
*M. Horodecki, P. Horodecki, R. Horodecki, [http://dx.doi.org/10.1016/S0375-9601(96)00706-2 "Separability of Mixed States: Necessary and Sufficient Conditions"] , Physics Letters A 223, 1-8 (1996)
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