Positive partial transpose

The '''Peres-Horodecki criterion''' is a necessary condition for the joint [[density matrix]] $\backslash rho$ of two systems $A$ and $B$ to be [[separable states| 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 $\backslash rho$ is separable, then the partial transpose :$\backslash sigma\_\{m\backslash ,\; \backslash mu\backslash ,\; n\backslash ,\; \backslash nu\}\; :=\; \backslash rho\_\{n\backslash ,\; \backslash mu\backslash ,\; m\backslash ,\; \backslash nu\}$ of $\backslash rho$, taken in some basis $|m\backslash rangle\_A\; \backslash otimes\; |\backslash mu\backslash rangle\_B$, has non negative [[eigenvalues]]. In matrix notation, if we write a ''N'' X ''M'' mixed state �? as block matrix: : ,where each $\backslash rho\; \_\{i\; :\; j\}$ is ''M'' X ''M'' and ''n'' runs from 1 to ''N''. The partial transpose of �? is then given by : So �? is PPT if $(\; I\; \backslash otimes\; T\; )\; (\backslash rho)$ is positive, where ''T'' is the transposition map on matrices. That necessity of PPT for separability follows immediately from the fact that if �? is separable, then $(\; I\; \backslash otimes\; \backslash Phi\; )\; (\backslash rho)$ must be positive for all positive map Φ. The transposition map is clearly a positive map. Showing that being PPT is also sufficient for in 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 the [[Hahn-Banach theorem]] (see reference below). From the existence of entanglement witnesses, one can show that $(\; I\; \backslash otimes\; \backslash Phi\; )\; (\backslash rho)$ being positive for all positive map Φ is not only necessary but also sufficient for separability of �?. Furthermore, every positive map from the [[C*-algebra]] of $2\; \backslash times\; 2$ matrices to $2\; \backslash times\; 2$ or $3\; \backslash times\; 3$ matrices can be decomposed into a sum of [[Choi's theorem on completely positive maps|completely positive]] and completely copositive maps. In other words, every such map Λ can be written as :$\backslash Lambda\; =\; \backslash Lambda\; \_1\; +\; \backslash Lambda\; \_2\; \backslash circ\; T$ ,where $\backslash Lambda\_1$ and $\backslash Lambda\_2$ 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 propeties including the fact that thay are [[Bound entanglement| bound entangled]], i.e. they can not be [[entanglement distillation|distilled]] for quantum communication purposes. ==References== *[[Asher Peres]], ''Separability Criterion for Density Matrices'', Phys. Rev. Lett. 77, 1413–1415 (1996) *M. Horodecki, P. Horodecki, R. Horodecki, ''Separability of Mixed States: Necessary and Sufficient Conditions'', Physics Letters A 223, p. 1 (1996). {{FromWikipedia}} [[Category:Handbook of Quantum Information]] [[Category:Entanglement]]