PPT criterion

An important example of [[separability criteria]] via PnCP maps is the '''PPT criterion''', as stated in the following '''theorem''': '''If a state $\backslash varrho\; \backslash in\; S\_A\; \backslash otimes\; S\_B$ is separable, then $\backslash varrho^\{T\_B\}\; :=\; (id\_A\; \backslash otimes\; T\_B)[\backslash varrho]\; \backslash geq\; 0$, where $\backslash ;\; T\_B$ is the transposition on the second subsystem of the compound system $S\_A\; \backslash otimes\; S\_B$.''' Therefore if $\backslash varrho^\{T\_B\}\; :=\; (id\_A\; \backslash otimes\; T\_B)[\backslash varrho]$ is nonpositive, the state $\backslash varrho$ is entangled. In arbitrary dimensions of the two subsystems the PPT criterion provides only a necessary but not sufficient condition for separability, i.e. in "high" dimensions there exist states that remain positive under partial transposition (PPT states) even if they are entangled. If instead the bipartite system has dimension $d\_A\; d\_B\; \backslash leq\; 6$, i.e. the bipartite system is of the type $2\; \backslash otimes\; 2$ or $2\; \backslash otimes\; 3$ or $3\; \backslash otimes\; 2$, then the PPT criterion provides a necessary and sufficient condition for separability, as follows. '''Theorem: A state $\backslash varrho\_\{AB\}\; \backslash in\; S\_A\; \backslash otimes\; S\_B$ with $d\_A\; d\_B\; \backslash leq\; 6$ is ''separable'' if and only if $\backslash varrho\_\{AB\}^\{T\_B\}\; :=\; (id\_A\; \backslash otimes\; T\_B)[\backslash varrho\_\{AB\}]\; \backslash geq\; 0$, i.e. if and only if $\backslash varrho\_\{AB\}$ is PPT.''' == Related papers == * A. Peres, ''Separability criterion for density matrices'', Phys. Rev. Lett. '''77(8)''', 1413 (1996). * M. Horodecki, P. Horodecki, R. Horodecki, ''Separability of mixed states: necessary and sufficient conditions'', Phys. Lett. A '''223''', 1 (1996). {{stub}} [[Category:Entanglement]] [[Category:Handbook of Quantum Information]]