Theorem [PPT separability criterion]: If a state $\varrho \in S_A \otimes S_B$ is separable, then $\varrho^{T_B} := (id_A \otimes T_B)[\varrho] \geq 0 ,$ where $T_{B}$, is the transposition on the second subsystem of the compound system $S_A \otimes S_B$.

Therefore if $\varrho^{T_B} := (id_A \otimes T_B)[\varrho]$ is nonpositive, the state $\varrho$ 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 $d_A d_B \leq 6$, i.e., the bipartite system is of the type $2 \otimes 2$ or $2 \otimes 3$ or $3 \otimes 2$, then the PPT criterion provides a necessary and sufficient condition for separability, as follows.

Theorem: A state $\varrho_{AB} \in S_A \otimes S_B$ with $d_A d_B \leq 6$ is separable if and only if $\varrho_{AB}^{T_B} := (id_A \otimes T_B)[\varrho_{AB}] \geq 0$, i.e. if and only if $\varrho_{AB}$ 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).

Category:Entanglement Category:Handbook of Quantum Information