# PPT criterion

An important example of separability criteria via PnCP maps is the PPT criterion, as stated in the following theorem:

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  TB is the transposition on the second subsystem of the compound system SA ⊗ SB.

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 dAdB ≤ 6, i.e. the bipartite system is of the type 2 ⊗ 2 or 2 ⊗ 3 or 3 ⊗ 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 dAdB ≤ 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.

• 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).