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

If a state 𝜚 ∈ S_A ⊗ S_B is separable, then 𝜚^T_B := (id_A ⊗ T_B)[𝜚] ≥ 0, where  T_B is the transposition on the second subsystem of the compound system S_A ⊗ S_B.

Therefore if 𝜚^T_B := (id_A ⊗ T_B)[𝜚] is nonpositive, the state 𝜚 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_Ad_B ≤ 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 𝜚_AB ∈ S_A ⊗ S_B with d_Ad_B ≤ 6 is separable if and only if 𝜚_AB^T_B := (id_A ⊗ T_B)[𝜚_AB] ≥ 0, i.e. if and only if 𝜚_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