An important example of separability criteria via PnCP maps is the PPT criterion, as stated in the following theorem:
If a state 
is separable, then ![\varrho^{T_B} := (id_A \otimes T_B)[\varrho] \geq 0](/wiki/images/math/4/0/4/40479d6ee1fbfcc30c3dbe427c24088c.png)
, where 
is the transposition on the second subsystem of the compound system 
.
Therefore if ![\varrho^{T_B} := (id_A \otimes T_B)[\varrho]](/wiki/images/math/3/0/a/30aa8edd25bc68e6eae542c99767fd4d.png)
is nonpositive, the state 
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 
, i.e. the bipartite system is of the type 
or 
or 
, then the PPT criterion provides a necessary and sufficient condition for separability, as follows.
Theorem: A state 
with is separable if and only if ![\varrho_{AB}^{T_B} := (id_A \otimes T_B)[\varrho_{AB}] \geq 0](/wiki/images/math/d/b/9/db9dadd065909878d0d4d6ef1fb14f05.png)
, i.e. if and only if 
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).
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