Reduction criterion

The reduction criterion[1] is a separability criterion, that is a condition that all separable states have to satisfy and a violation of it is therefore a proof of entanglement.

Let ρA: = trB(ρAB) and ρB: = trA(ρAB). Then the reduction criterion states that for any separable ρAB, two derived operators must be positive semidefinite:

\rho_A \otimes I_B - \rho_{AB} \geq 0
I_A \otimes \rho_B - \rho_{AB} \geq 0.

The criterion comes from applying the positive but not completely positive (PnCP) map

L(\rho) = I \; tr(\rho) - \rho to one part of a bipartite system.

In general it is weaker than the PPT criterion, but any state violating it is always distillable. For states of dimension 2 \times 2 or 2 \times 3 it is equivalent to the PPT criterion and therefore also necessary and sufficient.

All entangled Werner states of local dimension above 3 satisfy the reduction criterion, but violate the stronger PPT criterion. All entangled isotropic states violate the reduction criterion.

References

[1] Reduction criterion of separability and limits for a class of distillation protocols, Michał Horodecki, Paweł Horodecki, Publication Phys. Rev. A 59, 4206 - 4216 (1999) pre-print

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