# Reduction criterion

The reduction criterion1 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:

ρA ⊗ IB − ρAB ≥ 0

IA ⊗ ρB − ρAB ≥ 0.

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

L(ρ) = Itr(ρ) − ρ 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 × 2 or 2 × 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.

\bibitem{quant-ph/9708015}

1. quant-ph/9708015