The reduction criterionquant-ph/9708015 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) = I tr(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}
Category:Entanglement Category:Handbook of Quantum Information