# 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 *ρ**A* : = *t**r**B*(*ρ**A**B*) and *ρ**B* : = *t**r**A*(*ρ**A**B*). Then the reduction criterion states that for any separable *ρ**A**B*, two derived operators must be positive semidefinite:

*ρ**A* ⊗ *I**B* − *ρ**A**B* ≥ 0

*I**A* ⊗ *ρ**B* − *ρ**A**B* ≥ 0.

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

*L*(*ρ*) = *I* *t**r*(*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

- quant-ph/9708015↩