Reduction criterion

The '''reduction criterion'''quant-ph/9708015 is a [[separability criteria|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 $\backslash rho\_A\; :=\; tr\_B(\backslash rho\_\{AB\})$ and $\backslash rho\_B\; :=\; tr\_A(\backslash rho\_\{AB\})$. Then the reduction criterion states that for any separable $\backslash rho\_\{AB\}$, two derived operators must be positive semidefinite: :$\backslash rho\_A\; \backslash otimes\; I\_B\; -\; \backslash rho\_\{AB\}\; \backslash geq\; 0$ :$I\_A\; \backslash otimes\; \backslash rho\_B\; -\; \backslash rho\_\{AB\}\; \backslash geq\; 0$. The criterion comes from applying the positive but not completely positive (PnCP) map :$L(\backslash rho)\; =\; I\; \backslash ;\; tr(\backslash rho)\; -\; \backslash rho$ to one part of a [[bipartite]] system. In general it is weaker than the [[PPT criterion]], but any state violating it is always [[Entanglement distillation|distillable]]. For states of dimension $2\; \backslash times\; 2$ or $2\; \backslash times\; 3$ it is equivalent to the PPT criterion and therefore also necessary and sufficient. All entangled [[Werner state]]s of local dimension above 3 satisfy the reduction criterion, but violate the stronger PPT criterion. All entangled [[isotropic state]]s violate the reduction criterion. \bibitem{quant-ph/9708015} [[Category:Entanglement]] [[Category:Handbook of Quantum Information]]