In quantum information theory, the reduction criterion is a necessary condition a mixed state must satisfy in order for it to be separable.
[2] Violation of the reduction criterion is closely related to the distillability of the state in question.
[1] Let H1 and H2 be Hilbert spaces of finite dimensions n and m respectively.
L(Hi) will denote the space of linear operators acting on Hi.
Consider a bipartite quantum system whose state space is the tensor product An (un-normalized) mixed state ρ is a positive linear operator (density matrix) acting on H. A linear map Φ: L(H2) → L(H1) is said to be positive if it preserves the cone of positive elements, i.e. A is positive implied Φ(A) is also.
So a mixed state ρ being separable implies Direct calculation shows that the above expression is the same as where ρ1 is the partial trace of ρ with respect to the second system.
The dual relation is obtained in the analogous fashion.
[3]: Theorem A.16 These bounds are satisfied by separable density matrices, while entangled states can violate them.
Entangled states exhibit a form of stochastic dependence stronger than the strongest classical dependence and in fact they violate Fréchet like bounds.
It is also worth mentioning that is possible to give a Bayesian interpretation of these bounds.