Reduction criterion

In quantum information theory, the reduction criterion is a necessary condition a mixed state must satisfy in order for it to be separable. In other words, the reduction criterion is a separability criterion. It was first proved and independently formulated in 1999. Violation of the reduction criterion is closely related to the distillability of the state in question. In quantum information theory, the reduction criterion is a necessary condition a mixed state must satisfy in order for it to be separable. In other words, the reduction criterion is a separability criterion. It was first proved and independently formulated in 1999. Violation of the reduction criterion is closely related to the distillability of the state in question. 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. From the one-to-one correspondence between positive maps and entanglement witnesses, we have that a state ρ is entangled if and only if there exists a positive map Φ such that is not positive. Therefore, if ρ is separable, then for all positive map Φ, Thus every positive, but not completely positive, map Φ gives rise to a necessary condition for separability in this way. The reduction criterion is a particular example of this. Suppose H1 = H2. Define the positive map Φ: L(H2) → L(H1) by It is known that Φ is positive but not completely positive. So a mixed state ρ being separable implies