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.

Details

Let H₁ and H₂ be Hilbert spaces of finite dimensions n and m respectively. L(H_i) will denote the space of linear operators acting on H_i. Consider a bipartite quantum syste 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 witness

Entanglement witness

In quantum information theory, an entanglement witness is an object of geometric nature which distinguishes an entangled state from separable ones.- Details :...

es, 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 necessariy 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


Direct calculation shows that the above expression is same as


where ρ₁ is the partial trace

Partial trace

In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function...

of ρ with respect to the second system. The dual relation


is obtained in the analogous fashion. The reduction criterion consists of the above two inequalities.