Reduction criterion

From Wikipedia, the free encyclopedia

In quantum information theory, the reduction criterion is a necessary condtion a mixed state must satisfy in order for it to be separable. In other words, the reduction criterion is a separability criterion.

[edit] Details

Let H₁ and H₂ be Hilber 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

$H = H_1 \otimes H_2.$

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

$(I \otimes \Phi)(\rho)$

is not positive. Therefore, if ρ is separable, then for all positive map Φ,

$(I \otimes \Phi)(\rho) \geq 0.$

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

$\Phi(A) = \operatorname{Tr}A - A.$