Reduction criterion

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".

Details

Let "H"1 and "H"2 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 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"("H"2) → "L"("H"1) 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)( ho)

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

:(I otimes Phi)( ho) 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 "H"1 = "H"2. Define the positive map Φ: "L"("H"2) → "L"("H"1) by

:Phi(A) = operatorname{Tr}A - A.

It is known that Φ is positive but not completely positive. So a mixed state "ρ" being separable implies

:(I otimes Phi) ( ho) geq 0.

Direct calculation shows that the above expression is same as

:I otimes ho_1 - ho geq 0

where "ρ"1 is the partial trace of "ρ" with respect to the second system. The dual relation

: ho_2 otimes I - ho geq 0

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


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