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"("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"("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.