Why Does the Kraus Map Need to be Completely Positive?

[Henriamaa]

In Quantum Computation we define a map that takes on density matrix to another. It is represented by some kraus matrices. I do not know why it has to be completely positive.

 

[kith]

I'm not active in this area, so take my remarks with a grain of salt.

A positive map $\Phi$ takes a density matrix to another density matrix. The difference between positive and completely positive maps is important for entangled states in combined systems, where the map acts only on one of the systems.

For a seperable state $\rho = \rho_1 \otimes \rho_2$, a new state $(\Phi \otimes I_2)\rho$ is always positive ($I_2$ is the identity in the second system). Counterintuitively, non-seperable states can yield negative eigenvalues when acted upon with a positive but not completely positive map.

Since we want our map to yield a valid density matrix in all cases, we require it to be completely positive, that is to say that $(\Phi \otimes I_2)\rho$ has to be positive for arbitrary systems 2 and states $\rho$.