Kraus representation

Every finite-dimensional completely positive map has $T(\rho )={\sum }_j{K}_j\rho K_j^{†}$. It is trace preserving exactly when ${\sum }_j{K}_j^{†}{K}_j=I$.

For a Kraus representation $\Phi (X)={\sum }_k{A}_{k}X{A}_k^{†}$, trace preservation is equivalent to the first identity, by cyclicity of the trace. The second says $\Phi (I)=I$, namely unitality, and is a distinct constraint. For $0<\gamma <1$, the amplitude damping channel has operators $A_0=\mathrm{diag}(1,\sqrt{1-\gamma })$ and $A_1=\sqrt{\gamma }|0⟩⟨1|$ and is trace-preserving but not unital. Taking the adjoints as Kraus operators instead gives a unital completely positive map whose output trace on $|0⟩⟨0|$ is $1+\gamma$.

The operators $K_j$ in a Kraus representation are Kraus operators. For a measurement outcome $i$, its associated Kraus operators determine both the outcome probability and the conditional post-measurement state.