Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups

We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator $L$ of a completely positive dynamical semigroup and any matrix $B$ there exists a unique matrix $K$ and a unique completely positive map $\mathrm{\Phi }$ such that (i) $L=K(\cdot )+(\cdot )K^{\ast }+\mathrm{\Phi }$, (ii) the superoperator $\mathrm{\Phi }(B^{\ast }(\cdot )B)$ has trace zero, and (iii) $\mathrm{t}\mathrm{r}(B^{\ast }K)$ is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some $B$-weighted inner product.