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Given any separable complex Hilbert space, any trace-class operator $B$ whichdoes not have purely imaginary trace, and any generator $L$ of anorm-continuous one-parameter semigroup of completely positive maps we provethat there exists a unique bounded operator $K$ and a unique completelypositive map $\Phi$ such that (i) $L=K(\cdot)+(\cdot)K^*+\Phi$, (ii) thesuperoperator $\Phi(B^*(\cdot)B)$ is trace class and has vanishing trace, and(iii) ${\rm tr}(B^*K)$ is a real number. Central to our proof is a modifiedversion of the Choi formalism which relates completely positive maps topositive semi-definite operators. We characterize when this correspondence isinjective and surjective, respectively, which in turn explains why the proofidea of our main result cannot extend to non-separable Hilbert spaces. Inparticular, we find examples of positive semi-definite operators which haveempty pre-image under the Choi formalism as soon as the underlying Hilbertspace is infinite-dimensional.

From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups