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

Given any separable complex Hilbert space, any trace-class operator Formula: see text which does not have purely imaginary trace, and any generator Formula: see text of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator Formula: see text and a unique completely positive map Formula: see text such that (i) Formula: see text, (ii) the superoperator Formula: see text is trace class and has vanishing trace, and (iii) Formula: see text is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.