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 Formula: see text of a completely positive dynamical semigroup and any matrix Formula: see text there exists a unique matrix Formula: see text and a unique completely positive map Formula: see text such that (i) Formula: see text, (ii) the superoperator Formula: see text has trace zero, and (iii) Formula: see text 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 Formula: see text-weighted inner product.