Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups

Abstract

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 $\Phi$ such that (i) $L=K(\cdot )+(\cdot )K^{\ast }+\Phi$, (ii) the superoperator $\Phi (B^{\ast }(\cdot )B)$ has trace zero, and (iii) $\mathrm{tr}(B^{\ast }K)$ 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.

19 pages

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Cited by in corpus (5)

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• A universal constraint for relaxation rates for quantum Markov generators: complete positivity and beyond
• Superoperator master equations for depolarizing dynamics
• The concept of minimal dissipation and the identification of work in autonomous systems: A view from classical statistical physics
• From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups