Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups

TL;DR

This paper extends the unique decomposition of generators of quantum-dynamical semigroups, allowing for a broader class of decompositions involving a matrix B and a completely positive map, with implications for understanding quantum dynamics.

Contribution

It generalizes the Gorini-Kossakowski-Sudarshan decomposition by introducing a B-weighted inner product and a broader class of decompositions for quantum dynamical generators.

Findings

Existence of a unique decomposition involving K and Φ for any generator L and matrix B.

The decomposition is orthogonal with respect to a B-weighted inner product.

Relation established between trace of CP maps, Kraus operators, and Choi matrix expectations.

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) $L=K(\cdot)+(\cdot)K^*+\Phi$, (ii) the superoperator $\Phi(B^*(\cdot)B)$ has trace zero, and (iii) ${\rm tr}(B^*K)$ 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.