Which differential equations correspond to the Lindblad equation?

TL;DR

This paper investigates the inverse problem of determining whether a given finite-dimensional 1ODE corresponds to a Lindblad master equation, providing a general solution and conditions for the existence of such a quantum dynamical representation.

Contribution

It offers a comprehensive method to identify Lindblad equations from 1ODEs, including a positivity test and properties relating the two forms.

Findings

Provides a general solution to the inverse problem

Includes a complete positivity test for the 1ODE parameters

Derives properties linking the master equation and 1ODE representations

Abstract

The Lindblad master equation can always be transformed into a first-order linear ordinary differential equation (1ODE) for the coherence vector. We pose the inverse problem: given a finite-dimensional, non-homogeneous 1ODE, does a corresponding Lindblad equation exist? If so, what are the corresponding Hamiltonian and Lindblad operators? We provide a general solution to this problem, including a complete positivity test in terms of the parameters of the 1ODE. We also derive a host of properties relating the two representations (master equation and 1ODE), which are of independent interest.