Which differential equations correspond to the Lindblad equation?

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.

23 pages, 1 figure

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• Assessing non-Markovian dynamics
• Completely Positive Maps and Classical Correlations
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• A Brief History of the GKLS Equation

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