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arXiv:2301.07887 · Physical Review Research

Which differential equations correspond to the Lindblad equation?

Victor Kasatkin, Larry Gu, Daniel A. Lidar

Published Nov 21, 2023• 7 citations
arXivHTMLPDF

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, nonhomogeneous 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) of independent interest.

DOI
10.1103/physrevresearch.5.043163

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The quantum damped harmonic oscillator is described by the master equation with usual Lindblad form. The equation has been solved completely by us in arXiv: 0710.2724 [quant-ph]. To construct the general solution a few facts of representation theory based on the Lie algebra su(1,1) were used. In this paper we treat a general model described by a master equation with generalized Lindblad form. Then we examine the algebraic structure related to some Lie algebras and construct the interesting approximate solution. Quantum Computation (Computer) is one of main subjects in Quantum Physics. To realize it we must overcome severe problems arising from Decoherence, so we need to study Quantum Open System to control decoherence (if possible). This paper is a series of [1] and [2], and we study dynamics of a quantum open system. First we explain our purpose in a short manner. See [3] as a general introduction to this subject. ∗E-mail address:
Master equations play a pivotal role in investigating open quantum systems. In particular, the Bloch-Redfield equation stands out due to its relation to a concrete physical environment. However, without further approximations it does not lead to a Lindblad master equation that guarantees that the density matrix stays completely positive, which has raised some concerns regarding its use. This study builds on previous efforts to transform the Bloch-Redfield framework into a mathematically robust Lindblad equation while fully preserving the effects that are lost within the secular approximation that is commonly used to guarantee positivity. These previous approaches introduced two potential deficiencies: The environment-induced energy shift can be non-Hermitian and some decay rates can be negative, violating the assumptions of Lindblad's theorem. Here we propose and evaluate straightforward solutions to both problems. Our approach offers an effective and general procedure for obtaining a Lindblad equation, derived from a concrete physical environment, while mitigating the unphysical dynamics present in the Bloch-Redfield equation.