Lindbladian reverse engineering for general non-equilibrium steady states: A scalable null-space approach

TL;DR

This paper introduces a scalable linear algebra method for reverse engineering Lindbladian dynamics from a given non-equilibrium steady state, enabling efficient reconstruction of open quantum system evolutions.

Contribution

The authors present a null-space based approach that simplifies Lindbladian reconstruction to a linear problem, with a necessary and sufficient condition for feasibility.

Findings

Method successfully reconstructs Lindbladians for various quantum systems.

Correlation matrix null-space determines the existence of Lindbladian solutions.

Approach scales linearly or quadratically with system size, enhancing computational efficiency.

Abstract

The study of open system dynamics is of paramount importance both from its fundamental aspects as well as from its potential applications in quantum technologies. In the simpler and most commonly studied case, the dynamics of the system can be described by a Lindblad master equation. However, identifying the Lindbladian that leads to general non-equilibrium steady states (NESS) is usually a non-trivial and challenging task. Here we introduce a method for reconstructing the corresponding Lindbaldian master equation given any target NESS, i.e., a \textit{Lindbladian Reverse Engineering} ($\mathcal{L}$RE) approach. The method maps the reconstruction task to a simple linear problem. Specifically, to the diagonalization of a correlation matrix whose elements are NESS observables and whose size scales linearly (at most quadratically) with the number of terms in the Hamiltonian (Lindblad jump operator) ansatz. The kernel (null-space) of the correlation matrix corresponds to Lindbladian solutions. Moreover, the map defines an iff condition for $\mathcal{L}$RE, which works as both a necessary and a sufficient condition; thus, it not only defines, if possible, Lindbladian evolutions leading to the target NESS, but also determines the feasibility of such evolutions in a proposed setup. We illustrate the method in different systems, ranging from bosonic Gaussian systems, dissipative-driven collective spins and random local spin models.