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

TL;DR

This work addresses the challenge of identifying Lindbladian dynamics that produce a given non-equilibrium steady state (NESS) by introducing Lindbladian reverse engineering (LRE). The core idea is to map the reconstruction to the kernel of a correlation matrix $M(\hat{\rho}_{\rm ss})$ constructed from NESS observables, so that finding a null eigenvector $M(\hat{\rho}_{\rm ss})|\varphi_{\mathcal{L}}\rangle=0$ yields the Lindbladian parameters in a $(J+K^2)$-dimensional vector. Importantly, the method provides an iff condition for reconstructability and a no-go criterion when the kernel is empty, while allowing post-processing to enforce complete positivity of the dissipator. Applied to bosonic Gaussian states, driven-dissipative collective spins, and random local spin models, the approach demonstrates how different operator bases can realize the same NESS and identifies regimes where the reconstruction is unique or degenerate. The framework offers a scalable, linear-reconstruction alternative to full process tomography and suggests avenues for discovering alternative, experimentally feasible Lindbladians and exploring dissipative phases and non-Markovian features.

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.

Figures (4)

• Figure 1: Lindbladian reverse engineering for the collective steady states of Eq.\ref{['DDM_ss']}, in the weak dissipative regime $\omega/\kappa = 2$. In all panels values below $\approx O(10^{-10})$ are beyond our numerical accuracy, and are interpreted as effectively null. We show in (a) the elements of the kernel eigenvector $\ket{\varphi_{\mathcal{L}} }$ for the correlation matrix $\hat{M} (\hat{\rho}_{\rm{ss}})$. The elements correspond to the set of parameters in the constructed Lindbladian. In panel (b) we show the two lowest eigenvalues of the correlation matrix $\hat{M} (\hat{\rho}_{\rm{ss}})$, displaying in the (inset-panel) only the second eigenvalue in a log-log scale, in order to highlight its gapless behavior. In panel (c) we show the eigenvalues of the dissipative matrix $\bm{\gamma}$ for the lowest eigenstate (kernel) of the correlation matrix.
• Figure 2: (a) The two lowest eigenvalues of the correlation matrix $\hat{M} (\hat{\rho}_{\rm{s}})$ in the random local model, for varying dissipative strength $\alpha_D$. (b) The norm difference between the steady obtained from the reverse engineered Lindbladian $\hat{\rho}_{s}^{M}$ and the exact one $\hat{\rho}_{s}$. All results here are averaged over $200$ random realizations of the system coefficients.
• Figure 3: (a) The smallest eigenvalue $\lambda_1(\hat{M}_\epsilon)$ of the correlation matrix $\hat{M}(\hat{\rho}_\epsilon)$ as a function of the perturbation $\epsilon$, for different sizes $N$ and in the strong dissipative regime, $\omega_{0}/\kappa=1/2$. (b) We show the norm difference between $\hat{\rho}_{s}^{\epsilon}$ and $\hat{\rho}_{s}$ in the strong dissipative regime. (c) The first eigenvalue of the correlation matrix at the weak dissipative regime, $\omega_{0}/\kappa=2$. (d) The norm difference between $\hat{\rho}_{s}^{\epsilon}$ and $\hat{\rho}_{s}$ and (inset panel) between $\hat{\rho}_{s}^{\epsilon,+}$ and $\hat{\rho}_{s}$, for $\omega_{0}/\kappa=2$.
• Figure 4: (a) The smallest eigenvalue $\lambda_1(\hat{M}_\epsilon)$ of the correlation matrix $\hat{M}(\hat{\rho}_\epsilon)$, for varying perturbation $\epsilon$ and dissipative strength $\alpha_D$. (b) The norm difference between the steady obtained from the reverse engineered Lindbladian, $\hat{\rho}_{s}^{\epsilon}$, and the exact one $\hat{\rho}_{s}$. All figures are shown on a logarithmic scale. The results presented here are averaged over 30 random realizations of the system coefficients.