Lindblad evolution as gradient flow

TL;DR

The paper presents a gradient-flow framework for Lindblad evolution in finite-dimensional quantum systems, showing that for a broad class of jump operators the dynamics can be written as ∂ρ/∂t = - ∂/∂ρ^T Φ(ρ) + R(ρ). It provides explicit expressions for the gradient potential Φ and the solenoidal term R, and demonstrates how the formalism reduces to a simple gradient flow in the space of Bloch vectors, with clear interpretability of entropy production and steady states. For Hermitian jumps, the evolution is a pure gradient flow driving toward the maximally mixed state, while non-Hermitian jumps require either a complexified gradient or an orthogonal Helmholtz-Hodge decomposition to recover a gradient-like picture; in all cases the steady states are determined by the potential. The framework is illustrated with qubit and qutrit examples, and extended to non-Hermitian cases via explicit constructions, including Riccati equations for the orthogonal decomposition, highlighting potential applications in state preparation and numerical optimization in open quantum systems.

Abstract

We give a simple argument that, for a large class of jump operators, the Lindblad evolution can be written as a gradient flow in the space of density operators acting on a Hilbert space of dimension $D$. We give explicit expressions for the (matrix-valued) eigenvectors and eigenvalues of the Lindblad evolution using this formalism. We argue that in many cases the interpretation of the evolution is simplified by passing from the complex $D^2$-dimensional space of density operators to the real $D^2-1$-dimensional space of Bloch vectors. When jump operators are non-Hermitian the evolution is not in general gradient flow, but we show that it nevertheless resembles gradient flow in two particular ways. Importantly, the steady states of Lindbladian evolution are still determined by the potential in all cases.