Lindblad evolution as gradient flow

TL;DR

This paper demonstrates that Lindblad evolution can often be expressed as a gradient flow in the space of density operators, providing explicit eigenstructure and simplifying interpretation via Bloch vectors.

Contribution

It introduces a formalism to represent Lindblad evolution as a gradient flow and explores its implications, including explicit eigenvector expressions and the role of steady states.

Findings

Lindblad evolution can be written as a gradient flow for many jump operators.

Explicit eigenvalues and eigenvectors of Lindblad evolution are derived.

Steady states are determined by a potential in all cases.

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.