Characterisation of gradient flows for a given functional

TL;DR

This paper investigates conditions under which a vector and co-vector field on a manifold can be related via a Riemannian metric, and applies this to characterize when certain quantum dynamical equations have a gradient flow structure.

Contribution

It provides necessary and sufficient conditions for representing a co-vector as a metric dual of a vector, and applies this to identify when Lindblad equations admit gradient flow formulations.

Findings

Existence of a Riemannian metric relating given vector and co-vector fields

Characterization of gradient flow structures in quantum Lindblad equations

Connection between BKM-detailed balance and gradient flow representation

Abstract

Let $X$ be a vector field and $Y$ be a co-vector field on a smooth manifold $M$. Does there exist a smooth Riemannian metric $g_{\alpha \beta}$ on $M$ such that $Y_\beta = g_{\alpha \beta} X^\alpha$? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.