# Characterisation of gradient flows for a given functional

## TL;DR

This paper explores when a co-vector field can be expressed as the gradient of a vector field on a manifold and applies the result to quantum systems.

## Contribution

The paper provides necessary and sufficient conditions for a co-vector field to be a gradient and applies it to quantum gradient flows.

## Key findings

- A smooth Riemannian metric exists if specific conditions are met on the manifold.
- Finite-dimensional ergodic Lindblad equations have a gradient flow structure under bkm-detailed balance.

## 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 \documentclass[12pt]{minimal}
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\begin{document}$$g_{\alpha \beta }$$\end{document}gαβ on M such that \documentclass[12pt]{minimal}
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\begin{document}$$Y_\beta = g_{\alpha \beta } X^\alpha $$\end{document}Yβ=gαβXα? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we provide a gradient-flow characterisation for dissipative quantum systems. Namely, we show that finite-dimensional ergodic Lindblad equations admit a gradient flow structure for the von Neumann relative entropy if and only if the condition of bkm-detailed balance holds.

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Source: https://tomesphere.com/paper/PMC11211160