A time-dependent variational principle for dissipative dynamics

TL;DR

This paper develops a variational method for approximating dissipative quantum dynamics described by Lindblad equations, focusing on mixed states and demonstrating its application to fermionic systems like the Hubbard model.

Contribution

It extends the time-dependent variational principle to dissipative systems with mixed states, exploring different information metrics and showing their equivalence for fermionic Gaussian states.

Findings

All monotone Riemannian metrics yield the same trajectory for fermionic Gaussian states.

The method provides a locally optimal approximation to Lindblad dynamics.

Application to the Hubbard model demonstrates practical utility.

Abstract

We extend the time-dependent variational principle to the setting of dissipative dynamics. This provides a locally optimal (in time) approximation to the dynamics of any Lindblad equation within a given variational manifold of mixed states. In contrast to the pure-state setting there is no canonical information geometry for mixed states and this leads to a family of possible trajectories --- one for each information metric. We focus on the case of the operationally motivated family of monotone riemannian metrics and show further, that in the particular case where the variational manifold is given by the set of fermionic gaussian states all of these possible trajectories coincide. We illustrate our results in the case of the Hubbard model subject to spin decoherence.