Purification of Lindblad dynamics, geometry of mixed states and geometric phases

TL;DR

This paper introduces a nonlinear Schrödinger equation that purifies Lindblad dynamics in open quantum systems, unifies various geometric phase definitions, and explores the complex geometric structures involved.

Contribution

It presents a novel nonlinear Schrödinger equation for Lindblad dynamics, unifies multiple geometric phase concepts, and analyzes the associated higher gauge geometric structure.

Findings

The nonlinear Schrödinger equation successfully purifies Lindblad dynamics.

It unifies different geometric phase definitions for open quantum systems.

The geometric structure is related to higher gauge theory and categorical bibundles.

Abstract

We propose a nonlinear Schr\"odinger equation in a Hilbert space enlarged with an ancilla such that the partial trace of its solution obeys to the Lindblad equation of an open quantum system. The dynamics involved by this nonlinear Schr\"odinger equation constitutes then a purification of the Lindbladian dynamics. This nonlinear equation is compared with other Schr\"odinger like equations appearing in the theory of open systems. We study the (non adiabatic) geometric phases involved by this purification and show that our theory unifies several definitions of geometric phases for open systems which have been previously proposed. We study the geometry involved by this purification and show that it is a complicated geometric structure related to an higher gauge theory, i.e. a categorical bibundle with a connective structure.