A Symmetry Approach to Geometric Phase for Quantum Ensembles

TL;DR

This paper introduces a novel geometric phase framework for quantum ensembles using symmetry-based tools, unifying phases for both closed and open quantum systems through a new connection on the purification bundle.

Contribution

It develops a new symmetry-based connection on the purification bundle that generalizes geometric phases for mixed quantum states in both unitary and open system evolutions.

Findings

Derives a new connection for mixed states using symmetry tools.

Unifies geometric phase concepts for closed and open quantum systems.

Provides a mathematical framework for analyzing quantum phases in ensembles.

Abstract

We use tools from the theory of dynamical systems with symmetries to stratify Uhlmann's standard purification bundle and derive a new connection for mixed quantum states. For unitarily evolving systems, this connection gives rise to the 'interferometric' geometric phase of Sj\"{o}qvist et al. [Phys. Rev. Lett. 85 2845 - 2849 (2000)], and for more generally evolving open systems it gives rise to the generalization of the interferometric geometric phase due to Tong et al. [Phys. Rev. Lett. 93 080405 (2004)].