Geometry along Evolution of Mixed Quantum States

TL;DR

This paper characterizes the geometry of mixed quantum states' evolution, deriving an explicit metric related to energy dispersion, and demonstrates its measurement and application in quantum systems.

Contribution

It provides a new explicit form for the mixed state geometric metric, linking it to energy dispersion and extending it to arbitrary decompositions of density operators.

Findings

Derived an explicit line element for mixed states.

Connected the metric to energy dispersion in unitary evolution.

Applied the metric to a thermal magnetic system.

Abstract

The metric underlying the mixed state geometric phase in unitary and nonunitary evolution [Phys. Rev. Lett. {\bf 85}, 2845 (2000); Phys. Rev. Lett. {\bf 93}, 080405 (2004)] is delineated. An explicit form for the line element is derived and shown to be related to an averaged energy dispersion in the case of unitary evolution. The line element is measurable in interferometry involving nearby internal states. Explicit geodesics are found in the single qubit case. It is shown how the Bures line element can be obtained by extending our approach to arbitrary decompositions of density operators. The proposed metric is applied to a generic magnetic system in a thermal state.