Local geometry and quantum geometric tensor of mixed states

TL;DR

This paper extends the quantum geometric tensor to mixed states, revealing its gauge-invariant form, connections to the Bures metric and Uhlmann form, and explores implications for quantum geometry and experiments.

Contribution

It introduces a gauge-invariant quantum geometric tensor for mixed states, linking it to known metrics and forms, and analyzes its properties and experimental relevance.

Findings

The real part of the QGT is the Bures metric.

The imaginary part of the QGT is the Uhlmann form, which vanishes for physical processes.

The Bures metric reduces to the Fubini-Study metric at zero temperature.

Abstract

The quantum geometric tensor (QGT) is a fundamental concept for characterizing the local geometry of quantum states. After casting the geometry of pure quantum states and extracting the QGT, we generalize the geometry to mixed quantum states via the density matrix and its purification. The gauge-invariant QGT of mixed states is derived, whose real and imaginary parts are the Bures metric and the Uhlmann form, respectively. In contrast to the imaginary part of the pure-state QGT that is proportional to the Berry curvature, the Uhlmann form vanishes identically for ordinary physical processes. Moreover, there exists a Pythagorean-like equation that links different local distances and reflect the underlying fibration. The Bures metric of mixed states is shown to reduce to the corresponding Fubini-Study metric of the ground state as temperature approaches zero, establishing a correspondence despite the different underlying fibrations. We also present two examples with contrasting local geometries and discuss experimental implications.