Mixed-State Berry Curvature in quantum multiparameter estimations

TL;DR

This paper introduces a new concept of mixed-state quantum Berry curvature, deriving explicit formulas for various states, and highlights its significance in multi-parameter quantum estimation.

Contribution

It defines and derives the mixed-state quantum Berry curvature, extending the concept from pure states and demonstrating its importance in quantum metrology.

Findings

Derived explicit formulas for mixed-state Berry curvature.

Obtained exact expression for an arbitrary qubit state.

Highlighted the role of quantum curvature in multi-parameter estimation.

Abstract

For pure states, the quantum Berry curvature was well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.