Quantum Geometric Tensor for Mixed States Based on the Covariant Derivative

TL;DR

This paper introduces a generalized quantum geometric tensor for mixed states using the purification bundle and covariant derivative, extending geometric analysis tools from pure to mixed quantum states.

Contribution

The authors define a mixed-state quantum geometric tensor that reduces to the traditional tensor for pure states, enabling unified geometric analysis of quantum states.

Findings

The generalized QGT reduces to the traditional QGT for pure states.

The real part of the MSQGT corresponds to the Bures metric.

The imaginary part of the MSQGT corresponds to the mean gauge curvature.

Abstract

The quantum geometric tensor (QGT) is a fundamental quantity for characterizing the geometric properties of quantum states and plays an essential role in elucidating various physical phenomena. The traditional QGT, defined only for pure states, has limited applicability in realistic scenarios where mixed states are common. To address this limitation, we generalize the definition of the QGT to mixed states using the purification bundle and the covariant derivative. Notably, our proposed definition reduces to the traditional QGT when mixed states approach pure states. In our framework, the real and imaginary parts of this generalized QGT correspond to the Bures metric and the mean gauge curvature, respectively, endowing it with a broad range of potential applications. Additionally, using our proposed mixed-state QGT (MSQGT), we derive the geodesic equation applicable to mixed states. This work establishes a unified framework for the geometric analysis of both pure and mixed states, thereby deepening our understanding of the geometric properties of quantum states.