Mathematical Foundation of the U$^N(1)$ Quantum Geometric Tensor

TL;DR

This paper develops the mathematical framework for the $ ext{U}^N(1)$ quantum geometric tensor of mixed states, including its geometric description, properties, and fundamental inequalities, enhancing understanding of quantum state geometry.

Contribution

It introduces a principal bundle-based description of the $ ext{U}^N(1)$ QGT, compares it with pure-state cases, and proves a key inequality with physical implications.

Findings

Established a principal bundle description for the $ ext{U}^N(1)$ QGT.

Derived a Pythagorean-like distance decomposition for mixed states.

Proved a fundamental inequality with physical significance.

Abstract

In this paper, we systematically establish the mathematical foundation for the $\text{U}^N(1)$ quantum geometric tensor (QGT) of mixed states Explicitly, we present a description based on the $\text{U}^N(1)$ principal bundle and derive a Pythagorean-like distance decomposition equation. Additionally, we offer a comprehensive comparison of its properties with those of the U(1) principal bundle description of the pure-state QGT. Finally, we prove a fundamental inequality for the $\text{U}^N(1)$ QGT and discuss its physical implication.