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

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.