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

Xin Wang, Xu-Yang Hou, Jia-Chen Tang, Hao Guo

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.

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

Abstract

In this paper, we systematically establish the mathematical foundation for the quantum geometric tensor (QGT) of mixed states Explicitly, we present a description based on the 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 QGT and discuss its physical implication.

Paper Structure

This paper contains 18 sections, 68 equations, 1 table.

Table of Contents

  1. Introduction
  2. The quantum distance and QGT for pure states
  3. Basic formalism
  4. Fibre bundle description
  5. The Sjöqvist distance and the U$^N(1)$ quantum geometric tensor
  6. Overview of the Basic formalism
  7. Mathematical foundation for U$^N(1)$ QGT
  8. Purification of density matrix
  9. Fibre bundle description
  10. Fundamental inequalities about the U$^N(1)$ QGT
  11. Examples of U$^N(1)$ QGT
  12. Bosonic coherent state
  13. Fermionic coherent state
  14. Conclusion
  15. Conflict of interest
  16. ...and 3 more sections