中国物理B ›› 2026, Vol. 35 ›› Issue (2): 20203-020203.doi: 10.1088/1674-1056/adf4b0

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Double Wilczek-Zee connection and mixed-state quantum geometric tensor

Xiaoguang Wang(王晓光)1, Xiao-Ming Lu(陆晓铭)2, Jing Liu(刘京)3,†, Wenkui Ding(丁文魁)1,‡, and Libin Fu(傅立斌)4   

  1. 1 Zhejiang Key Laboratory of Quantum State Control and Optical Field Manipulation, Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China;
    2 School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China;
    3 Center for Theoretical Physics and School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China;
    4 Graduate School of China Academy of Engineering Physics, Beijing 100193, China
  • 收稿日期:2025-06-09 修回日期:2025-07-21 接受日期:2025-07-28 发布日期:2026-02-09
  • 通讯作者: Jing Liu, Wenkui Ding E-mail:liujing@hainanu.edu.cn;wenkuiding@zstu.edu.cn
  • 基金资助:
    Project supported by Quantum Science and Technology– National Science and Technology Major Project (Grant No. 2024ZD0301000), the National Natural Science Foundation of China (Grant No. 12305031), the Hangzhou Joint Fund of the Natural Science Foundation of Zhejiang Province, China (Grant No. LHZSD24A050001), and the Science Foundation of Zhejiang Sci-Tech University (Grant Nos. 23062088-Y and 23062153-Y).

Double Wilczek-Zee connection and mixed-state quantum geometric tensor

Xiaoguang Wang(王晓光)1, Xiao-Ming Lu(陆晓铭)2, Jing Liu(刘京)3,†, Wenkui Ding(丁文魁)1,‡, and Libin Fu(傅立斌)4   

  1. 1 Zhejiang Key Laboratory of Quantum State Control and Optical Field Manipulation, Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China;
    2 School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China;
    3 Center for Theoretical Physics and School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China;
    4 Graduate School of China Academy of Engineering Physics, Beijing 100193, China
  • Received:2025-06-09 Revised:2025-07-21 Accepted:2025-07-28 Published:2026-02-09
  • Contact: Jing Liu, Wenkui Ding E-mail:liujing@hainanu.edu.cn;wenkuiding@zstu.edu.cn
  • Supported by:
    Project supported by Quantum Science and Technology– National Science and Technology Major Project (Grant No. 2024ZD0301000), the National Natural Science Foundation of China (Grant No. 12305031), the Hangzhou Joint Fund of the Natural Science Foundation of Zhejiang Province, China (Grant No. LHZSD24A050001), and the Science Foundation of Zhejiang Sci-Tech University (Grant Nos. 23062088-Y and 23062153-Y).

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摘要： The Wilczek-Zee connection (WZC) is a key concept in the study of topology of quantum systems. Here, we introduce the double Wilczek-Zee connection (DWZC) which naturally appears in the pure-state quantum geometric tensor (QGT), another important concept in the field of quantum geometry. The DWZC is Hermitian with respect to the two integer indices, just like the original Hermitian WZC. Based on the symmetric logarithmic derivative operator, we propose a mixed-state quantum geometric tensor. Using the symmetric properties of the DWZC, we find that the real part of the QGT is connected to the real part of the DWZC and the square of eigenvalue differences of the density matrix, whereas the imaginary part can be given in terms of the imaginary part of the DWZC and the cube of the eigenvalue differences. For density matrices with full rank or no full rank, the QGT can be given in terms of real and imaginary parts of the DWZC.

关键词: quantum geometry, Wilczek-Zee connection, quantum geometric tensor

Abstract: The Wilczek-Zee connection (WZC) is a key concept in the study of topology of quantum systems. Here, we introduce the double Wilczek-Zee connection (DWZC) which naturally appears in the pure-state quantum geometric tensor (QGT), another important concept in the field of quantum geometry. The DWZC is Hermitian with respect to the two integer indices, just like the original Hermitian WZC. Based on the symmetric logarithmic derivative operator, we propose a mixed-state quantum geometric tensor. Using the symmetric properties of the DWZC, we find that the real part of the QGT is connected to the real part of the DWZC and the square of eigenvalue differences of the density matrix, whereas the imaginary part can be given in terms of the imaginary part of the DWZC and the cube of the eigenvalue differences. For density matrices with full rank or no full rank, the QGT can be given in terms of real and imaginary parts of the DWZC.

Key words: quantum geometry, Wilczek-Zee connection, quantum geometric tensor

中图分类号:  (Geometry, differential geometry, and topology)

  • 02.40.-k
04.60.Pp (Loop quantum gravity, quantum geometry, spin foams)