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

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.