Quantum geometric tensor of mixed states based on Uhlmann's approach

The quantum geometric tensor (QGT) measures the difference between adjacent quantum states and reveals the underlying local geometry. Its real part gives a metric and its imaginary part is the Berry curvature. Therefore, the QGT has broad applications in quantum information science as well as topological quantum matter. To generalize the QGT to mixed states, we implement purification of the density matrix and follow Uhlmann's formulation of fibration to derive the QGT from the local geometry. The gauge-invariant form of the QGT has a real part corresponding to the Bures metric of mixed states and an imaginary part that vanishes for typical systems. Importantly, the QGT of mixed states may not approach the QGT of the corresponding pure states due to their different underlying fibrations. We present examples showing such possibilities and discuss the implications.