Sjöqvist quantum geometric tensor of finite-temperature mixed states

The quantum geometric tensor (QGT) reveals local geometric properties and associated topological information of quantum states. Here a generalization of the QGT to mixed quantum states at finite temperatures based on the Sjöqvist distance is developed. The resulting Sjöqvist QGT is invariant under gauge transformations of individual spectrum levels. A Pythagorean-like relation connects the distances and gauge transformations, which clarifies the role of the parallel-transport condition. The real part of the QGT naturally decomposes into a sum of the Fisher-Rao metric and Fubini-Study metrics, allowing a distinction between different contributions to the quantum distance. The imaginary part of the QGT is proportional to the weighted summation of the Berry curvatures, which leads to a geometric phase for mixed states under certain conditions. We present three examples of different dimensions to illustrate the temperature dependence of the QGT and a discussion on possible implications.