Fidelity and fidelity susceptibility based on Hilbert-Schmidt inner product

Abstract

We reinvestigate the fidelity based on Hilbert-Schmidt inner product and give a simplified form. The geometric meaning of the fidelity is clarified. We then give the analytic expression of the fidelity susceptibility in both Hilbert and Liouville space. By using the reconstruction of symmetric logarithmic derivative in Liouville space, we present the time derivative of fidelity susceptibility with the normalized density vector representation.