Efficient fidelity estimation: Alternative derivation and related applications

In [Phys. Rev. A 107, 012427 (2023)], A. J. Baldwin and J. A. Jones proved that Uhlmann-Jozsa's fidelity between two quantum states $\rho$ and $\sigma$, i.e., $F(\rho,\sigma)~:=~(Tr\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^2$, can be written in a simplified form as $F(\rho,\sigma) = (Tr\sqrt{\rho\sigma})^2$. In this article, we give an alternative proof of this result, using a function power series expansion and the properties of the trace function. Our approach not only reinforces the validity of the simplified expression but also facilitates the exploration of novel dissimilarity functions for quantum states and more complex trace functions of a density operator.