Nearly tight lower bounds for estimating quantum functionals: Uhlmann fidelity, trace distance, and von Neumann entropy

In this paper, we present a unified framework for proving lower bounds for estimating functionals of quantum states. We therefore resolve several open problems by establishing lower bounds that match known upper bounds: we show that it requires $\widetilde{\Omega}(N^2)$ samples to estimate the Uhlmann fidelity, trace distance, and von Neumann entropy. Moreover, they immediately imply matching query lower bounds of $\widetilde{\Omega}(N)$ by quantum sample-to-query lifting. These lower bounds imply the near-optimality of a dozen quantum algorithms since 2016.