The Sample Complexity of Fidelity Estimation to a Known Rank-$r$ Reference State Is $\tilde{\Theta }(r^2/{\epsilon }^2)$

Gye Jin Lee, Sunghyeon Jo

We settle the sample complexity of estimating the root Uhlmann fidelity $F(\rho ,\sigma )=\mathrm{tr}\sqrt{\sqrt{\sigma }\rho \sqrt{\sigma }}$ between an unknown state $\rho$ and a known rank-$r$ reference s…