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

Gye Jin Lee, Sunghyeon Jo

We settle the sample complexity of estimating the root Uhlmann fidelity $F(\ddot{I},\ddot{I}ƒ)=\mathrm{tr}\sqrt{\sqrt{\ddot{I}}ƒ\ddot{I}\sqrt{\ddot{I}}ƒ}$ between an unknown state $\ddot{I}$ and a known rank-$r$ refer…