The Sample Complexity of Fidelity Estimation to a Known Rank-$r$ Reference State Is $\widetildeΘ(r^2/\varepsilon^2)$

Abstract

We settle the sample complexity of estimating the root Uhlmann fidelity $F(ρ,σ)=\operatorname{tr}\sqrt{\sqrtσρ\sqrtσ}$ between an unknown state $ρ$ and a known rank-$r$ reference state $σ$. Writing $S(r,\varepsilon)$ for the sample complexity at additive error $\varepsilon$, we resolve the open problem posed by Wang by closing, up to logarithmic factors, the gap between the previously known bounds $Ω(r/\varepsilon^2)$ and $O(r^2/\varepsilon^2)$. We prove $S(r,\varepsilon)=\widetildeΘ(r^2/\varepsilon^2)$ for all $0<\varepsilon\le\varepsilon_0$, where $\varepsilon_0>0$ is a universal constant. The lower bound already holds on a $2r$-dimensional system when $σ$ is maximally mixed on a fixed $r$-dimensional subspace, and for a hard family of states that do not commute with $σ$. The proof combines exact spectral moment matching, a radially size-biased doubly correlated Wishart model, and the Cauchy identity, reducing state indistinguishability to a long-cycle estimate for a weighted random permutation. A direct-sum embedding and binomial thinning yield the optimal $1/\varepsilon^2$ dependence. We also prove a near-quadratic lower bound $\widetildeΩ(r^2)$ for quantum spectrum estimation at constant accuracy. Combined with the recent $O(r^2(\log\log r/\log r)^2)$ upper bound, this determines the polynomial order of the sample complexity in this regime and establishes a near-quadratic barrier.