Quantum state certification

We consider the problem of quantum state certification, where one is given $n$ copies of an unknown $d$-dimensional quantum mixed state $ρ$, and one wants to test whether $ρ$ is equal to some known mixed state $σ$ or else is $ε$-far from $σ$. The goal is to use notably fewer copies than the $Ω(d^2)$ needed for full tomography on $ρ$ (i.e., density estimation). We give two robust state certification algorithms: one with respect to fidelity using $n = O(d/ε)$ copies, and one with respect to trace distance using $n = O(d/ε^2)$ copies. The latter algorithm also applies when $σ$ is unknown as well. These copy complexities are optimal up to constant factors.