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(d2) 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/ε2) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.