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 -far from σ. The goal is to use notably fewer copies than the Ω(d) 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/ ) copies, and one with respect to trace distance using n = O(d/ ) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.