Quantum state certification
Costin Bădescu, Ryan W. O’Donnell, John C. Wright · 2019
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 є-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/є) copies, and one with respect to trace distance using n = O(d/є2) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.