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. • Theory of computation → Quantum computation theory; Sample complexity and generalization bounds. KEYWORDS quantum property testing
Recently, REF obtained tight bounds on the copy complexity of testing whether an unknown density matrix is equal to a known density matrix.
Quantum state certification
STOC 2019
Physics Computer Science Mathematics