Testing identity of collections of quantum states: sample complexity analysis

We study the problem of testing identity of a collection of unknown quantum states given sample access to this collection, each state appearing with some known probability. We show that for a collection of d-dimensional quantum states of cardinality N, the sample complexity is O(√Nd/ε2), with a matching lower bound, up to a multiplicative constant. The test is obtained by estimating the mean squared Hilbert-Schmidt distance between the states, thanks to a suitable generalization of the estimator of the Hilbert-Schmidt distance between two unknown states by Buadescu, O'Donnell, and Wright (https://dl.acm.org/doi/10.1145/3313276.3316344).