Abstract: 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(\\sqrt{N}d\/\\epsilon^2), which is optimal up to a 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 B\u0103descu, O’Donnell, and Wright.<\/p>\n