Testing identity of collections of quantum states: sample complexity analysis

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( √ N d/ǫ 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 Bȃdescu, O'Donnell, and Wright (https://dl.acm.org/doi/10.1145/3313276.3316344).

Universal algorithms for quantum data learning

“…, Uρ l U † ). For instance, estimating the distances between states ρ i ∈ S is used in quantum state certification protocols [33][34][35], which aim at giving statistical guarantees that a source of quantum states is truly iid by checking whether all ρ i are equal. Certifying that a source produces identical states in this way is much more resource efficient in comparison to a protocol based on quantum state tomography.…”

“…If we are free to ask for copies of ρ i at will (the so-called query model [34]) then the sample complexity is still Θ(d/ 2 ). If, on the other hand, ρ i are sampled from a distribution p i (the so-called sampling model [35]) then the sample complexity increases to Θ( √ ld/ 2 ).…”