Testing identity of collections of quantum states: sample complexity analysis

Quantum 7, 1105 (2023).

https://doi.org/10.22331/q-2023-09-11-1105

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 $O(sqrt{N}d/epsilon^2)$, 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 Bădescu, O’Donnell, and Wright [13].