# Testing identity of collections of quantum states: sample complexity analysis

arXiv: https://arxiv.org/abs/2103.14511 · HTML: https://mlsanity.com/paper/2103.14511/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Mar 2021.

## Built on by (most cited first, 1 of 1)

- [Non-iid hypothesis testing: from classical to quantum](https://mlsanity.com/paper/2510.06147/) (2025)

## Abstract (arXiv)

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).