Testing identity of collections of quantum states: sample complexity analysis

TL;DR

This paper analyzes the sample complexity for testing the identity of collections of unknown quantum states, establishing both upper and lower bounds and proposing an effective testing method based on distance estimation.

Contribution

It provides tight bounds on the sample complexity for quantum state identity testing and introduces a generalized estimator for the Hilbert-Schmidt distance.

Findings

Sample complexity is O(√N d / ε²) for N states of dimension d.

Matching lower bounds confirm the optimality of the sample complexity.

A generalized estimator effectively measures the mean squared Hilbert-Schmidt distance.

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 $O(\sqrt{N}d/\epsilon^2)$, {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 B\u{a}descu, O'Donnell, and Wright (https://dl.acm.org/doi/10.1145/3313276.3316344).