Constant-sized robust self-tests for states and measurements of unbounded dimension

TL;DR

This paper introduces constant-sized robust self-tests for quantum states and measurements of unbounded dimension, extending previous work and providing new tools for certifying high-dimensional quantum systems.

Contribution

It develops a novel algebraic framework for robust self-testing, including an analogue of the Gowers-Hatami theorem, and presents the first constant-sized self-tests for odd-dimensional maximally entangled states and high-rank measurements.

Findings

Robust self-tests for correlations from measuring maximally entangled states.

Extension of Gowers-Hatami approach to an algebraic framework.

First constant-sized self-tests for odd-dimensional maximally entangled states.

Abstract

We consider correlations, $p_{n,x}$, arising from measuring a maximally entangled state using $n$ measurements with two outcomes each, constructed from $n$ projections that add up to $xI$. We show that the correlations $p_{n,x}$ robustly self-test the underlying states and measurements. To achieve this, we lift the group-theoretic Gowers-Hatami based approach for proving robust self-tests to a more natural algebraic framework. A key step is to obtain an analogue of the Gowers-Hatami theorem allowing to perturb an "approximate" representation of the relevant algebra to an exact one. For $n=4$, the correlations $p_{n,x}$ self-test the maximally entangled state of every odd dimension as well as 2-outcome projective measurements of arbitrarily high rank. The only other family of constant-sized self-tests for strategies of unbounded dimension is due to Fu (QIP 2020) who presents such self-tests for an infinite family of maximally entangled states with even local dimension. Therefore, we are the first to exhibit a constant-sized self-test for measurements of unbounded dimension as well as all maximally entangled states with odd local dimension.