Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities

TL;DR

This paper introduces a new analytical method to derive robust self-testing bounds for quantum states using Bell inequalities, improving existing bounds and achieving tight results for the GHZ state.

Contribution

The authors develop a novel technique for analytic self-testing bounds, providing improved and tight bounds for the CHSH and Mermin inequalities in quantum systems.

Findings

Improved bounds for singlet self-testing with CHSH inequality.

Tight bounds for GHZ state self-testing with Mermin inequality.

Method applicable to other quantum scenarios.

Abstract

Self-testing refers to the phenomenon that certain extremal quantum correlations (almost) uniquely identify the quantum system under consideration. For instance observing the maximal violation of the CHSH inequality certifies that the two parties share a singlet. While self-testing results are known for several classes of states, in many cases they are only applicable if the observed statistics are almost perfect, which makes them unsuitable for practical applications. Practically relevant self-testing bounds are much less common and moreover they all result from a single numerical method (with one exception which we discuss in detail). In this work we present a new technique for proving analytic self-testing bounds of practically relevant robustness. We obtain improved bounds for the case of self-testing the singlet using the CHSH inequality (in particular we show that non-trivial fidelity with the singlet can be achieved as long as the violation exceeds $\beta^{*} = (16 + 14 \sqrt{2})/17 \approx 2.11$). In case of self-testing the tripartite GHZ state using the Mermin inequality we derive a bound which not only improves on previously known results but turns out to be tight. We discuss other scenarios to which our technique can be immediately applied.