# Self-testing properties of Gisin's elegant Bell inequality

## TL;DR

This paper investigates the self-testing capabilities of Gisin's elegant Bell inequality, showing it cannot certify states and measurements in a device-independent manner unlike the CHSH inequality, due to operator conjugation issues.

## Contribution

It provides a complete characterization of scenarios with maximal violation of Gisin's elegant Bell inequality and explains why it lacks self-testing properties.

## Key findings

- Gisin's elegant Bell inequality does not exhibit self-testing.
- A full characterization of maximal violation scenarios is provided.
- The difficulty arises from distinguishing an operator from its complex conjugate.

## Abstract

An experiment in which the Clauser-Horne-Shimony-Holt inequality is maximally violated is self-testing (i.e., it certifies in a device-independent way both the state and the measurements). We prove that an experiment maximally violating Gisin's elegant Bell inequality is not similarly self-testing. The reason can be traced back to the problem of distinguishing an operator from its complex conjugate. We provide a complete and explicit characterization of all scenarios in which the elegant Bell inequality is maximally violated. This enables us to see exactly how the problem plays out.