Self-testing properties of Gisin's elegant Bell inequality

Understand

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.