Self-testing properties of Gisin's elegant Bell inequality

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.

Figure 1

Alice's and Bob's measurement eigenstates form two dual Platonic solids inscribed in Bloch spheres. Alice's eigenstates sit at the corners of an octahedron, Bob's eigenstates can be grouped into two dual sets of SIC vectors which sit at the corners of a cube. (a) The octahedron in Alice's Bloch sphere. (b) The cube in Bob's Bloch sphere.