Device-independent certification of two bits of randomness from one entangled bit and Gisin's elegant Bell inequality

We prove that as conjectured by Acín et al. [Phys. Rev. A 93, 040102(R) (2016)], two bits of randomness can be certified in a device-independent way from one bit of entanglement using the maximal quantum violation of Gisin's elegant Bell inequality. This suggests a surprising connection between maximal entanglement, complete sets of mutually unbiased bases, and elements of symmetric informationally complete positive operator-valued measures, on one side, and the optimal way of certifying maximal randomness, on the other.

Figure 1

The source simultaneously emits two systems, one to each side. Buttons represent possible measurements. Light bulbs represent possible outcomes. Alice and Bob wants to certify in a device-independent way that the two bits produced when Alice presses her button 4 are actually random (i.e., unpredictable even for an adversary who manufactured the devices).