Optimal randomness certification from one entangled bit

By performing local projective measurements on a two-qubit entangled state one can certify in a device-independent way up to one bit of randomness. We show here that general measurements, defined by positive-operator-valued measures, can certify up to two bits of randomness, which is the optimal amount of randomness that can be certified from an entangled bit. General measurements thus provide an advantage over projective ones for device-independent randomness certification.

Figure 1

Lower bound on the randomness ($-{log}_2$ of the guessing probability) as a function of the visibility $v$ ranging from 0.85 to 1.0. The correlations are of the form $v\mathbf{q}+(1-v)\mathbf{r}$, where $\mathbf{q}$ are the quantum correlations yielding the maximum violation of the respective inequalities, and $\mathbf{r}$ denotes the completely unbiased correlations that assign the same probability to each measurement outcome. The figure was obtained using level 2 of the NPA hierarchy and thus only represents a lower bound on the maximal randomness (note, for instance, that we do not recover the optimal values of two bits for $v=1$). Better noise resistance than the one provided by these curves may thus be obtained by performing a more complex analysis.