Fig. 1: Asymptotic randomness-generation rates R_c and R_q as functions of the depolarization noise d.

For illustration purpose, here we simulate the result at a trial according to either the X-basis or Z-basis measurement on the depolarized single-photon state \((1-d/2)\left|0\right\rangle \left\langle 0\right|+d/2\left|1\right\rangle \left\langle 1\right|\), where \(\left|0\right\rangle\) and \(\left|1\right\rangle\) are the two eigenstates (in the single-photon subspace) of the Z-basis measurement and d quantifies the depolarization noise. At each trial the X-basis measurement is selected with probability P_X = 0.9999, and so the imbalance τ = (P_X − P_Z)/2 is exactly known. Our method can certify randomness without assuming that the state and measurements are fully characterized. Instead, our method requires only an upper bound δ on the misalignment angle between the two measurement bases and a lower bound q_1,lb on the probability of a single photon in a practical photon source. For the ideal case (the red curves), we set q_1,lb = 1 and δ = 0, while for the practical case (the blue curves), we set q_1,lb = 0.95 and δ = 5^∘.