Generation of Fresh and Pure Random Numbers for Loophole-Free Bell Tests

We demonstrate extraction of randomness from spontaneous-emission events less than 36 ns in the past, giving output bits with excess predictability below $10^{-5}$ and strong metrological randomness assurances. This randomness generation strategy satisfies the stringent requirements for unpredictable basis choices in current “loophole-free Bell tests” of local realism [Hensen et al., Nature (London) 526, 682 (2015); Giustina et al., this issue, Phys. Rev. Lett. 115, 250401 (2015); Shalm et al., preceding Letter, Phys. Rev. Lett. 115, 250402 (2015)].

Figure 1

Space-time diagram for the production of random numbers in a loophole-free Bell scenario. As shown, up to $k$ raw bits can be generated in an interval that is spacelike separated from both (i) the pair generation and (ii) the distant measurement. Laser pulses () with random phases ${\varphi }_i$ are converted into raw random bits $d_i$ and extracted bits $x_i$ by a running xor $(\oplus )$ calculation.

Figure 2

Random number generation for loophole-free Bell tests. (a) Experimental schematic. Laser pulses with random phases ${\varphi }_i$ from a distributed feedback laser diode (DFB LD) are converted to random powers $p_i$ by an UMZI detected with a linear photoreceiver (PIN PD) to give analog voltages $v_i$. These are one-bit digitized with a comparator and $\mathrm{D}$-type flip-flop to give raw bits $d_i$ and summed modulo 2 with an xor gate to give extracted bits $x_i$. The output value $x_{i+k}$ includes the parity of $k$ raw bits, $d_{i+1}$ to $d_{i+k}$, due to pulses spacelike separated from the distant measurement and from the entanglement production. (b),(c) “Persistence mode” visualization of $v(t)$ statistics. Warmer colors show greater frequency; teal histogram on the left axis describes the voltages sampled inside the time window indicated in gray. (b) Noninterfering pulses obtained by blocking the long interferometer path; (c) interfering pulses. (d) Two-state machine describing the randomness extraction.

Figure 3

Histogram (points) of analog signals $v$ showing an arcsine distribution, and prediction (line) from a Monte Carlo simulation of Eq. (2) using measured rms deviations for all noise sources $v_S$, $v_L$, $v_{\mathrm{PD}}$, $v_H$, and $v_{\text{ref}}$, and a fitted visibility $\mathcal{V}=0.955$. Voltage scale is offset due to ac coupling.

Figure 4

Observed frequencies (spot size) of field amplitude $|\mathcal{E}|$ and resulting phase dispersion $\delta {\varphi }_{\mathrm{rms}}$ over 50 ps, measured by heterodyne detection with gain $\overline{A}$ for continuous currents (colors, left to right) 15, 16, 16.5, 17, 17.5, 19 mA. Gray line shows $\delta {\varphi }_{\mathrm{rms}}\propto |\mathcal{E}{|}^{-1}$ scaling of spontaneous-emission-driven phase diffusion.

Figure 5

Measured two-point correlations (blue points) $|4{\mathrm{\Gamma }}_x(k)|$ for extracted bits $x$ computed on 300 Gbits, showing exponential approach to ideal behavior. Error bars show $\pm 1\sigma$ statistical error in the correlation measurement. Horizontal bars show ${\epsilon }_{\text{max}}^k$, the metrologically derived upper bound on predictability, for ordinary (green), digitizer paranoid (orange), and fully paranoid (red) distrust levels and (left to right) $1\sigma$, $2\sigma$, and $6\sigma$ confidence. (Lower left) Histogram of $4{\mathrm{\Gamma }}_x(4)$ computed on 1000 runs of 1 Gbits each, is normal by the Pearson ${\chi }^2$ test ($p=0.27$). Curve shows ideal distribution.