Semi-Device-Independent Heterodyne-Based Quantum Random-Number Generator

Randomness is a fundamental feature of quantum mechanics, which is an invaluable resource for both classical and quantum technologies. Practical quantum random-number generators (QRNGs) usually need to trust their devices, but their security can be jeopardized in the case of imperfections or malicious external actions. In this work, we present a robust implementation of a semi-device-independent QRNG that guarantees both security and fast generation rates. The system works in a prepare-and-measure scenario, where measurement and source are uncharacterized, but a bound on the energy of the prepared states is assumed. Our implementation exploits heterodyne detection, which offers increased generation rate and improved long-term stability compared to alternative measurement strategies. In particular, due to the tomographic properties of heterodyne measurement, we can compensate for fast phase fluctuations via postprocessing, avoiding complex active phase-stabilization systems. As a result, our scheme combines high security and speed with a simple setup featuring only commercial-off-the-shelf components, making it an attractive solution in many practical scenarios.

Figure 1

General idea of the semi-DI QRNG protocol.

Figure 2

(a) For $x=0$ and $x=1$, the blue and red states are prepared, respectively. In the left panel states with phase $\varphi =0$ are prepared; on the right panel, the prepared states have phase $\varphi =\theta$. (b) homodyne detection along $X$ quadrature. When $\theta \ne 0$ the two states become less distinguishable. (c) Heterodyne measurement: for a binary outcome, if the received state is located on the left side with respect to the linear classifier then $b=0$, otherwise $b=1$. The state distinguishability does not vary with $\theta$.

Figure 3

Experimental setup, which consists of two sections, preparation and measurement. A coherent state is generated by a cw laser and sent to the interferometer. One arm, with $1\mathrm{\%}$ of the light (purple path), is employed to prepare the signal, and the other one with $99\mathrm{\%}$ of the light (green route), is the local oscillator. In each path, $10\mathrm{\%}$ of light is transmitted to PM for monitoring the power. The two paths are combined on the ${90}^{\circ }$ optical hybrid, which is followed by a pair of balanced detectors implementing the heterodyne measurement. An FPGA controls the phase modulator and the synchronization with the oscilloscope.

Figure 4

Relative phase $\varphi$ between signal and LO as a function of time. The system shows a drift of about ${32}^{\circ }/$s, while for time scales comparable with the chunk size no drift is observed (see inset). The shaded area in the inset shows the $\pm 1\sigma$ confidence interval, demonstrating that fast fluctuations are due to statistical noise.

Figure 5

Conditional min-entropy as a function of the mean photon number. The orange curve is the numerical predictions obtained by SDP. The green curve shows the numerical results of SDP when inefficiencies are considered and shows good agreement with experimental data (blue points).

Figure 6

In this figure, the conditional min-entropy is plotted as a function of the mean photon number for homodyne and heterodyne detection with different phases and no losses ($\eta =1$). It becomes clear that, when $\theta ={90}^{\circ }$, the min-entropy goes to zero, consequently no random bit can be extracted. This happens due to the fact that when $\theta ={90}^{\circ }$, the two input states show the same distribution in the $X$ quadrature, as if only one state is sent.

Figure 7

In this figure, the conditional min-entropy is plotted as a function of the phase $\theta$ for homodyne detection in the lossless case ($\eta =1$). For each value of $\theta$, we choose the $\mu$ value that maximizes the min-entropy.