# Characterization of a Quantum Random Number Generator Based on Vacuum Fluctuations
## 3 Materials and Methods  
A schematic representation of the implemented QRNG is shown in Figure 2. A continuous-wave laser at 1550.92 nm is used as the LO, and an optical bandpass filter (OBF) is applied to suppress its sideband frequencies. Posteriorly, the LO is split by a 90/10 BS (BS1) such that an optical power meter (OPM) in the 10% output arm allows real-time monitoring of the power level without tampering with the setup. The LO is then split by a second BS (BS2) with one of its input ports blocked to assure the purity of the vacuum state. After the BS2, the resulting output beams are detected by a pair of photodiodes on a ThorLabs PDB450C balanced amplified photodetector with an output bandwidth of 45 MHz. A VOA is placed in one of the output arms to allow precise adjustments of the BS2 splitting ratio. To better account for the differences in the responsivity of the photodiodes, the attenuation was set, such that the photocurrents measured were indistinguishable. The subsequent signal is amplified with a TIA and finally digitized by an 8-bit Picoscope 6403D ADC with a \(\pm\)50 mV acquisition range. As previously described, the homodyne measurements also contain unwanted noise, such as electronic noise that is intrinsic to the electronic components in the experimental setup. Consequently, the RNs yielded by the QRNG are never truly random but contain simultaneously quantum and classical contributions. A randomness extraction process is thus required to suppress these contributions and attain secure true RNs since all classical noise can, in principle, be known and manipulated by an adversary. Consequently, the raw measurements are at last subjected to a post-processing algorithm that outputs a sequence of uniformly distributed random bits. This process requires obtaining a uniform distribution, quantifying the entropy that originates from quantum phenomena, \(H_{\text{quant}}\), and applying a randomness extractor. Initially, to obtain a sequence of uniformly distributed bits, a binning process is applied to the Gaussian distribution of homodyne measurements. Here, a bit sequence of length \(n\) is assigned to all outcomes that fall in the same bin and, therefore, \(n\)-bit unbiased RNs  
Figure 2: Schematic representation of the implemented QRNG. A laser is split by a BS with one of its input ports blocked (BS2) and the subsequent photocurrents are subtracted. The resulting signal, \(\text{i}_{1}-\text{i}_{2}\), is amplified, digitized, and processed to obtain a string of unbiased random bits. are produced. The sequence length merely depends on the number of bins used, such that \(2^{n}\) bins are required for a sequence of \(n\) bits [27; 28]. The total entropy, \(H_{\text{total}}\), yielded by a bit sequence obtained through the binning process is given by the Shannon entropy, for a uniform distribution \(X\):  
\[H(X)_{\text{total}}=-\sum_{i=1}p_{i}\log_{2}p_{i}=\log_{2}(p_{i})=n, \tag{14}\]  
where \(p_{i}\) is the probability of finding a sample on the \(i\)-th bin [40]. This same process can also be applied to assess the contribution of electronic noise. As such, measurements of these fluctuations are taken by removing the LO and the same binning is superimposed to its distribution. The entropy contribution originating from the electronic noise is given by:  
\[H_{\text{elec}}=-\sum_{i=1}^{2^{n}}p_{i}^{\text{e}}\log_{2}p_{i}^{\text{e}}, \tag{15}\]  
where \(p_{i}^{\text{e}}\) is the probability of finding an electronic noise measurement in each bin. Although this quantity is relatively constant, periodic evaluations of the noise level can be made to ensure that it does not increase [27; 41]. This constitutes the major contribution to the fraction of entropy originating from classical effects, \(H_{\text{class}}\).