Semi-device-independent randomness certification on discretized continuous-variable platforms

TL;DR

This work addresses certifying quantum randomness in continuous-variable optics under semi-device-independent assumptions by enforcing a dimension bound through restricting preparations to the two-level Fock subspace. It develops a PAM framework with dimension witnesses and maps CV measurement schemes, particularly homodyne binning and displacement‑based photodetection, to dichotomic data for randomness certification. The authors derive analytical and numerical min‑entropy bounds, demonstrate robust witness violations under losses and misaligned reference frames, and show that simple displacement or hybrid measurement configurations can achieve near‑maximal quantum violations with practical efficiency requirements. The results point to a scalable, low‑complexity route to CV QRNGs, enabling robust, device‑independent‑like randomness generation with standard optical components and resilience against common experimental imperfections.

Abstract

Randomness is fundamental for secure communication and information processing. While continuous-variable optical systems offer an attractive platform for this task, certifying genuine quantum randomness in such setups remains challenging. We present a semi-device-independent scheme for randomness certification tailored to continuous-variable implementations, where the dimension assumption is operationally implemented by restricting state preparations to the two-level Fock subspace. Using standard homodyne and displacement-based measurements, we show that simple optical setups can achieve dimension-witness violations that certify positive min-entropy, even in the presence of realistic losses and misaligned reference frames. These results demonstrate that practical and scalable quantum randomness generation is achievable with minimal experimental complexity on continuous-variable platforms.

Figures (8)

• Figure 1: Prepare-and-measure scenario. Alice’s device prepares a quantum state $\rho_x$ upon pressing a coloured button $x\in\{\text{green},\text{red},\text{yellow}\}$ and sends the system to Bob. In his lab, Bob chooses a measurement labeled by $y\in\{\text{blue},\text{red}\}$ and records an outcome $b$. By repeating the procedure many times, the parties estimate the conditional statistics $p(b| x,y)$, which are later analyzed to assess their (non)classical nature and the presence of certifiable randomness.
• Figure 2: Causal structure of the PAM scenario. Inputs $X$ and $Y$ select the preparation device $A$ and the measurement device $B$. The devices may share pre-established classical randomness modelled by a latent variable $\lambda$ (arrows $\lambda \to A$ and $\lambda \to B$). Within each run the only communication is from $A$ to $B$ (edge $A \to B$, bounded to dimension $d$).
• Figure 3: Tilt dependence of violation and efficiency. (a) Optimal values of $S_3(w)$ as a function of the tilt parameter $w$ for different measurement setups. The results for both qubit (dashed green) and bit (dashed gray) systems using general measurements are also included for comparison. (b) Critical detection efficiency $\eta_{\mathrm{crt}}$ required to violate the tilted inequality $S_3(w)$ in the $(3,2,2,2)$ scenario, shown as a function of $w$. The color code denotes the measurement setups as follows: HH (brown), DD (magenta), HD (violet), DH (yellow). The measurement labels follow the same notation as in Table \ref{['tab: witness_values']}.
• Figure 4: Witness violation versus detection efficiency. Maximum attainable value of the PAM witnesses (left to right) $i \in {S_3, S_4, S_{33,1}, S_{33,2}}$ as a function of the displacement detection efficiency $\eta$. Losses are modeled on the displacement stage while homodyne is treated as lossless. Each curve corresponds to a measurement setup (DD/HD/DH for two settings; DDD/DDH/$\ldots$ for three). A violation occurs above the classical bound, and the intersection defines the critical efficiency $\eta_{\rm crit}$. Dashed lines indicate the values of $\eta_{>{\rm HH}}$ at which hybrid or displacement configurations overtake the constant HH curve, marking the efficiency thresholds beyond which they outperform full homodyne detection.
• Figure 5: Min-entropy versus witness violation. Min-entropy $H_{\infty}$ of the uniform average guessing probability $\overline{p}^{(u)}_{\mathrm{guess}} = 1/6 \sum_{x,y} \max_b p(b|x,y)$ as a function of the normalized value of the tilted witness $S_3(w)$ for several values of $w$. The normalization is such that $0$ corresponds to the classical bound and $1$ to the quantum maximal violation. The solid lines show the min-entropy obtained from the optimized uniform average guessing probability, while the dashed lines correspond to the analytical lower bound given in Eq. \ref{['eq: hmin_lower_bound']}. The maximal min-entropies for the rightmost points of each curve follows. For the optimized values, $H_{\infty}^{\max} \approx 0.382$ for $w = 0.10$ and $w = 0.90$; $H_{\infty}^{\max} \approx 0.318$ for $w = 0.30$ and $w = 0.70$; $H_{\infty}^{\max} \approx 0.288$ for $w = 0.50$. For the analytical lower bound, $H_{\infty}^{\max} \approx 0.228$ for all $w$ values.