Device-independent Shannon entropy certification

TL;DR

This work develops a device-independent QRNG certification framework based on Shannon entropy, contrasting it with min-entropy approaches and using the Navascués-Pironio-Acín hierarchy to bound quantum correlations. The authors propose a two-stage method: first bound outcome probabilities via an SDP (level $Q_2$) and then solve a nonlinear entropy optimization under these bounds using COBYQA with basin-hopping. Across several Bell inequalities, they quantify which certificates maximize Shannon entropy under varying noise, revealing that no single inequality is universally optimal; BC$_3$ often performs best at moderate noise, while CHSH can dominate at low noise and modulations vary with parameters. They additionally derive analytic lower bounds on $H$ from per-outcome bounds and CHSH-based guessing probabilities, and they extend the analysis to a three-dimensional qudit scenario using the BM functional, highlighting both the potential and limits of Shannon-entropy certification in DI QRNGs. Overall, the work provides a practical, relax-and-bound DI framework for Shannon-entropy randomness certification with insights into convexity, constraints, and qudit extensions, enabling improved DI security proofs and extractor design.

Abstract

Quantum technologies promise information processing and communication technology advancements, including random number generation (RNG). Using Bell inequalities, a user of a quantum RNG hardware can certify that the values provided by an untrusted device are truly random. This problem has been extensively studied for von Neumann and min-entropy as a measure of randomness. However, in this paper, we analyze the feasibility of such verification for Shannon entropy. We investigate how the usability of various Bell inequalities differs depending on the presence of noise. Moreover, we present the benefit of certification for Shannon compared to min-entropy, as well as the tight analytical lower bound for Shannon entropy in randomness certification.

Figures (12)

• Figure 1: (color online) Shannon entropy-based min-entropy sequence lower bounds based on \ref{['eq:prep1']}, per event in the sequence, compared to the min-entropy for a single event. Three statistical distance bounds $\varepsilon$ has been analyzed: (a) $\varepsilon = 10^{-8}$; (b) $\varepsilon = 10^{-12}$; (c) $\varepsilon = 10^{-16}$. In (a), for $k \approx 30k$ (and $k \approx 45k$ in (b), and $k \approx 62.5k$ for (c)), this lower bound (divided per event) exceeds the maximal single-event min-entropy value for CHSH.
• Figure 2: Brief description of our randomness certification method using Shannon entropy.
• Figure 3: (color online) Lower bounds on Shannon entropy for randomness certification provided by CHSH inequality, as well as \ref{['eq:modchsh']}, as a function of the level of noise.
• Figure 4: (color online) Lower bounds on Shannon entropy for randomness certification provided by the family of Braunstein-Caves inequalities, as a function of the level of noise.
• Figure 5: (color online) Lower bounds of Shannon entropy for randomness certification provided by the $I_1$ and $I_2$ inequalities, as a function of the level of noise.

Theorems & Definitions (1)

• Proposition 1