More entropy from shorter experiments using polytope approximations to the quantum set

TL;DR

This work tackles the finite-size limitations of device-independent QRNG by introducing two general-purpose algorithms that iteratively refine outer polytope approximations to the quantum set, guided by the device’s typical behavior and cryptographic intuition. By embedding these refinements into the probability estimation (PE) framework, the authors derive significantly tighter certified entropy bounds and demonstrate substantial entropy gains across bipartite and tripartite DI-QRNG protocols, including randomness amplification, with fewer device uses. They validate their approach on simulated and real experimental data and release a Python toolkit to construct polytope approximations and compute PE-based entropy bounds, enabling practical deployment. The method offers a scalable, ready-to-use route to higher entropy rates in real-world DI-QRNG and can be extended to semi-DI or more complex multi-party scenarios, subject to computational constraints like vertex enumeration.

Abstract

We introduce a systematic method for constructing polytope approximations to the quantum set in a variety of device-independent quantum random number generation (DI-QRNG) protocols. Our approach relies on two general-purpose algorithms that iteratively refine an initial outer-polytope approximation, guided by typical device behaviour and cryptographic intuition. These refinements strike a balance between computational tractability and approximation effectiveness. By integrating these approximations into the probability estimation (PE) framework [Zhang et al., PRA 2018], we obtain significantly improved certified entropy bounds in the finite-size regime. We test our method on various bipartite and tripartite DI-QRNG protocols, using both simulated and experimental data. In all cases, it yields notably higher entropy rates with fewer device uses than the existing techniques. We further extend our analysis to the more demanding task of randomness amplification, demonstrating major performance gains without added complexity. These results offer an effective and ready-to-use method to prove security-with improved certified entropy rates-in the most common practical DI-QRNG protocols. Our algorithms and entropy certification with PE tools are publicly available under a non-commercial license at https://github.com/CQCL/PE_polytope_approximation.

Figures (6)

• Figure 1: Schematic of DI randomness certification protocol. An entropy source $\mathbf{Z}$ is used to sequentially challenge $n$ times a quantum device, composed of non-communicating sub-units, which for each challenge $z_i$ return an answer $c_i$. The data $\mathbf{(c,z)}$ is evaluated by a pre-defined linear entropy witness $W$. If an entropy larger than a (pre-defined) threshold $t$ is witnessed, the protocol produces a certified entropy source with at least $t-k$ bits. The constant $k$ is typically a small number that depends on the witness design and on a chosen security parameter $\varepsilon$ (see for instance \ref{['eq:extractsmoothH']}).
• Figure 2: Illustrations of algorithms NearV (\ref{['alg:closestEP']}) and MaxGP (\ref{['alg:gp']}). The blue region represents an approximation to the quantum set $\mathcal{Q}$ given by some level of the NPA hierarchy while the yellow lines represent the boundaries of a polytope $\mathcal{P}_{\rm in}$ containing $\mathcal{Q}$. Our aim is to generate a finer outer-polytope approximation $\mathcal{P}_Q$ to $\mathcal{Q}$ taking into account the typical conditional behaviour $\vec{p}$ (black dot). (a) NearV (Algorithm \ref{['alg:closestEP']}): Randomly choose one vertex $\vec{v}_{\rm near}$ from the set of $m$ nearest non-quantum vertices to $\vec{p}$ and generate a quantum Bell inequality $(\vec{b},\beta)$ (half-space delimited by the orange line, tangent to $\mathcal{Q}$) defined by the vector between $\vec{v}_{\rm near}$ and its closest $\vec{q}\in\mathcal{Q}$ (the grey dotted line). The output polytope $\mathcal{P}_Q$ is the intersection of $\mathcal{P}_{\rm in}$ with the quantum Bell inequality $(\vec{b},\beta)$. (b) MaxGP (Algorithm \ref{['alg:gp']}): Solve optimisation \ref{['eq:GP_A_givenxbar']} for $\vec{p}$ and a set of allowed behaviours $\mathcal{P}_{\rm in}$ and obtains the optimal adversarial guessing strategies $\{\vec{\mu}_\lambda\}_\lambda$ (in the illustration, red points with $|E|=2$). The algorithm uses each $\vec{\mu}_\lambda$ to generate a quantum Bell inequality (delimited by a red line) in a similar method to Algorithm \ref{['alg:closestEP']}. The new polytope $\mathcal{P}_Q$ is the intersection of $\mathcal{P}_{\rm in}$ with at least one quantum Bell inequality.
• Figure 3: Extractable entropy rates from noisy CHSH correlations for different levels of white noise. Solid lines are obtained through the PE technique using different sets of allowed conditional behaviours: (orange) $\mathcal{P}_Q$ from 10 iterations of NearV (Algorithm \ref{['alg:closestEP']} with $m=10$); (blue) $\mathcal{P}_Q$ from 10 iterations of MaxGP (Algorithm \ref{['alg:gp']}); (green) $\mathcal{P}_Q=\mathcal{NS}$; and (red) $\mathcal{P}_Q=\mathcal{NS} \cap \textrm{CHSH} \leq 2\sqrt 2$, which coincides with set used in knill2020generation. The purple dashed line is obtained using the EAT approach arnon2018practical and the grey dotted line represents the method pironio2013securitynieto2014using using the Azuma-Hoeffding inequality. The figure is produced using level 2 of the NPA hierarchy.
• Figure 4: Extractable entropy rates from noisy asymmetric CHSH correlations ($\alpha=8$) for different levels of white noise. Solid lines are obtained through the PE technique using sets of allowed conditional behaviours: (orange) $\mathcal{P}_Q$ from 10 iterations of NearV (Algorithm \ref{['alg:closestEP']} with $m=10$); (blue) $\mathcal{P}_Q$ from 10 iterations of MaxPG (Algorithm \ref{['alg:gp']}); (red) - $\mathcal{P}_Q=\mathcal{NS} \cap \textrm{CHSH} \leq 2\sqrt 2$ (set defined according to the approach in knill2020generation); and (green) - $\mathcal{P}_Q=\mathcal{NS} \cap \textrm{CHSH} \leq 2\sqrt 2 \cap \textrm{CHSH}_{\alpha=8}\leq TB$ (Tsirelson's bound) (set used in bierhorst2020tsirelson). The purple dashed line is obtained using the EAT approach arnon2018practical, and the grey dotted line uses the method in pironio2013securitynieto2014using, which uses the Azuma-Hoeffding inequality. The figure is produced using level 2 of the NPA hierarchy.
• Figure 5: Extractable entropy rates from noisy Mermin correlations obtained from Table \ref{['tab:Mermin_data']} as a function of the number of rounds $n$. Solid lines correspond to lower bounds computed with the PE technique using $\mathcal{P}_Q$ obtained through: (orange) NearV algorithm (Algorithm \ref{['alg:closestEP']} with $m=1$); (blue) MaxGP algorithm (Algorithm \ref{['alg:gp']}); (green) $\mathcal{P}_Q=\mathcal{NS}$; (red) $\mathcal{P}_Q=\mathcal{NS} \cap (\textrm{lifted-CHSH}\leq 2\sqrt{2}$), where the lifted-CHSH is a generalisation of CHSH for 3 parties pironio2005lifting. The grey dotted line is obtained by approach pironio2013security using analytical optimal $P_{\rm guess}$ as a function of the Mermin value Woodhead2018randomnessversus. The figure is produced using level 2 of the NPA hierarchy.

Theorems & Definitions (22)

• Definition 1
• Corollary 1
• Theorem 1: Theorem 23 in knill2020generation
• Definition 2
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof