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

TL;DR

This paper presents a new method for constructing polytope approximations of the quantum set that improves entropy bounds in device-independent quantum random number generation, reducing the number of device uses needed for high entropy.

Contribution

It introduces algorithms for polytope approximation of the quantum set that enhance entropy certification in DI-QRNG protocols, applicable to both simulated and experimental data.

Findings

Higher entropy rates achieved with fewer device uses.

Significant improvements in randomness amplification performance.

Method is practical and publicly available.

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.