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Extensive search of Shannon entropy-based randomness certification protocols

Robert Okuła, Piotr Mironowicz

TL;DR

This work addresses the problem of device-independent randomness certification by evaluating a vast catalog of Bell expressions for a $(4,3)$ measurement configuration using Shannon entropy as the primary randomness metric. It introduces a scalable framework that models the source as a Werner state $\rho = p \tfrac{I}{4} + (1-p)|\psi\rangle\langle\psi|$, computes Tsirelson bounds via the NPA SDP hierarchy, and estimates entropy with both analytic upper-bound distributions $P_H$ and nonlinear optimization. The study identifies five notable protocols with favorable entropy across white-noise levels $p\in\{10^{-6},0.1,0.2\}$ and reveals that analytic entropy bounds generally upper-bound the true randomness, necessitating nonlinear refinement for precise quantification. Additionally, the authors extend the analysis to self-testing for boxes and quantify the Flex metric, finding limited box-certifiability at very low noise and its loss under higher noise, thereby informing robust DI-RNG protocol design.

Abstract

Quantum technologies offer significant advancements in information processing and communication, notably in the domain of random number generation (RNG). The use of Bell inequalities enables users to certify the randomness of outputs produced by untrusted quantum RNG devices. We present a method for quantitatively analyzing Bell expressions used to certify randomness in quantum systems. Using this method, we conducted a comprehensive analysis on more than half a million Bell expressions involving configurations with four measurement settings for one party and three for the other. We identified five notable examples based on entropy scores under varying levels of white noise. As an extension of these results, we further incorporate the concept of self-testing for boxes (Banacki et al 2022, New J. Phys. 24 083003), enabling a more comprehensive characterization of quantum correlations through the evaluation of $Boxes(α, B)$ and the corresponding measure $Flex(α, B)$.

Extensive search of Shannon entropy-based randomness certification protocols

TL;DR

This work addresses the problem of device-independent randomness certification by evaluating a vast catalog of Bell expressions for a measurement configuration using Shannon entropy as the primary randomness metric. It introduces a scalable framework that models the source as a Werner state , computes Tsirelson bounds via the NPA SDP hierarchy, and estimates entropy with both analytic upper-bound distributions and nonlinear optimization. The study identifies five notable protocols with favorable entropy across white-noise levels and reveals that analytic entropy bounds generally upper-bound the true randomness, necessitating nonlinear refinement for precise quantification. Additionally, the authors extend the analysis to self-testing for boxes and quantify the Flex metric, finding limited box-certifiability at very low noise and its loss under higher noise, thereby informing robust DI-RNG protocol design.

Abstract

Quantum technologies offer significant advancements in information processing and communication, notably in the domain of random number generation (RNG). The use of Bell inequalities enables users to certify the randomness of outputs produced by untrusted quantum RNG devices. We present a method for quantitatively analyzing Bell expressions used to certify randomness in quantum systems. Using this method, we conducted a comprehensive analysis on more than half a million Bell expressions involving configurations with four measurement settings for one party and three for the other. We identified five notable examples based on entropy scores under varying levels of white noise. As an extension of these results, we further incorporate the concept of self-testing for boxes (Banacki et al 2022, New J. Phys. 24 083003), enabling a more comprehensive characterization of quantum correlations through the evaluation of and the corresponding measure .

Paper Structure

This paper contains 12 sections, 19 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Randomness
  3. Randomness extraction and certification
  4. Entropy as the measure of randomness
  5. Bounding the probabilities
  6. Calculating entropy
  7. Results
  8. General analysis
  9. Selected protocols
  10. Boxes self-testing
  11. Conclusions
  12. Acknowledgements

Figures (3)

  • Figure 1: (color online) Difference between the Shannon entropy values obtained from \ref{['eq:hipo']} and those derived from nonlinear optimization, illustrating the overestimation of entropy by \ref{['eq:hipo']} at higher noise levels.
  • Figure 2: (color online) Certification of randomness using the Bell inequalities defined in Table \ref{['eq:best']}, for two measures of randomness. The values are obtained using the standard certification approach, which relies on the nonlinear optimization, constrained by the NPA-obtained bounds, as opposed to \ref{['eq:hipo']}.
  • Figure 3: (color online) A distribution of the Bell certificates, as definied by \ref{['eq:protocol']}. The certificates are classified based on the value of the measure of randomness (Shannon entropy) that the certificate yields. The histograms are presented for the white noise value $p = 1\mathrm{e}{-6}$, for both Shannon and min-entropy.