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Necessary and Sufficient Condition for Randomness Certification from Incompatibility.

Yi Li1,2,3, Yu Xiang1, Jordi Tura4,5

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Summary
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This study identifies the necessary quantum resources for certified randomness using measurement incompatibility. It shows that specific measurement compatibility structures prevent randomness certification, guiding the development of more robust quantum random number generators.

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Area of Science:

  • Quantum Information Theory
  • Foundations of Quantum Mechanics

Background:

  • Certified randomness generation relies on Bell nonlocality or Einstein-Podolsky-Rosen (EPR) steering from uncharacterized devices.
  • Standard spot-checking protocols may require specific quantum resources beyond basic nonlocality for guaranteed randomness.

Purpose of the Study:

  • To establish the necessary and sufficient conditions for certified randomness in bipartite systems.
  • To identify the minimum quantum resources required for randomness certification.
  • To develop practical methods for detecting the possibility of certified randomness.

Main Methods:

  • Formulating the condition for certified randomness in terms of measurement incompatibility.
  • Analyzing measurement compatibility structures, specifically hypergraphs and star subgraphs.
  • Generalizing results to the Bell scenario using chained Bell inequalities.

Main Results:

  • Certified randomness is possible if and only if the correlations do not stem from a measurement compatibility structure isomorphic to a star subgraph.
  • A star subgraph structure, where a central measurement is compatible with peripheral ones, precludes certified randomness.
  • Violation of any chained Bell inequality confirms the absence of such a structure, validating randomness certification.

Conclusions:

  • The incompatibility structure of measurements is crucial for generating certified random numbers.
  • This work provides a framework for identifying the minimum quantum resources needed for reliable randomness certification.
  • Chained Bell inequalities serve as effective witnesses for randomness certification.