A Single Entangled System Is an Unbounded Source of Nonlocal Correlations and of Certified Random Numbers

The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities – entanglement, nonlocality and randomness – relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: Given a single entangled system, how many nonlocal correlations in a sequence can we obtain? And from this single entangled system, how many certified random numbers is it possible to generate? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from a single entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by nearmaximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme. 1998 ACM Subject Classification G.3 Probability and Statistics ∗ This work is supported by the ERC CoG QITBOX and AdG OSYRIS, the AXA Chair in Quantum Information Science, Spanish MINECO (QIBEQI and SEV-2015-0522), Fundación Cellex, Generalitat de Catalunya (SGR 875 and Cerca Program). † M.J. acknowledges support from the Marie Curie COFUND action through the ICFOnest program. ‡ R. A. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 705109. § M.J.H. acknowledges support from the EPSRC (through the NQIT Quantum Hub), the FQXi Large Grant Thermodynamic vs information theoretic entropies in probabilistic theories, and the hospitality of the Department of Computer Science at the University of Oxford. ¶ P.W. acknowledges computational resources granted by the High Performance Computing Center North (SNIC 2015/1-162 and SNIC 2016/1-320). © Florian J. Curchod, Markus Johansson, Remigiusz Augusiak, Matty J. Hoban, Peter Wittek, and Antonio Acín; licensed under Creative Commons License CC-BY 12th Conference on the Theory of Quantum Computation, Communication, and Cryptography (TQC 2017). Editor: Mark M. Wilde; Article No. 1; pp. 1:1–1:23 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 1:2 Single Entangled System Is an Unbounded Source of Nonlocality and Randomness