Self-testing with finite statistics enabling the certification of a quantum network link

Jean-Daniel Bancal1,2,3, Kai Redeker4, Pavel Sekatski3,2, Wenjamin Rosenfeld4,5, and Nicolas Sangouard1,3

1Université Paris-Saclay, CEA, CNRS, Institut de Physique Théorique, 91191, Gif-sur-Yvette, France
2Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
3Quantum Optics Theory Group, Universität Basel, CH-4056 Basel, Switzerland
4Fakultät für Physik, Ludwig-Maximilians-Universität, 80799 München, Germany
5Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany

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Self-testing is a method to certify devices from the result of a Bell test. Although examples of noise tolerant self-testing are known, it is not clear how to deal efficiently with a finite number of experimental trials to certify the average quality of a device without assuming that it behaves identically at each run. As a result, existing self-testing results with finite statistics have been limited to guarantee the proper working of a device in just one of all experimental trials, thereby limiting their practical applicability. We here derive a method to certify through self-testing that a device produces states on average close to a Bell state without assumption on the actual state at each run. Thus the method is free of the I.I.D. (independent and identically distributed) assumption. Applying this new analysis on the data from a recent loophole-free Bell experiment, we demonstrate the successful distribution of Bell states over 398 meters with an average fidelity of $\geq$55.50% at a confidence level of 99%. Being based on a Bell test free of detection and locality loopholes, our certification is evidently device-independent, that is, it does not rely on trust in the devices or knowledge of how the devices work. This guarantees that our link can be integrated in a quantum network for performing long-distance quantum communications with security guarantees that are independent of the details of the actual implementation.

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[5] Davide Orsucci, Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski, "How post-selection affects device-independent claims under the fair sampling assumption", arXiv:1908.11123.

[6] Xavier Valcarce, Pavel Sekatski, Davide Orsucci, Enky Oudot, Jean-Daniel Bancal, and Nicolas Sangouard, "What is the minimum CHSH score certifying that a state resembles the singlet?", arXiv:1910.04606.

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