# How post-selection affects device-independent claims under the fair sampling assumption

Davide Orsucci1, Jean-Daniel Bancal1,2, Nicolas Sangouard1, and Pavel Sekatski1

1Quantum Optics Theory Group, Universität Basel, CH-4056 Basel, Switzerland
2Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

### Abstract

Device-independent certifications employ Bell tests to guarantee the proper functioning of an apparatus from the sole knowledge of observed measurement statistics, i.e. without assumptions on the internal functioning of the devices. When these Bell tests are implemented with devices having too low efficiency, one has to post-select the events that lead to successful detections and thus rely on a fair sampling assumption. The question that we address in this paper is what remains of a device-independent certification under fair sampling. We provide an intuitive description of post-selections in terms of $filters$ and define the fair sampling assumption as a property of these filters, equivalent to the definition introduced in Ref. [1]. When this assumption is fulfilled, the post-selected data is reproduced by an ideal experiment where lossless devices measure a $filtered$ state which can be obtained from the $actual$ state via local probabilistic maps. Trusted conclusions can thus be obtained on the quantum properties of this filtered state and the corresponding measurement statistics can reliably be used, e.g., for randomness generation or quantum key distribution. We also explore a stronger notion of fair sampling leading to the conclusion that the post-selected data is a fair representation of the data that would be obtained with lossless detections. Furthermore, we show that our conclusions hold in cases of small deviations from exact fair sampling. Finally, we describe setups previously or potentially used in Bell-type experiments under fair sampling and identify the underlying device-specific assumptions.

The year 2015 marked a significant step forward in the field of quantum information, as the long-standing goal of performing loophole-free experimental violations of a Bell inequality was finally achieved. These experiments have shown, beyond any reasonable doubt, that quantum mechanics cannot be explained by any local hidden-variable classical theory. But this was not the end of the story, as we are now entering the era where Bell tests could find application as a technological tool in a very exciting new spin-off of quantum information: device-independent certifications.

Device-independence is yet another surprising feature of quantum mechanics. Sometimes it is possible to guarantee that a set of devices are operating according to their intended specifications without ever needing to look inside'' to discover how exactly do they work. The core idea is that a violation --by a sufficiently large margin-- of a Bell inequality is enough to certify that entanglement must be present inside the devices that are used in the Bell test. These Bell tests can then be incorporated in more involved protocol that allow to perform practically useful tasks, such as the production of certified truly random numbers, or secure distribution of keys using only untrusted quantum devices.

There is however a difficult technological challenge ahead, namely, for many of these protocols to work the Bell inequality has to be violated by a large margin --the experimentally observed value of the Bell operator has to be much higher than the classically reachable value. This puts an extra burden in performing these protocols, to the extent that, for example, as of today there has been no experimental demonstration of a fully device-independent quantum key distribution protocol. The main obstacle to these goals is to reach a sufficiently high photon preparation and detector efficiency. The situation is similar to the status of experimental Bell tests in the past, when low efficiency was one of the main limiting factors to go fully loophole-free. Nonetheless, many Bell experiments with photons yielding a large Bell violation have been performed prior to 2015, by using a so-called fair sampling'' assumption. The idea is that, according to this assumption, the non-detection events are locally random and do not depend on the quantum state of the incoming photon, therefore post-selection of the cases where the detectors click results in a unbiased sample from the original data.

The question that we address in our work, then, is whether it is possible to use the fair sampling assumption also in the new context of device-independent certifications to allow the implementation of experiments that would otherwise be impossible with the currently existing technology. The question is delicate, as one has to clearly distinguish between the properties of the devices that are assumed to be known, in order to satisfy fair sampling, from the properties that we want to certify in a device-independent fashion. In our work we show that such a distinction is possible and, also, that only a partial physical characterization of the device is required to be confident that fair sampling holds. We introduce a entirely quantum and general formulation of fair-sampling in which the proof of security and correctness is particularly simple and appealing, stemming from an equivalence between the experiment in the lab'' and an ideal quantum experiment using lossless photon sources and detectors. Moreover, we perform a careful analysis of what happens in the case in which the assumption is only approximately satisfied, which will always happen in any real-world situation. We believe that our study of approximate fair sampling could be particularly useful for the analysis of upcoming experiments, since this aspect has been neglected in several recent proof-of-principle demonstrations of device-independent protocols that have assumed that fair sampling holds exactly.

### ► References

[1] D. W. Berry, H. Jeong, M. Stobińska, and T. C. Ralph, Fair-sampling assumption is not necessary for testing local realism, Physical Review A 81(1), 012109 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.012109

[2] R. H. Hadfield, Single-photon detectors for optical quantum information applications, Nature Photonics 3(12), 696–705 (2009).
https:/​/​doi.org/​10.1038/​nphoton.2009.230

[3] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters 23(15), 880–884 (1969).
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[4] J. F. Clauser, and M. A. Horne, Experimental consequences of objective local theories, Physical Review D 10(2), 526–535 (1974).
https:/​/​doi.org/​10.1103/​PhysRevD.10.526

[5] P. M. Pearle, Hidden-variable example based upon data rejection, Physical Review D 2(8), 1418–1425 (1970).
https:/​/​doi.org/​10.1103/​PhysRevD.2.1418

[6] A. Garg, and N. D. Mermin, Detector inefficiencies in the Einstein-Podolsky-Rosen experiment, Physical Review D 35(12), 3831–3835 (1987).
https:/​/​doi.org/​10.1103/​PhysRevD.35.3831

[7] S. Massar, and S. Pironio, Violation of local realism versus detection efficiency, Physical Review A 68(6), 062109 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.68.062109

[8] E. Pomarico, B. Sanguinetti, P. Sekatski, H. Zbinden, and N. Gisin, Experimental amplification of an entangled photon: what if the detection loophole is ignored? New Journal of Physics 13(6), 063031 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​13/​6/​063031

[9] I. Gerhardt, Q. Liu, A. Lamas-Linares, J. Skaar, V. Scarani, V. Makarov, and C. Kurtsiefer, Experimentally faking the violation of Bell’s inequalities, Physical Review Letters 107(17), 170404 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.170404

[10] J. Romero, D. Giovannini, D. S. Tasca, S. M. Barnett, and M. J. Padgett, Tailored two-photon correlation and fair-sampling: a cautionary tale, New Journal of Physics 15(8), 083047 (2013).
https:/​/​doi.org/​10.1088/​1367-2630/​15/​8/​083047

[11] J. Jogenfors, A. M. Elhassan, J. Ahrens, M. Bourennane, and J. Å. Larsson. Hacking the Bell test using classical light in energy-time entanglement-based quantum key distribution, Science Advances 1(11), 1500793 (2015).

[12] J. Jogenfors, Breaking the Unbreakable: Exploiting Loopholes in Bell’s Theorem to Hack Quantum Cryptography, Linköping University Electronic Press (2017).
https:/​/​doi.org/​10.3384/​diss.diva-140912

[13] B. Hensen et. al., Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682 (2015).
https:/​/​doi.org/​10.1038/​nature15759

[14] L. K. Shalm et. al., Strong loophole-free test of local realism, Physical Review Letters 115(25), 250402 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250402

[15] M. Giustina et. al., Significant-loophole-free test of Bell’s theorem with entangled photons, Physical Review Letters 115(25), 250401 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250401

[16] W. Rosenfeld, D. Burchardt, R. Garthoff, K. Redeker, N. Ortegel, M. Rau, and H. Weinfurter, Event-ready Bell test using entangled atoms simultaneously closing detection and locality loopholes, Physical Review Letters 119(1), 010402 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.010402

[17] M. H. Li, C. Wu, Y. Zhang, W. Z. Liu, B. Bai, Y. Liu, W. Zhang, Q. Zhao, H. Li, Z. Wang, L. You, W. J. Munro, J. Yin, J. Zhang, C.-Z. Peng, X. Ma, Q. Zhang, J. Fan, and J.-W. Pan, Test of local realism into the past without detection and locality loopholes, Physical Review Letters 121(8), 080404 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.080404

[18] D. Mayers, and A. Yao, Self testing quantum apparatus, arXiv:quant-ph/​0307205.
arXiv:quant-ph/0307205

[19] I. Šupić, and J. Bowles, Self-testing of quantum systems: a review, arxiv:quant-ph/​1904.10042.
arXiv:1904.10042

[20] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-independent security of quantum cryptography against collective attacks, Physical Review Letters 98(23), 230501 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.230501

[21] S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, Device-independent quantum key distribution secure against collective attacks, New Journal of Physics 11(4), 045021 (2009).
https:/​/​doi.org/​10.1088/​1367-2630/​11/​4/​045021

[22] M. McKague, T. H. Yang, and V. Scarani, Robust self-testing of the singlet, Journal of Physics A: Mathematical and Theoretical 45(45), 455304 (2012).
https:/​/​doi.org/​10.1088/​1751-8113/​45/​45/​455304

[23] J. Kaniewski, Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities, Physical Review Letters 117(7), 070402 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.070402

[24] P. Sekatski, J.-D. Bancal, S. Wagner, and N. Sangouard, Certifying the building blocks of quantum computers from Bell’s theorem, Physical Review Letters 121(18), 180505 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.180505

[25] J.-D. Bancal, K. Redeker, P. Sekatski, W. Rosenfeld, and N. Sangouard, Device-independent certification of an elementary quantum network link, arXiv:quant-ph/​1812.09117.
arXiv:1812.09117

[26] R. Colbeck, and A. Kent, Private randomness expansion with untrusted devices, Journal of Physics A: Mathematical and Theoretical 44(9), 095305 (2011).
https:/​/​doi.org/​10.1088/​1751-8113/​44/​9/​095305

[27] A. Acín, and L. Masanes, Certified randomness in quantum physics, Nature 540, 213 (2016).
https:/​/​doi.org/​10.1038/​nature20119

[28] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Violation of Bell's inequality under strict Einstein locality conditions, Physical Review Letters 81(23), 5039 (1998).
https:/​/​doi.org/​10.1103/​PhysRevLett.81.5039

[29] D. L. Moehring, M. J. Madsen, B. B. Blinov, and C. Monroe, Experimental Bell inequality violation with an atom and a photon, Physical Review Letters 93(9), 090410 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.93.090410

[30] S. Gómez, A. Mattar, I. Machuca, E. S. Gómez, D. Cavalcanti, O. J. Farías, A. Acín, and G. Lima, Experimental investigation of partially entangled states for device-independent randomness generation and self-testing protocols, Physical Review A 99(3), 032108 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.032108

[31] K. T. Goh, C. Perumangatt, Z. X. Lee, A. Ling, and V. Scarani, Experimental comparison of tomography and self-testing in certifying entanglement, Physical Review A 100(2), 022305 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.022305

[32] E. Polino, I. Agresti, D. Poderini, G. Carvacho, G. Milani, G. B. Lemos, R. Chaves, and F. Sciarrino, Device independent certification of a quantum delayed choice experiment, arXiv:quant-ph/​1806.00211.
https:/​/​doi.org/​10.1103/​PhysRevA.100.022111
arXiv:1806.00211

[33] I. Agresti, D. Poderini, L. Guerini, M. Mancusi, G. Carvacho, L. Aolita, D. Cavalcanti, R. Chaves, and F. Sciarrino, Experimental device-independent certified randomness generation with an instrumental causal structure, arXiv:quant-ph/​1905.02027.
arXiv:1905.02027

[34] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Reviews of Modern Physics 86(2), 419–478 (2014).
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[35] Q. Y. He, E. G. Cavalcanti, M. D. Reid, and P. D. Drummond, Bell inequalities for continuous-variable measurements, Physical Review A 81(6), 062106 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.062106

[36] M. A. Nielsen, and I. Chuang, Quantum computation and quantum information, Cambridge University Press (2002).

[37] D. Rosset, R. Ferretti-Schöbitz, J.-D. Bancal, N. Gisin, and Y.-C. Liang, Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses, Physical Review A 86(6), 062325 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.062325

[38] R. Schmied, J.-D. Bancal, B. Allard, M. Fadel, V. Scarani, P. Treutlein, and N. Sangouard, Bell Correlations in a Bose-Einstein Condensate, Science 352, 441 (2016).
https:/​/​doi.org/​10.1103/​PhysRevD.10.526

[39] N. Gisin, Hidden quantum nonlocality revealed by local filters, Physics Letters A 210(3), 151-156 (1996).
https:/​/​doi.org/​10.1016/​S0375-9601(96)80001-6

[40] R. Rabelo, M. Ho, D. Cavalcanti, N. Brunner, and V. Scarani, Device-independent certification of entangled measurements, Physical Review Letters 107(5), 050502 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.050502

[41] J.-D. Bancal, N. Sangouard, and P. Sekatski, Noise-resistant device-independent certification of Bell state measurements, Physical Review Letters 121(25), 250506 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.250506

[42] M. O. Renou, J. Kaniewski, and N. Brunner, Self-testing entangled measurements in quantum networks, Physical Review Letters 121(25), 250507 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.250507

[43] D. Cavalcanti, and P. Skrzypczyk, Quantum steering: a review with focus on semidefinite programming, Reports on Progress in Physics 80(2), 024001 (2016).
https:/​/​doi.org/​10.1088/​1361-6633/​80/​2/​024001

[44] J. Barrett, R. Colbeck, and A. Kent, Memory attacks on device-independent quantum cryptography, Physical Review Letters 110(1), 010503 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.010503

[45] J. Müller-Quade, and R. Renner, Composability in quantum cryptography, New Journal of Physics 11(8), 085006 (2009).
https:/​/​doi.org/​10.1088/​1367-2630/​11/​8/​085006

[46] L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Hacking commercial quantum cryptography systems by tailored bright illumination, Nat. Photonics 4(10), 686-689 (2010).
https:/​/​doi.org/​10.1038/​nphoton.2010.214

[47] S. Pironio, A. Acín, S. Massar, A. B. de La Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random numbers certified by Bell’s theorem, Nature 464, 1021 (2010).
https:/​/​doi.org/​10.1038/​nature09008

[48] Z. Cao, H. Zhou, X. Yuan, and X. Ma, Source-independent quantum random number generation, Physical Review X 6(1), 011020 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.011020

[49] C. A. Miller, and Y. Shi, Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices, Journal of the ACM 63(4), 33 (2016).
https:/​/​doi.org/​10.1145/​2885493

[50] C. C. W. Lim, C. Portmann, M. Tomamichel, R. Renner, and N. Gisin, Device-independent quantum key distribution with local Bell test, Physical Review X 3(3), 031006 (2013).
https:/​/​doi.org/​10.1103/​PhysRevX.3.031006

[51] U. Vazirani, and T. Vidick, Fully device independent quantum key distribution, Physical Review Letters 113(14), 140501 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.140501

[52] R. Arnon-Friedman, R. Renner, and T. Vidick, Simple and tight device-independent security proofs, SIAM Journal on Computing 48(1), 181-225 (2019).
https:/​/​doi.org/​10.1137/​18M1174726

[53] A. Aspect, J. Dalibard, and G. Roger, Experimental test of Bell's inequalities using time-varying analyzers, Physical Review Letters 49(25), 1804 (1982).
https:/​/​doi.org/​10.1103/​PhysRevLett.49.1804

[54] Scully, M.O. and Zubairy, M.S., Quantum optics, Cambridge University Press (1999).

[55] F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: a review, , 016001 (2018).
https:/​/​doi.org/​10.1088Reports%20on%20Progress%20in%20Physics%20textbf82(1)

[56] S. Slussarenko, and G. J. Pryde, Photonic quantum information processing: a concise review, arXiv:quant-ph/​1907.06331.
https:/​/​doi.org/​10.1063/​1.5115814
arXiv:1907.06331

### Cited by

[1] Zhihao Bian, A. S. Majumdar, C. Jebarathinam, Kunkun Wang, Lei Xiao, Xiang Zhan, Yongsheng Zhang, and Peng Xue, "Experimental demonstration of one-sided device-independent self-testing of any pure two-qubit entangled state", Physical Review A 101 2, 020301 (2020).

[2] Santiago Tarrago Velez, Vivishek Sudhir, Nicolas Sangouard, and Christophe Galland, "Bell Correlation Spectroscopy of Collective Molecular Vibrations", arXiv:1912.04502.

The above citations are from SAO/NASA ADS (last updated successfully 2020-04-07 18:35:28). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2020-04-07 18:35:26).