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.
 H.-J. Briegel, W. Dur, J. I. Cirac, and P. Zoller, Quantum repeaters: the role of imperfect local operations in quantum communication, Phys. Rev. Lett. 81, 5932 (1998).
 M. Epping, H. Kampermann, and D. Bruss, Large-scale quantum networks based on graphs, New J. Phys. 18, 053036 (2016).
 S. Das, S. Khatri, and J. P. Dowling, Robust quantum network architectures and topologies for entanglement distribution, Physical Review A 97, 012335 (2018).
 D. N. Matsukevich, T. Chanelière, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, Entanglement of remote atomic qubits, Phys. Rev. Lett. 96, 030405 (2006).
 C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, Functional quantum nodes for entanglement distribution over scalable quantum networks, Science 316, 1316 (2007).
 Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, Experimental demonstration of a bdcz quantum repeater node, Nature (London) 454, 1098 (2008).
 D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. Matsukevich, L.-M. Duan, and C. Monroe, Entanglement of single-atom quantum bits at a distance, Nature (London) 449, 68 (2007).
 D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenck, and C. Monroe, Bell inequality violation with two remote atomic qubits, Phys. Rev. Lett. 100, 150404 (2008).
 K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, Entangling macroscopic diamonds at room temperature, Science 334, 1253 (2011).
 S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, An elementary quantum network of single atoms in optical cavities, Nature (London) 484, 195 (2012).
 J. Hofmann, M. Krug, N. Ortegel, L. Gérard, M. Weber, W. Rosenfeld, and H. Weinfurter, Heralded entanglement between widely separated atoms, Science 337, 72 (2012).
 H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, Heralded entanglement between solid-state qubits separated by three metres, Nature 497, 86 (2013).
 J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
 D. Mayers and A. Yao, Quantum cryptography with imperfect apparatus, Proceeding FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science , 503 (1998).
 S. Pironio, A. Acin, S. Massar, A. Boyer 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).
 C. Bamps and S. Pironio, Sum-of-squares decompositions for a family of clauser-horne-shimony-holt-like inequalities and their application to self-testing, Phys. Rev. A 91, 052111 (2015).
 M. Tomamichel and E. Hänggi, The link between entropic uncertainty and nonlocality, J. Phys. A: Math. Theor. 46, 055301 (2013).
 C.-E. Bardyn, T. C. H. Liew, S. Massar, M. McKague, and V. Scarani, Device-independent state estimation based on Bell's inequalities, Phys. Rev. A 80, 062327 (2009).
 C. Miller and Y. Shi, Optimal robust self-testing by binary nonlocal xor games, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2013 22, 254 (2013).
 T. H. Yang, T. Vértesi, J.-D. Bancal, V. Scarani, and M. Navascués, Robust and versatile black-box certification of quantum devices, Phys. Rev. Lett. 113, 040401 (2014).
 J. Kaniewski, Analytic and nearly optimal self-testing bounds for the clauser-horne-shimony-holt and mermin inequalities, Phys. Rev. Lett. 117, 070402 (2016).
 I. Šupić, A. Coladangelo, R. Augusiak, and A. Acín, Self-testing multipartite entangled states through projections onto two systems, New Journal of Physics 20, 083041 (2018).
 F. Magniez, D. Mayers, M. Mosca, and H. Ollivier, Self-testing of quantum circuits, Proceedings of the 33rd International Colloquium on Automata, Lang Lecture Notes in Computer Science No. 4052, edited by Bugliesi , 72 (2006).
 J.-D. Bancal, M. Navascues, V. Scarani, T. Vertesi, and T. H. Yang, Physical characterization of quantum devices from nonlocal correlations, Phys. Rev. A 91, 022115 (2015).
 P. Sekatski, J.-D. Bancal, S. Wagner, and N. Sangouard, Certifying the building blocks of quantum computers from Bell's theorem, Phys. Rev. Lett. 121, 180505 (2018).
 T. R. Tan, Y. Wan, S. Erickson, P. Bierhorst, D. Kienzler, S. Glancy, E. Knill, D. Leibfried, and D. J. Wineland, Chained Bell inequality experiment with high-efficiency measurements, Phys. Rev. Lett. 118, 130403 (2017).
 X. Valcarce, P. Sekatski, D. Orsucci, E. Oudot, J.-D. Bancal, and N. Sangouard, What is the minimum CHSH score certifying that a state resembles the singlet? Quantum 4, 246 (2020).
 B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature (London) 526, 682 (2015).
 L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, Strong loophole-free test of local realism, Phys. Rev. Lett. 115, 250402 (2015).
 M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-c. A. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, Significant-loophole-free test of Bell's theorem with entangled photons, Phys. Rev. Lett. 115, 250401 (2015).
 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, Phys. Rev. Lett. 119, 010402 (2017).
 J. S. Bell, On the einstein podolsky rosen paradox, Physics 1, 195 (1964).
 M. Fürst, H. Weier, S. Nauerth, D. G. Marangon, C. Kurtsiefer, and H. Weinfurter, High speed optical quantum random number generation, Opt. Express 18, 13029–13037 (2010).
 D. Rosset, J.-D. Bancal, and N. Gisin, Classifying 50 years of Bell inequalities, J. Phys. A: Math Theor. 47, 424022 (2014).
 B. Hensen, N. Kalb, M. S. Blok, A. E. Dréau, A. Reiserer, R. F. L. Vermeulen, R. N. Schouten, M. Markham, D. J. Twitchen, K. Goodenough, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, Loophole-free Bell test using electron spins in diamond: second experiment and additional analysis, Scientific Reports 6, 30289 (2016).
 J. Hofmann, Heralded atom-atom entanglement, PhD thesis (2014).
 K. Redeker, Entanglement of single rubidium atoms: From a Bell test towards applications, PhD thesis (2020).
 W. Hoeffding, On the distribution of the number of successes in independent trials, Annals of Math. Stat. 27, 713 (1956).
 M. McKague, Self-testing in parallel, New J. Phys. 18, 045013 (2016b).
 Xavier Valcarce, Julian Zivy, Nicolas Sangouard, and Pavel Sekatski, "Self-testing two-qubit maximally entangled states from generalized Clauser-Horne-Shimony-Holt tests", Physical Review Research 4 1, 013049 (2022).
 Dian Wu, Qi Zhao, Xue-Mei Gu, Han-Sen Zhong, You Zhou, Li-Chao Peng, Jian Qin, Yi-Han Luo, Kai Chen, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan, "Robust Self-Testing of Multiparticle Entanglement", Physical Review Letters 127 23, 230503 (2021).
 Gaëtan Gras, Davide Rusca, Hugo Zbinden, and Félix Bussières, "Countermeasure Against Quantum Hacking Using Detection Statistics", Physical Review Applied 15 3, 034052 (2021).
 Aleksandra Gočanin, Ivan Šupić, and Borivoje Dakić, "Sample-Efficient Device-Independent Quantum State Verification and Certification", PRX Quantum 3 1, 010317 (2022).
 Joshua Morris, Valeria Saggio, Aleksandra Gočanin, and Borivoje Dakić, "Quantum Verification and Estimation with Few Copies", Advanced Quantum Technologies 5 5, 2100118 (2022).
 Marie Ioannou, Maria Ana Pereira, Davide Rusca, Fadri Grünenfelder, Alberto Boaron, Matthieu Perrenoud, Alastair A. Abbott, Pavel Sekatski, Jean-Daniel Bancal, Nicolas Maring, Hugo Zbinden, and Nicolas Brunner, "Receiver-Device-Independent Quantum Key Distribution", Quantum 6, 718 (2022).
 Dian Wu, Qi Zhao, Can Wang, Liang Huang, Yang-Fan Jiang, Bing Bai, You Zhou, Xue-Mei Gu, Feng-Ming Liu, Ying-Qiu Mao, Qi-Chao Sun, Ming-Cheng Chen, Jun Zhang, Cheng-Zhi Peng, Xiao-Bo Zhu, Qiang Zhang, Chao-Yang Lu, and Jian-Wei Pan, "Closing the Locality and Detection Loopholes in Multiparticle Entanglement Self-Testing", Physical Review Letters 128 25, 250401 (2022).
 Yukun Wang, Xinjian Liu, Shaoxuan Wang, Haoying Zhang, and Yunguang Han, "Robust one-sided self-testing of two-qubit states via quantum steering", Physical Review A 106 4, 042424 (2022).
 Ivan Šupić and Joseph Bowles, "Self-testing of quantum systems: a review", Quantum 4, 337 (2020).
 Francesco Graffitti, Alexander Pickston, Peter Barrow, Massimiliano Proietti, Dmytro Kundys, Denis Rosset, Martin Ringbauer, and Alessandro Fedrizzi, "Measurement-Device-Independent Verification of Quantum Channels", Physical Review Letters 124 1, 010503 (2020).
 Koon Tong Goh, Chithrabhanu Perumangatt, Zhi Xian Lee, Alexander Ling, and Valerio Scarani, "Experimental comparison of tomography and self-testing in certifying entanglement", Physical Review A 100 2, 022305 (2019).
 Davide Orsucci, Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski, "How post-selection affects device-independent claims under the fair sampling assumption", Quantum 4, 238 (2020).
 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?", Quantum 4, 246 (2020).
 Masahito Hayashi and Takeshi Koshiba, "Quantum verifiable protocol for secure modulo zero-sum randomness", Quantum Information Processing 21 8, 291 (2022).
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