The semi-device-independent approach provides a framework for prepare-and-measure quantum protocols using devices whose behavior must not be characterized nor trusted, except for a single assumption on the dimension of the Hilbert space characterizing the quantum carriers. Here, we propose instead to constrain the quantum carriers through a bound on the mean value of a well-chosen observable. This modified assumption is physically better motivated than a dimension bound and closer to the description of actual experiments. In particular, we consider quantum optical schemes where the source emits quantum states described in an infinite-dimensional Fock space and model our assumption as an upper bound on the average photon number in the emitted states. We characterize the set of correlations that may be exhibited in the simplest possible scenario compatible with our new framework, based on two energy-constrained state preparations and a two-outcome measurement. Interestingly, we uncover the existence of quantum correlations exceeding the set of classical correlations that can be produced by devices behaving in a purely pre-determined fashion (possibly including shared randomness). This feature suggests immediate applications to certified randomness generation. Along this line, we analyze the achievable correlations in several prepare-and-measure optical schemes with a mean photon number constraint and demonstrate that they allow for the generation of certified randomness. Our simplest optical scheme works by the on-off keying of an attenuated laser source followed by photocounting. It opens the path to more sophisticated energy-constrained semi-device-independent quantum cryptography protocols, such as quantum key distribution.
 D. Mayers and A. Yao, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Los Alamitos, 1998) pp. 503-509, arXiv:quant-ph/9809039.
 A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Physical Review Letters 98, 230501 (2007), arXiv:quant-ph/0702152.
 C. A. Miller and Y. Shi, in Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC '14 (ACM, New York, NY, USA, 2014) pp. 417-426, arXiv:1402.0489 [quant-ph].
 N. Brunner, S. Pironio, A. Acín, N. Gisin, A. A. Méthot, and V. Scarani, Physical Review Letters 100, 210503 (2008), arXiv:0802.0760 [quant-ph].
 T. Lunghi, J. B. Brask, C. C. W. Lim, Q. Lavigne, J. Bowles, A. Martin, H. Zbinden, and N. Brunner, Physical Review Letters 114, 150501 (2015), arXiv:1410.2790 [quant-ph].
 R. Cleve, P. Hoyer, B. Toner, and J. Watrous, in Proc. Annu. IEEE Conf. Comput. Complex. (IEEE, 2004) pp. 236-249, arXiv:quant-ph/0404076.
 A. C. Doherty, B. Toner, Y. C. Liang, and S. Wehner, in Proc. Annu. IEEE Conf. Comput. Complex. (IEEE, 2008) pp. 199-210, arXiv:0803.4373 [quant-ph].
 T. Van Himbeeck et al., in preparation.
 S. Pironio, A. Acín, S. Massar, A. Boyer de La Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Nature 464, 1021 (2010), arXiv:0911.3427 [quant-ph].
 J. B. Brask, A. Martin, W. Esposito, R. Houlmann, J. Bowles, H. Zbinden, and N. Brunner, Physical Review A 7, 054018 (2017), arXiv:1612.06566 [quant-ph].
 Jonatan Bohr Brask, Anthony Martin, William Esposito, Raphael Houlmann, Joseph Bowles, Hugo Zbinden, and Nicolas Brunner, "Megahertz-Rate Semi-Device-Independent Quantum Random Number Generators Based on Unambiguous State Discrimination", Physical Review Applied 7 5, 054018 (2017).
 Mathieu Bozzio, Eleni Diamanti, and Frédéric Grosshans, "Semi-device-independent quantum money with coherent states", arXiv:1812.09256 (2018).
 Yukun Wang, Ignatius William Primaatmaja, Emilien Lavie, Antonios Varvitsiotis, and Charles Ci Wen Lim, "Characterising the correlations of prepare-and-measure quantum networks", npj Quantum Information 5 1, 17 (2019).
 Yanbao Zhang, Emanuel Knill, and Peter Bierhorst, "Certifying quantum randomness by probability estimation", Physical Review A 98 4, 040304 (2018).
 Armin Tavakoli, Jędrzej Kaniewski, Tamás Vértesi, Denis Rosset, and Nicolas Brunner, "Self-testing quantum states and measurements in the prepare-and-measure scenario", Physical Review A 98 6, 062307 (2018).
 Marco Avesani, Davide G. Marangon, Giuseppe Vallone, and Paolo Villoresi, "Source-device-independent heterodyne-based quantum random number generator at 17 Gbps", Nature Communications 9 1, 5365 (2018).
 Mathieu Bozzio, Eleni Diamanti, and Frédéric Grosshans, "Semi-device-independent quantum money with coherent states", Physical Review A 99 2, 022336 (2019).
 Alastair A Abbott, Cristian S Calude, Michael J Dinneen, and Nan Huang, "Experimentally probing the algorithmic randomness and incomputability of quantum randomness", Physica Scripta 94 4, 045103 (2019).
 Jiajun Ma, Aishwarya Hakande, Xiao Yuan, and Xiongfeng Ma, "Coherence as a resource for source-independent quantum random-number generation", Physical Review A 99 2, 022328 (2019).
The above citations are from Crossref's cited-by service (last updated 2019-03-20 09:24:10) and SAO/NASA ADS (last updated 2019-03-20 09:24:11). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.