Tilted Hardy paradoxes for device-independent randomness extraction

Shuai Zhao1, Ravishankar Ramanathan1, Yuan Liu1, and Paweł Horodecki2,3

1Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
2International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
3Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The device-independent paradigm has had spectacular successes in randomness generation, key distribution and self-testing, however most of these results have been obtained under the assumption that parties hold trusted and private random seeds. In efforts to relax the assumption of measurement independence, Hardy's non-locality tests have been proposed as ideal candidates. In this paper, we introduce a family of tilted Hardy paradoxes that allow to self-test general pure two-qubit entangled states, as well as certify up to $1$ bit of local randomness. We then use these tilted Hardy tests to obtain an improvement in the generation rate in the state-of-the-art randomness amplification protocols for Santha-Vazirani (SV) sources with arbitrarily limited measurement independence. Our result shows that device-independent randomness amplification is possible for arbitrarily biased SV sources and from almost separable states. Finally, we introduce a family of Hardy tests for maximally entangled states of local dimension $4, 8$ as the potential candidates for DI randomness extraction to certify up to the maximum possible $2 \log d$ bits of global randomness.

We introduce a family of tilted Hardy paradoxes that enable the self-testing of general pure two-qubit entangled states and the certification of up to $1$ bit of local randomness. Utilizing these tilted Hardy tests, we achieve enhanced generation rates in the state-of-the-art randomness amplification protocols for Santha-Vazirani (SV) sources with arbitrarily limited measurement independence. Our findings show that device-independent randomness amplification is possible for arbitrarily biased SV sources and from almost separable states.

► BibTeX data

► References

[1] Albert Einstein, Boris Podolsky, and Nathan Rosen. ``Can quantum-mechanical description of physical reality be considered complete?'' Phys. Rev. 47, 777 (1935).
https:/​/​doi.org/​10.1103/​PhysRev.47.777

[2] Erwin Schrödinger. ``Discussion of probability relations between separated systems.'' Cambridge University Press. (1935).
https:/​/​doi.org/​10.1017/​S0305004100013554

[3] Jonathan Barrett, Lucien Hardy, and Adrian Kent. ``No signaling and quantum key distribution.'' Phys. Rev. Lett. 95, 010503 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.95.010503

[4] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. ``Device-independent security of quantum cryptography against collective attacks.'' Phys. Rev. Lett. 98, 230501 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.230501

[5] Stefano Pironio, Antonio Acín, Serge Massar, A. Boyer de La Giroday, Dzmitry N. Matsukevich, Peter Maunz, Steven Olmschenk, David Hayes, Le Luo, T. Andrew Manning, and C. Monroe. ``Random numbers certified by Bell’s theorem.'' Nature 464, 1021–1024 (2010) (2010).
https:/​/​doi.org/​10.1038/​nature09008

[6] Stefano Pironio and Serge Massar. ``Security of practical private randomness generation.'' Phys. Rev. A 87, 012336 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.012336

[7] Dominic Mayers and Andrew Yao. ``Quantum cryptography with imperfect apparatus.'' Proceedings 39th Annual Symposium on Foundations of Computer Science, pages 503–509 (1998).
https:/​/​doi.org/​10.1109/​SFCS.1998.743501

[8] Dominic Mayers and Andrew Yao. `` Self testing quantum apparatus.'' Quantum Info. Comput. 4(4), 273–286 (2004).
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0307205
arXiv:quant-ph/0307205

[9] Ivan Šupić and Joseph Bowles. ``Self-testing of quantum systems: a review.'' Quantum 4, 337 (2020).
https:/​/​doi.org/​10.22331/​q-2020-09-30-337

[10] Koon Tong Goh, Chithrabhanu Perumangatt, Zhi Xian Lee, Alexander Ling, and Valerio Scarani. ``Experimental comparison of tomography and self-testing in certifying entanglement.'' Phys. Rev. A 100, 022305 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.022305

[11] Roger Colbeck and Renato Renner. ``Free randomness can be amplified.'' Nat. Phys. 8, 450–453 (2012).
https:/​/​doi.org/​10.1038/​nphys2300

[12] Rodrigo Gallego, Lluis Masanes, Gonzalo De La Torre, Chirag Dhara, Leandro Aolita, and Antonio Acín. ``Full randomness from arbitrarily deterministic events.'' Nat. Commun. 4, 2654 (2013).
https:/​/​doi.org/​10.1038/​ncomms3654

[13] Ravishankar Ramanathan, Fernando G. S. L. Brandão, Karol Horodecki, Michał Horodecki, Paweł Horodecki, and Hanna Wojewódka. ``Randomness amplification under minimal fundamental assumptions on the devices.'' Phys. Rev. Lett. 117, 230501 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.230501

[14] Fernando G. S. L. Brandão, Ravishankar Ramanathan, Andrzej Grudka, Karol Horodecki, Michał Horodecki, Paweł Horodecki, Tomasz Szarek, and Hanna Wojewódka. ``Realistic noise-tolerant randomness amplification using finite number of devices.'' Nat. Commun. 7, 11345 (2016).
https:/​/​doi.org/​10.1038/​ncomms11345

[15] Ravishankar Ramanathan, Michał Horodecki, Hammad Anwer, Stefano Pironio, Karol Horodecki, Marcus Grünfeld, Sadiq Muhammad, Mohamed Bourennane, and Paweł Horodecki. ``Practical no-signalling proof randomness amplification using Hardy paradoxes and its experimental implementation.'' arXiv:1810.11648 (2018).
https:/​/​doi.org/​10.48550/​arXiv.1810.11648
arXiv:1810.11648

[16] Max Kessler and Rotem Arnon-Friedman. ``Device-independent randomness amplification and privatization.'' IEEE Journal on Selected Areas in Information Theory 1(2), 568–584 (2020).
https:/​/​doi.org/​10.1109/​JSAIT.2020.3012498

[17] Miklos Santha and Umesh V. Vazirani. ``Generating quasi-random sequences from semi-random sources.'' Journal of Computer and System Sciences 33(1), 75–87 (1986).
https:/​/​doi.org/​10.1016/​0022-0000(86)90044-9

[18] Antonio Acín, Serge Massar, and Stefano Pironio. ``Randomness versus nonlocality and entanglement.'' Phys. Rev. Lett. 108, 100402 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.100402

[19] Cédric Bamps and Stefano 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).
https:/​/​doi.org/​10.1103/​PhysRevA.91.052111

[20] Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. ``All pure bipartite entangled states can be self-tested.'' Nat. Commun. 8, 15485 (2017).
https:/​/​doi.org/​10.1038/​ncomms15485

[21] Cédric Bamps, Serge Massar, and Stefano Pironio. ``Device-independent randomness generation with sublinear shared quantum resources.'' Quantum 2, 86 (2018).
https:/​/​doi.org/​10.22331/​q-2018-08-22-86

[22] Florian J. Curchod, Markus Johansson, Remigiusz Augusiak, Matty J. Hoban, Peter Wittek, and Antonio Acín. ``Unbounded randomness certification using sequences of measurements.'' Phys. Rev. A 95, 020102 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.020102

[23] Gilles Pütz, Denis Rosset, Tomer Jack Barnea, Yeong-Cherng Liang, and Nicolas Gisin. ``Arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality.'' Phys. Rev. Lett. 113, 190402 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.190402

[24] Ravishankar Ramanathan, Yuan Liu, and Paweł Horodecki. ``Large violations in Kochen Specker contextuality and their applications.'' New J. Phys. 24, 033035 (2022).
https:/​/​doi.org/​10.1088/​1367-2630/​ac3a84

[25] Lucien Hardy. ``Nonlocality for two particles without inequalities for almost all entangled states.'' Phys. Rev. Lett. 71, 1665 (1993).
https:/​/​doi.org/​10.1103/​PhysRevLett.71.1665

[26] Rafael Rabelo, Law Yun Zhi, and Valerio Scarani. ``Device-independent bounds for Hardy’s experiment.'' Phys. Rev. Lett. 109, 180401 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.180401

[27] Hong-Wei Li, Marcin Pawłowski, Ramij Rahaman, Guang-Can Guo, and Zheng-Fu Han. ``Device-and semi–device-independent random numbers based on noninequality paradox.'' Phys. Rev. A 92, 022327 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.022327

[28] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. ``Proposed experiment to test local hidden-variable theories.'' Phys. Rev. Lett. 23, 880 (1969).
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[29] Miguel Navascués, Stefano Pironio, and Antonio Acín. ``A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations.'' New J. Phys. 10 073013 (2008).
https:/​/​doi.org/​10.1088/​1367-2630/​10/​7/​073013

[30] Danilo Boschi, S Branca, Francesco De Martini, and Lucien Hardy. ``Ladder proof of nonlocality without inequalities: Theoretical and experimental results.'' Phys. Rev. Lett. 79, 2755 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.79.2755

[31] Ravishankar Ramanathan, Monika Rosicka, Karol Horodecki, Stefano Pironio, Michał Horodecki, and Paweł Horodecki. ``Gadget structures in proofs of the Kochen-Specker theorem.'' Quantum 4, 308 (2020).
https:/​/​doi.org/​10.22331/​q-2020-08-14-308

[32] Ravishankar Ramanathan, Paweł Horodecki, and Michał Banacki. ``No-signaling-proof randomness extraction from public weak sources.'' arXiv:2108.08819 (2021).
https:/​/​doi.org/​10.48550/​arXiv.2108.08819
arXiv:2108.08819

[33] Paul Moritz Cohn. ``Basic algebra: groups, rings and fields''. Springer London (2012).
https:/​/​doi.org/​10.1007/​978-0-85729-428-9

[34] Camille Jordan. ``Essai sur la géométrie à $ n $ dimensions.'' Bulletin de la S. M. F. 3, 103-174 (1875).
https:/​/​doi.org/​10.24033/​bsmf.90

[35] Ravishankar Ramanathan, Dardo Goyeneche, Sadiq Muhammad, Piotr Mironowicz, Marcus Grünfeld, Mohamed Bourennane, and Paweł Horodecki. ``Steering is an essential feature of non-locality in quantum theory.'' Nat. Commun. 9, 4244 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-06255-5

Cited by

[1] Abhishek Sadhu and Siddhartha Das, "Testing of quantum nonlocal correlations under constrained free will and imperfect detectors", Physical Review A 107 1, 012212 (2023).

[2] Ravishankar Ramanathan, "Finite Device-Independent Extraction of a Block Min-Entropy Source against Quantum Adversaries", arXiv:2304.09643, (2023).

[3] Yuan Liu, Ho Yiu Chung, and Ravishankar Ramanathan, "Investigations of the boundary of quantum correlations and device-independent applications", arXiv:2309.06304, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-03-02 11:13:07). 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 2024-03-02 11:13:05).