Device-independent certification of tensor products of quantum states using single-copy self-testing protocols

Ivan Šupić1, Daniel Cavalcanti2, and Joseph Bowles2

1Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

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


Self-testing protocols are methods to determine the presence of shared entangled states in a device independent scenario, where no assumptions on the measurements involved in the protocol are made. A particular type of self-testing protocol, called parallel self-testing, can certify the presence of copies of a state, however such protocols typically suffer from the problem of requiring a number of measurements that increases with respect to the number of copies one aims to certify. Here we propose a procedure to transform single-copy self-testing protocols into a procedure that certifies the tensor product of an arbitrary number of (not necessarily equal) quantum states, without increasing the number of parties or measurement choices. Moreover, we prove that self-testing protocols that certify a state and rank-one measurements can always be parallelized to certify many copies of the state. Our results suggest a method to achieve device-independent unbounded randomness expansion with high-dimensional quantum states.

► BibTeX data

► References

[1] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. Device-independent security of quantum cryptography against collective attacks. Physical Review Letters, 98(23):230501, 2007. doi:10.1103/​PhysRevLett.98.230501.

[2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86:419–478, Apr 2014. doi:10.1103/​RevModPhys.86.419.

[3] John Stewart Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195–200, 1964. URL: https:/​/​​record/​111654.

[4] Spencer Breiner, Amir Kalev, and Carl A. Miller. Parallel self-testing of the GHZ state with a proof by diagrams. In Peter Selinger and Giulio Chiribella, editors, Proceedings of the 15th International Conference on Quantum Physics and Logic, Halifax, Canada, 3-7th June 2018, volume 287 of Electronic Proceedings in Theoretical Computer Science, pages 43–66. Open Publishing Association, 2019. doi:10.4204/​EPTCS.287.3.

[5] Jean-Daniel Bancal, Miguel Navascués, Valerio Scarani, Tamás Vértesi, and Tzyh Haur Yang. Physical characterization of quantum devices from nonlocal correlations. Phys. Rev. A, 91:022115, Feb 2015. doi:10.1103/​PhysRevA.91.022115.

[6] 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, May 2015. doi:10.1103/​PhysRevA.91.052111.

[7] Joseph Bowles, Ivan Šupić, Daniel Cavalcanti, and Antonio Acín. Device-independent entanglement certification of all entangled states. Phys. Rev. Lett., 121:180503, Oct 2018. doi:10.1103/​PhysRevLett.121.180503.

[8] Joseph Bowles, Ivan Šupić, Daniel Cavalcanti, and Antonio Acín. Self-testing of Pauli observables for device-independent entanglement certification. Phys. Rev. A, 98:042336, Oct 2018. doi:10.1103/​PhysRevA.98.042336.

[9] Andrea Coladangelo, Alex B Grilo, Stacey Jeffery, and Thomas Vidick. Verifier-on-a-leash: new schemes for verifiable delegated quantum computation, with quasilinear resources. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 247–277. Springer, 2019. doi:10.1007/​978-3-030-17659-4_9.

[10] Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. All pure bipartite entangled states can be self-tested. Nature Communications, 8:15485, may 2017. doi:10.1038/​ncomms15485.

[11] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23:880–884, Oct 1969. doi:10.1103/​PhysRevLett.23.880.

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

[13] Matthew Coudron and Anand Natarajan. The parallel-repeated magic square game is rigid, 2016. arXiv:1609.06306. URL: https:/​/​​abs/​1609.06306.

[14] Andrea Coladangelo. Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH and the magic square game. Quantum Information and Computation, 17(9-10):831–865, 2017. doi:10.26421/​QIC17.9-10-6.

[15] Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick. Test for a large amount of entanglement, using few measurements. Quantum, 2:92, September 2018. doi:10.22331/​q-2018-09-03-92.

[16] Matthew Coudron and Henry Yuen. Infinite randomness expansion with a constant number of devices. In Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 427–436, New York, NY, USA, 2014. ACM. doi:10.1145/​2591796.2591873.

[17] Marissa Giustina, Marijn A. M. Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-Åke Larsson, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Morgan W. Mitchell, Jörn Beyer, Thomas Gerrits, Adriana E. Lita, Lynden K. Shalm, Sae Woo Nam, Thomas Scheidl, Rupert Ursin, Bernhard Wittmann, and Anton Zeilinger. Significant-loophole-free test of bell's theorem with entangled photons. Phys. Rev. Lett., 115:250401, Dec 2015. doi:10.1103/​PhysRevLett.115.250401.

[18] 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, and et al. Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526(7575):682–686, 2015. doi:10.1038/​nature15759.

[19] Rahul Jain, Carl A. Miller, and Yaoyun Shi. Parallel device-independent quantum key distribution. IEEE Transactions on Information Theory, 66(9):5567–5584, Sep 2020. doi:10.1109/​tit.2020.2986740.

[20] Amir Kalev and Carl A Miller. Rigidity of the magic pentagram game. Quantum Science and Technology, 3(1):015002, 2018. doi:10.1088/​1367-2630/​18/​2/​025021.

[21] Wen-Zhao Liu, Ming-Han Li, Sammy Ragy, Si-Ran Zhao, Bing Bai, Yang Liu, Peter J Brown, Jun Zhang, Roger Colbeck, Jingyun Fan, et al. Device-independent randomness expansion against quantum side information. Nature Physics, pages 1–4, 2021. doi:10.1038/​s41567-020-01147-2.

[22] Matthew McKague. Self-testing in parallel with CHSH. Quantum, 1:1, April 2017. doi:10.22331/​q-2017-04-25-1.

[23] Dominic Mayers and Andrew Yao. Self testing quantum apparatus. Quantum Info. Comput., 4:273, 2004. doi:10.26421/​QIC4.4-3.

[24] Miguel Navascués, Stefano Pironio, and Antonio Acín. Bounding the set of quantum correlations. Phys. Rev. Lett., 98:010401, Jan 2007. doi:10.1103/​PhysRevLett.98.010401.

[25] Miguel Navascués, Stefano Pironio, and Antonio Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics, 10(7):073013, 2008. doi:10.1088/​1367-2630/​10/​7/​073013.

[26] Anand Natarajan and Thomas Vidick. A quantum linearity test for robustly verifying entanglement. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 1003–1015, New York, NY, USA, 2017. ACM. doi:10.1145/​3055399.3055468.

[27] Anand Natarajan and Thomas Vidick. Low-degree testing for quantum states, and a quantum entangled games pcp for qma. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 731–742, Oct 2018. doi:10.1109/​FOCS.2018.00075.

[28] Dimiter Ostrev and Thomas Vidick. The structure of nearly-optimal quantum strategies for the CHSH (n) XOR games. Quantum Information & Computation, 16(13-14), pp.(13-14):1191–1211, 2016. doi:10.26421/​QIC16.13-14-6.

[29] Stefano Pironio, Antonio Acín, Serge Massar, A Boyer de La Giroday, Dzimitry N Matsukevich, Peter Maunz, Steven Olmschenk, David Hayes, Le Luo, T Andrew Manning, et al. Random numbers certified by Bell’s theorem. Nature, 464(7291):1021–1024, 2010. doi:10.1038/​nature09008.

[30] Sandu Popescu and Daniel Rohrlich. Which states violate Bell's inequality maximally? Physics Letters A, 169(6):411–414, 1992. doi:10.1016/​0375-9601(92)90819-8.

[31] Ben W. Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496:456, 2013. doi:10.1038/​nature12035.

[32] Lynden K. Shalm, Evan Meyer-Scott, Bradley G. Christensen, Peter Bierhorst, Michael A. Wayne, Martin J. Stevens, Thomas Gerrits, Scott Glancy, Deny R. Hamel, Michael S. Allman, Kevin J. Coakley, Shellee D. Dyer, Carson Hodge, Adriana E. Lita, Varun B. Verma, Camilla Lambrocco, Edward Tortorici, Alan L. Migdall, Yanbao Zhang, Daniel R. Kumor, William H. Farr, Francesco Marsili, Matthew D. Shaw, Jeffrey A. Stern, Carlos Abellán, Waldimar Amaya, Valerio Pruneri, Thomas Jennewein, Morgan W. Mitchell, Paul G. Kwiat, Joshua C. Bienfang, Richard P. Mirin, Emanuel Knill, and Sae Woo Nam. Strong loophole-free test of local realism. Phys. Rev. Lett., 115:250402, Dec 2015. doi:10.1103/​PhysRevLett.115.250402.

[33] Shubhayan Sarkar, Debashis Saha, Jedrzej Kaniewski, and Remigiusz Augusiak. Self-testing quantum systems of arbitrary local dimension with the minimal number of measurements, 2019. arXiv:1909.12722.

[34] Stephen J. Summers and Reinhard F. Werner. Maximal violation of Bell's inequalities is generic in quantum field theory. Communications in Mathematical Physics, 110(2):247–259, 1987. doi:10.1007/​BF01207366.

[35] Boris Tsirelson. Some results and problems on quantum Bell-type inequalities. Hadronis Journal Supplement, 8:329–45, 1993. URL: https:/​/​​naid/​10026857475/​en/​.

[36] Ivan Šupić and Joseph Bowles. Self-testing of quantum systems: a review. Quantum, 4:337, Sep 2020. doi:10.22331/​q-2020-09-30-337.

[37] Thomas Vidick. Parallel DIQKD from parallel repetition, 2017. arXiv:1703.08508.

[38] Xingyao Wu, Jean-Daniel Bancal, Matthew McKague, and Valerio Scarani. Device-independent parallel self-testing of two singlets. Phys. Rev. A, 93:062121, 2016. doi:10.1103/​PhysRevA.93.062121.

[39] Yukun Wang, Xingyao Wu, and Valerio Scarani. All the self-testings of the singlet for two binary measurements. New Journal of Physics, 18(2):025021, 2016. doi:10.1088/​1367-2630/​18/​2/​025021.

[40] Tzyh Haur Yang and Miguel Navascués. Robust self-testing of unknown quantum systems into any entangled two-qubit states. Phys. Rev. A, 87:050102, May 2013. doi:10.1103/​PhysRevA.87.050102.

[41] Tzyh Haur Yang, Tamás Vértesi, Jean-Daniel Bancal, Valerio Scarani, and Miguel Navascués. Robust and versatile black-box certification of quantum devices. Phys. Rev. Lett., 113:040401, Jul 2014. doi:10.1103/​PhysRevLett.113.040401.

Cited by

[1] Shubhayan Sarkar, Debashis Saha, Jędrzej Kaniewski, and Remigiusz Augusiak, "Self-testing quantum systems of arbitrary local dimension with minimal number of measurements", arXiv:1909.12722, npj Quantum Information 7 1, 151 (2021).

[2] Iris Agresti, Beatrice Polacchi, Davide Poderini, Emanuele Polino, Alessia Suprano, Ivan Šupić, Joseph Bowles, Gonzalo Carvacho, Daniel Cavalcanti, and Fabio Sciarrino, "Experimental Robust Self-Testing of the State Generated by a Quantum Network", PRX Quantum 2 2, 020346 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-22 10:14:06) and SAO/NASA ADS (last updated successfully 2021-10-22 10:14:07). The list may be incomplete as not all publishers provide suitable and complete citation data.