# Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems

1Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
2QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
3ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
4Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

### Abstract

Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve $d$ measurements settings, each having $d$ outcomes for an arbitrary prime number $d\geq 3$. We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.

Imagine a black box that produces random bits: 1, 0, 0, 0, 1, 0, 1, 1... We cannot open this box nor peek inside in order to see what is going on. Is there somebody inside tossing a coin every time the box produces a bit? Is there a written list of numbers known only to the box and the secret service of your country?

Let's analyze the situation: Having only access to the bits produced we can just perform statistics and play mathematics on them. And solely from the statistics of the box it is clearly impossible to say anything about its contents. End of story. Or, is it, really?

One of the most striking features of quantum physics is that, in some very particular situations, one can solve the above riddle. Through an experiment known as a Bell test, two or more boxes can become correlated in a very special way, which rules out all the above explanations, a phenomenon known as non-locality. When that Bell test gives the maximal score (maximal nonlocality), one can sometimes actually know what is inside the box!!! This is known as self-testing.

This statement may seem rather bold or puzzling. Of course, one could say that, if there is an atom of Caesium or an ion of Calcium inside the box that, in the process of generating the bits, behave exactly the same way, then it is impossible to tell them apart! But that is precisely the beauty of self-testing: it identifies all the states and measurements that behave the same way.

Pairs of particles prepared in the most quantum-correlated states are the easiest ones to self-test. These states are called maximally entangled and in their simpler version (qubits) the problem has been solved a long time ago: If the boxes inspect "uniformly" the state (performing the so-called mutually unbiased bases measurements), the correlations winning the Bell test with the best possible score are revealed. Then a mathematical proof shows that these states and measurements are the only ones that could have produced these correlations, thus revealing the relevant contents of the boxes.

However, for states of the same kind, albeit with more levels (qudits), designing a Bell test preserving these nice properties remains elusive. Here we manage to prove it for pairs of boxes that can produce an odd prime number of results each, and we provide a mathematical proof of the self-test when the number of possible results is three.

### ► References

[1] O. Andersson, P. Badziąg, I. Bengtsson, I. Dumitru, and A. Cabello. Self-testing properties of Gisin's elegant Bell inequality. Phys. Rev. A, 96: 032119, 2017. DOI:10.1103/​PhysRevA.96.032119.
https:/​/​doi.org/​10.1103/​PhysRevA.96.032119

[2] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett., 98: 230501, 2007. DOI:10.1103/​PhysRevLett.98.230501.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.230501

[3] R. Arnon-Friedman, F. Dupuis, O. Fawzi, R. Renner, and T. Vidick. Practical device-independent quantum cryptography via entropy accumulation. Nat. Commun., 9: 459, 2018. DOI:10.1038/​s41467-017-02307-4.
https:/​/​doi.org/​10.1038/​s41467-017-02307-4

[4] A. Acín, N. Gisin, and L. Masanes. From Bell's theorem to secure quantum key distribution. Phys. Rev. Lett., 97: 120405, 2006. DOI:10.1103/​PhysRevLett.97.120405.
https:/​/​doi.org/​10.1103/​PhysRevLett.97.120405

[5] A. Acín, S. Massar, and S. Pironio. Randomness versus nonlocality and entanglement. Phys. Rev. Lett., 108: 100402, 2012. DOI:10.1103/​PhysRevLett.108.100402.
https:/​/​doi.org/​10.1103/​PhysRevLett.108.100402

[6] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan. A new proof for the existence of mutually unbiased bases. Algorithmica, 34: 512, 2002. DOI:10.1007/​s00453-002-0980-7.
https:/​/​doi.org/​10.1007/​s00453-002-0980-7

[7] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419, 2014. DOI:10.1103/​RevModPhys.86.419.
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[8] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1: 195, 1964.

[9] J. Barrett, L. Hardy, and A. Kent. No signaling and quantum key distribution. Phys. Rev. Lett., 95: 010503, 2005. DOI:10.1103/​PhysRevLett.95.010503.
https:/​/​doi.org/​10.1103/​PhysRevLett.95.010503

[10] J. Barrett, A. Kent, and S. Pironio. Maximally nonlocal and monogamous quantum correlations. Phys. Rev. Lett., 97: 170409, 2006. DOI:10.1103/​PhysRevLett.97.170409.
https:/​/​doi.org/​10.1103/​PhysRevLett.97.170409

[11] 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. DOI:10.1103/​PhysRevA.80.062327.
https:/​/​doi.org/​10.1103/​PhysRevA.80.062327

[12] H. Buhrman and S. Massar. Causality and Tsirelson's bounds. Phys. Rev. A, 72: 052103, 2005. DOI:10.1103/​PhysRevA.72.052103.
https:/​/​doi.org/​10.1103/​PhysRevA.72.052103

[13] 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. DOI:10.1103/​PhysRevA.91.052111.
https:/​/​doi.org/​10.1103/​PhysRevA.91.052111

[14] N. Brunner, S. Pironio, A. Acín, N. Gisin, A. A. Méthot, and V. Scarani. Testing the dimension of Hilbert spaces. Phys. Rev. Lett., 100: 210503, 2008. DOI:10.1103/​PhysRevLett.100.210503.
https:/​/​doi.org/​10.1103/​PhysRevLett.100.210503

[15] J. Bouda, M. Pawłowski, M. Pivoluska, and M. Plesch. Device-independent randomness extraction from an arbitrarily weak min-entropy source. Phys. Rev. A, 90: 032313, 2014. DOI:10.1103/​PhysRevA.90.032313.
https:/​/​doi.org/​10.1103/​PhysRevA.90.032313

[16] M. Bavarian and P. W. Shor. Information causality, Szemerédi-Trotter and algebraic variants of CHSH. Proc. Conference on Innovations in Theoretical Computer Science, 2015. DOI:10.1145/​2688073.2688112.
https:/​/​doi.org/​10.1145/​2688073.2688112

[17] S.-L. Chen, C. Budroni, Y.-C. Liang, and Y.-N. Chen. Natural framework for device-independent quantification of quantum steerability, measurement incompatibility, and self-testing. Phys. Rev. Lett., 116: 240401, 2016. DOI:10.1103/​PhysRevLett.116.240401.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.240401

[18] D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett., 88: 040404, 2002. DOI:10.1103/​PhysRevLett.88.040404.
https:/​/​doi.org/​10.1103/​PhysRevLett.88.040404

[19] A. Coladangelo, K. T. Goh, and V. Scarani. All pure bipartite entangled states can be self-tested. Nat. Commun., 8: 15485, 2017. DOI:10.1038/​ncomms15485.
https:/​/​doi.org/​10.1038/​ncomms15485

[20] 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. DOI:10.1103/​PhysRevLett.23.880.
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[21] R. Colbeck and A. Kent. Private randomness expansion with untrusted devices. J. Phys. A: Math. Theor., 44: 095305, 2011. DOI:10.1088/​1751-8113/​44/​9/​095305.
https:/​/​doi.org/​10.1088/​1751-8113/​44/​9/​095305

[22] R. Colbeck. Quantum and relativistic protocols for secure multi-party computation. PhD thesis, University of Cambridge, 2006.

[23] A. Coladangelo. Generalization of the Clauser-Horne-Shimony-Holt inequality self-testing maximally entangled states of any local dimension. Phys. Rev. A, 98: 052115, 2018. DOI:10.1103/​PhysRevA.98.052115.
https:/​/​doi.org/​10.1103/​PhysRevA.98.052115

[24] D. Cavalcanti and P. Skrzypczyk. Quantitative relations between measurement incompatibility, quantum steering, and nonlocality. Phys. Rev. A, 93: 052112, 2016. DOI:10.1103/​PhysRevA.93.052112.
https:/​/​doi.org/​10.1103/​PhysRevA.93.052112

[25] A. Coladangelo and J. Stark. Robust self-testing for linear constraint system games. 2017.
arXiv:1709.09267

[26] A. Coladangelo and J. Stark. Unconditional separation of finite and infinite-dimensional quantum correlations. 2018.
arXiv:1804.05116

[27] K. Dykema, V. I. Paulsen, and J. Prakash. Non-closure of the set of quantum correlations via graphs. 2017.
arXiv:1709.05032

[28] J. I. de Vicente. Simple conditions constraining the set of quantum correlations. Phys. Rev. A, 92: 032103, 2015. DOI:10.1103/​PhysRevA.92.032103.
https:/​/​doi.org/​10.1103/​PhysRevA.92.032103

[29] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47: 777, 1935. DOI:10.1103/​PhysRev.47.777.
https:/​/​doi.org/​10.1103/​PhysRev.47.777

[30] K. T. Goh, J. Kaniewski, E. Wolfe, T. Vértesi, X. Wu, Y. Cai, Y.-C. Liang, and V. Scarani. Geometry of the set of quantum correlations. Phys. Rev. A, 97: 022104, 2018. DOI:10.1103/​PhysRevA.97.022104.
https:/​/​doi.org/​10.1103/​PhysRevA.97.022104

[31] S.-W. Ji, J. Lee, J. Lim, K. Nagata, and H.-W. Lee. Multisetting Bell inequality for qudits. Phys. Rev. A, 78: 052103, 2008. DOI:10.1103/​PhysRevA.78.052103.
https:/​/​doi.org/​10.1103/​PhysRevA.78.052103

[32] J. Kaniewski. Self-testing of binary observables based on commutation. Phys. Rev. A, 95: 062323, 2017. DOI:10.1103/​PhysRevA.95.062323.
https:/​/​doi.org/​10.1103/​PhysRevA.95.062323

[33] J. Kaniewski and S. Wehner. Device-independent two-party cryptography secure against sequential attacks. New J. Phys., 18: 055004, 2016. DOI:10.1088/​1367-2630/​18/​5/​055004.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​5/​055004

[34] Y.-C. Liang, C.-W. Lim, and D.-L. Deng. Reexamination of a multisetting Bell inequality for qudits. Phys. Rev. A, 80: 052116, 2009. DOI:10.1103/​PhysRevA.80.052116.
https:/​/​doi.org/​10.1103/​PhysRevA.80.052116

[35] J. Lim, J. Ryu, S. Yoo, C. Lee, J. Bang, and J. Lee. Genuinely high-dimensional nonlocality optimized by complementary measurements. New J. Phys., 12: 103012, 2010. DOI:10.1088/​1367-2630/​12/​10/​103012.
https:/​/​doi.org/​10.1088/​1367-2630/​12/​10/​103012

[36] T. Moroder, J.-D. Bancal, Y.-C. Liang, M. Hofmann, and O. Gühne. Device-independent entanglement quantification and related applications. Phys. Rev. Lett., 111: 030501, 2013. DOI:10.1103/​PhysRevLett.111.030501.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.030501

[37] M. McKague. Self-testing graph states. Theory of Quantum Computation, Communication, and Cryptography. TQC 2011. Lecture Notes in Computer Science, 6745: 104, 2014. DOI:10.1007/​978-3-642-54429-3_7.
https:/​/​doi.org/​10.1007/​978-3-642-54429-3_7

[38] M. McKague and M. Mosca. Generalized self-testing and the security of the 6-state protocol. Theory of Quantum Computation, Communication, and Cryptography. TQC 2010. Lecture Notes in Computer Science, 6519: 113, 2011. DOI:10.1007/​978-3-642-18073-6_10.
https:/​/​doi.org/​10.1007/​978-3-642-18073-6_10

[39] C. A. Miller and Y. Shi. Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices. J. ACM, 63: 33, 2016. DOI:10.1145/​2885493.
https:/​/​doi.org/​10.1145/​2885493

[40] D. Mayers and A. Yao. Quantum cryptography with imperfect apparatus. Proceedings 39th Annual Symposium on Foundations of Computer Science, 1998. DOI:10.1109/​SFCS.1998.743501.
https:/​/​doi.org/​10.1109/​SFCS.1998.743501

[41] D. Mayers and A. Yao. Self testing quantum apparatus. Quant. Inf. Comp., 4: 273, 2004.

[42] M. McKague, T. H. Yang, and V. Scarani. Robust self-testing of the singlet. J. Phys. A: Math. Theor., 45: 455304, 2012. DOI:10.1088/​1751-8113/​45/​45/​455304.
https:/​/​doi.org/​10.1088/​1751-8113/​45/​45/​455304

[43] 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. Random numbers certified by Bell's theorem. Nature, 464: 1021, 2010. DOI:10.1038/​nature09008.
https:/​/​doi.org/​10.1038/​nature09008

[44] S. Popescu and D. Rohrlich. Which states violate Bell's inequality maximally? Phys. Lett. A, 169: 411, 1992. DOI:10.1016/​0375-9601(92)90819-8.
https:/​/​doi.org/​10.1016/​0375-9601(92)90819-8

[45] S. Popescu and D. Rohrlich. Quantum nonlocality as an axiom. Found. Phys., 24: 379, 1994. DOI:10.1007/​BF02058098.
https:/​/​doi.org/​10.1007/​BF02058098

[46] J. Ribeiro, G. Murta, and S. Wehner. Fully general device-independence for two-party cryptography and position verification. 2016.
arXiv:1609.08487

[47] J. Ribeiro, G. Murta, and S. Wehner. Fully device-independent conference key agreement. Phys. Rev. A, 97: 022307, 2018. DOI:10.1103/​PhysRevA.97.022307.
https:/​/​doi.org/​10.1103/​PhysRevA.97.022307

[48] J. Ribeiro, L. P. Thinh, J. Kaniewski, J. Helsen, and S. Wehner. Device independence for two-party cryptography and position verification with memoryless devices. Phys. Rev. A, 97: 062307, 2018. DOI:10.1103/​PhysRevA.97.062307.
https:/​/​doi.org/​10.1103/​PhysRevA.97.062307

[49] B. W. Reichardt, F. Unger, and U. Vazirani. Classical command of quantum systems. Nature, 496: 456, 2013. DOI:10.1038/​nature12035.
https:/​/​doi.org/​10.1038/​nature12035

[50] I. Šupić, R. Augusiak, A. Salavrakos, and A. Acín. Self-testing protocols based on the chained Bell inequalities. New J. Phys., 18: 035013, 2016. DOI:10.1088/​1367-2630/​18/​3/​035013.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​3/​035013

[51] A. Salavrakos, R. Augusiak, J. Tura, P. Wittek, A. Acín, and S. Pironio. Bell inequalities tailored to maximally entangled states. Phys. Rev. Lett., 119: 040402, 2017. DOI:10.1103/​PhysRevLett.119.040402.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.040402

[52] J. Silman, A. Chailloux, N. Aharon, I. Kerenidis, S. Pironio, and S. Massar. Fully distrustful quantum bit commitment and coin flipping. Phys. Rev. Lett., 106: 220501, 2011. DOI:10.1103/​PhysRevLett.106.220501.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.220501

[53] I. Šupić, A. Coladangelo, R. Augusiak, and A. Acín. Self-testing multipartite entangled states through projections onto two systems. New J. Phys., 20: 083041, 2018. DOI:10.1088/​1367-2630/​aad89b.

[54] W. Son, J. Lee, and M. S. Kim. Generic Bell inequalities for multipartite arbitrary dimensional systems. Phys. Rev. Lett., 96: 060406, 2006. DOI:10.1103/​PhysRevLett.96.060406.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.060406

[55] W. Slofstra. The set of quantum correlations is not closed. 2017.
arXiv:1703.08618

[56] S. J. Summers and R. F. Werner. Maximal violation of Bell's inequalities is generic in quantum field theory. Commun. Math. Phys., 110: 247, 1987. DOI:10.1007/​BF01207366.
https:/​/​doi.org/​10.1007/​BF01207366

[57] B. S. Tsirelson. Quantum analogues of the Bell inequalities. The case of two spatially separated domains. J. Soviet Math., 36: 557, 1987. DOI:10.1007/​BF01663472.
https:/​/​doi.org/​10.1007/​BF01663472

[58] B. S. Tsirelson. Some results and problems on quantum Bell-type inequalities. Hadronic J. Suppl., 8: 329, 1993.

[59] U. Vazirani and T. Vidick. Certifiable quantum dice: or, true random number generation secure against quantum adversaries. Proceedings 44th Annual ACM Symposium on Theory of Computing, 2012. DOI:10.1145/​2213977.2213984.
https:/​/​doi.org/​10.1145/​2213977.2213984

[60] U. Vazirani and T. Vidick. Fully device-independent quantum key distribution. Phys. Rev. Lett., 113: 140501, 2014. DOI:10.1103/​PhysRevLett.113.140501.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.140501

[61] Y. Wang, X. Wu, and V. Scarani. All the self-testings of the singlet for two binary measurements. New J. Phys., 18: 025021, 2016. DOI:10.1088/​1367-2630/​18/​2/​025021.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​025021

[62] T. H. Yang and M. Navascués. Robust self-testing of unknown quantum systems into any entangled two-qubit states. Phys. Rev. A, 87: 050102(R), 2013. DOI:10.1103/​PhysRevA.87.050102.
https:/​/​doi.org/​10.1103/​PhysRevA.87.050102

### Cited by

[1] Debashis Saha, Rafael Santos, and Remigiusz Augusiak, "Sum-of-squares decompositions for a family of noncontextuality inequalities and self-testing of quantum devices", Quantum 4, 302 (2020).

[2] F. Baccari, R. Augusiak, I. Šupić, J. Tura, and A. Acín, "Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing", arXiv:1812.10428, Physical Review Letters 124 2, 020402 (2020).

[3] Jędrzej Kaniewski, "Weak form of self-testing", arXiv:1910.00706, Physical Review Research 2 3, 033420 (2020).

[4] Ivan Šupić and Joseph Bowles, "Self-testing of quantum systems: a review", Quantum 4, 337 (2020).

[5] Xingjian Zhang and Qi Zhao, "Simultaneous certification of entangled states and measurements in bounded dimensional semiquantum games", Physical Review Research 2 3, 033400 (2020).

[6] J. Tura, A. Aloy, F. Baccari, A. Acín, M. Lewenstein, and R. Augusiak, "Optimization of device-independent witnesses of entanglement depth from two-body correlators", Physical Review A 100 3, 032307 (2019).

[7] C. Jebarathinam, Jui-Chen Hung, Shin-Liang Chen, and Yeong-Cherng Liang, "Maximal violation of a broad class of Bell inequalities and its implication on self-testing", Physical Review Research 1 3, 033073 (2019).

[8] 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.

[9] Remigiusz Augusiak, Alexia Salavrakos, Jordi Tura, and Antonio Acín, "Bell inequalities tailored to the Greenberger-Horne-Zeilinger states of arbitrary local dimension", arXiv:1907.10116.

[10] Albert Aloy, Matteo Fadel, and Jordi Tura, "The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing", arXiv:2001.04440.

[11] David Cui, Arthur Mehta, Hamoon Mousavi, and Seyed Sajjad Nezhadi, "A generalization of CHSH and the algebraic structure of optimal strategies", arXiv:1911.01593.

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-21 20:12:14) and SAO/NASA ADS (last updated successfully 2020-10-21 20:12:15). The list may be incomplete as not all publishers provide suitable and complete citation data.