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Quantum bounds for compiled XOR games and $d$-outcome CHSH games

Published 8 Mar 2024 in quant-ph | (2403.05502v2)

Abstract: Nonlocal games play a crucial role in quantum information theory and have numerous applications in certification and cryptographic protocols. Kalai et al. (STOC 2023) introduced a procedure to compile a nonlocal game into a single-prover interactive proof, using a quantum homomorphic encryption scheme, and showed that their compilation method preserves the classical bound of the game. Natarajan and Zhang (FOCS 2023) then showed that the quantum bound is preserved for the specific case of the CHSH game. Extending the proof techniques of Natarajan and Zhang, we show that the compilation procedure of Kalai et al. preserves the quantum bound for two classes of games: XOR games and d-outcome CHSH games. We also establish that, for any pair of qubit measurements, there exists an XOR game such that its optimal winning probability serves as a self-test for that particular pair of measurements.

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References (44)
  1. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett., 98:230501, Jun 2007. doi:10.1103/PhysRevLett.98.230501.
  2. Fast verification of any remote procedure call: Short witness-indistinguishable one-round proofs for NP. In International Colloquium on Automata, Languages, and Programming, pages 463–474. Springer, 2000. doi:10.1007/3-540-45022-X_39.
  3. Practical device-independent quantum cryptography via entropy accumulation. Nature Communications, 9(1):459, 2018. doi:10.1038/s41467-017-02307-4.
  4. Logical Bell inequalities. Phys. Rev. A, 85:062114, Jun 2012. doi:10.1103/PhysRevA.85.062114.
  5. Certified randomness in quantum physics. Nature, 540(7632):213–219, 2016. doi:10.1038/nature20119.
  6. Bell nonlocality. Rev. Mod. Phys., 86:419–478, Apr 2014. doi:10.1103/RevModPhys.86.419.
  7. J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1:195–200, Nov 1964. doi:10.1103/PhysicsPhysiqueFizika.1.195.
  8. Simple tests of quantumness also certify qubits. In Helena Handschuh and Anna Lysyanskaya, editors, Advances in Cryptology - CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Santa Barbara, CA, USA, August 20-24, 2023, Proceedings, Part V, volume 14085 of Lecture Notes in Computer Science, pages 162–191. Springer, 2023. doi:10.1007/978-3-031-38554-4_6.
  9. (Leveled) fully homomorphic encryption without bootstrapping. ACM Transactions on Computation Theory (TOCT), 6(3):1–36, 2014. doi:10.1145/2090236.2090262.
  10. Zvika Brakerski. Quantum FHE (almost) as secure as classical. In Annual International Cryptology Conference, pages 67–95. Springer, 2018. doi:10.1007/978-3-319-96878-0_3.
  11. Custom bell inequalities from formal sums of squares, 2023. arXiv:2308.08601.
  12. Efficient fully homomorphic encryption from (standard) LWE. SIAM Journal on computing, 43(2):831–871, 2014. doi:10.1109/FOCS.2011.12.
  13. Bell inequalities for arbitrarily high-dimensional systems. Physical review letters, 88(4):040404, 2002. doi:10.1103/PhysRevLett.88.040404.
  14. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23(15):880, 1969. doi:10.1103/PhysRevLett.23.880.
  15. A Computational Tsirelson’s Theorem for the Value of Compiled XOR Games. Cryptology ePrint Archive, Paper 2024/348, 2024. URL: https://eprint.iacr.org/2024/348.
  16. Roger Colbeck. Quantum And Relativistic Protocols For Secure Multi-Party Computation. PhD thesis, University of Cambridge, 2007. doi:10.48550/arXiv.0911.3814.
  17. All tight correlation Bell inequalities have quantum violations. Phys. Rev. Res., 2:012044, Feb 2020. doi:10.1103/PhysRevResearch.2.012044.
  18. Arthur Fine. Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett., 48:291–295, Feb 1982. doi:10.1103/PhysRevLett.48.291.
  19. Craig Gentry. Fully homomorphic encryption using ideal lattices. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 169–178, 2009. doi:10.1145/1536414.1536440.
  20. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC ’19. ACM, June 2019. doi:10.1145/3313276.3316366.
  21. Quantum Advantage from Any Non-local Game. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, page 1617–1628, 2023. doi:10.1145/3564246.3585164.
  22. How to delegate computations: the power of no-signaling proofs. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 485–494, 2014. doi:10.1145/3456867.
  23. Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems. Quantum, 3:198, 2019. doi:10.22331/q-2019-10-24-198.
  24. Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. Emerging applications of algebraic geometry, pages 157–270, 2009. doi:10.1007/978-0-387-09686-5_7.
  25. Quantum correlations in the minimal scenario. Quantum, 7:947, March 2023. doi:10.22331/q-2023-03-16-947.
  26. Urmila Mahadev. Classical homomorphic encryption for quantum circuits. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 332–338, 2018. doi:10.1109/FOCS.2018.00039.
  27. Urmila Mahadev. Classical Homomorphic Encryption for Quantum Circuits. SIAM Journal on Computing, 52(6):FOCS18–189–FOCS18–215, 2023. doi:10.1137/18M1231055.
  28. Self testing quantum apparatus. Quantum Info. Comput., 4(4):273–286, 2004. doi:10.48550/arXiv.quant-ph/0307205.
  29. Bounding the set of quantum correlations. Phys. Rev. Lett., 98:010401, Jan 2007. doi:10.1103/PhysRevLett.98.010401.
  30. A glance beyond the quantum model. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2115):881–890, 2010. doi:10.1098/rspa.2009.0453.
  31. Bounding the quantum value of compiled nonlocal games: from CHSH to BQP verification, 2023. doi:10.48550/arXiv.2303.01545.
  32. Dimiter Ostrev. The structure of nearly-optimal quantum strategies for the non-local xor games. Quantum Information & Computation, 16(13-14):1191–1211, 2016. doi:10.26421/QIC16.13-14-6.
  33. Random numbers certified by Bell’s theorem. Nature, 464(7291):1021–1024, 2010. doi:10.1038/nature09008.
  34. On data banks and privacy homomorphisms. Foundations of secure computation, 4(11):169–180, 1978.
  35. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2):120–126, 1978. doi:10.1145/359340.359342.
  36. Bell inequalities tailored to maximally entangled states. Phys. Rev. Lett., 119:040402, Jul 2017. doi:10.1103/PhysRevLett.119.040402.
  37. Self-testing of quantum systems: a review. Quantum, 4:337, September 2020. doi:10.22331/q-2020-09-30-337.
  38. On the relation between Bell’s inequalities and nonlocal games. Physics Letters A, 372(21):3796–3800, 2008. doi:10.1016/j.physleta.2008.03.001.
  39. Self-testing quantum systems of arbitrary local dimension with minimal number of measurements. npj Quantum Information, 7(1), October 2021. doi:10.1038/s41534-021-00490-3.
  40. Semidefinite programming relaxations for quantum correlations, 2024. doi:10.48550/arXiv.2307.02551.
  41. Boris Tsirelson. Quantum analogues of the Bell inequalities. the case of two spatially separated domains. Journal of Soviet Mathematics, 36:557–570, 1987. doi:10.1007/BF01663472.
  42. Fully Device-Independent Quantum Key Distribution. Phys. Rev. Lett., 113:140501, Sep 2014. doi:10.1103/PhysRevLett.113.140501.
  43. Device-independent quantum key distribution with arbitrarily small nonlocality, 2023. arXiv:2309.09650.
  44. Stephanie Wehner. Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities. Phys. Rev. A, 73:022110, Feb 2006. doi:10.1103/PhysRevA.73.022110.
Citations (3)
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Summary

  • The paper demonstrates that Kalai et al.'s compilation method preserves quantum bounds in both XOR and d-outcome CHSH games.
  • The paper employs a sum-of-squares decomposition and self-testing analysis to validate that the compiled framework retains quantum advantages.
  • The paper highlights implications for quantum cryptography by establishing that compiled nonlocal games reliably certify quantum resources in secure protocols.

Analysis of Quantum Bounds for Compiled XOR Games and d-outcome CHSH Games

This paper presents a comprehensive analysis of quantum bounds in the context of nonlocal games, specifically focusing on XOR games and d-outcome CHSH games. The authors extend existing methodologies to demonstrate that the compilation procedure outlined by Kalai et al. successfully preserves the quantum bounds for these classes of games, further contributing to the broader understanding of quantum resources in cryptographic and certification protocols.

Introduction to Nonlocal Games

Nonlocal games serve as fundamental tools in quantum information theory, highlighting differences between classical and quantum resources. The paper bases its analysis on the procedure developed by Kalai et al. that compiles nonlocal games into interactive proof systems using quantum homomorphic encryption. Previous research by Natarajan and Zhang confirmed that this method maintains the quantum bound in CHSH games. This paper expands on those results to investigate XOR games and d-outcome CHSH games, two significant subclasses in the field of nonlocal games.

XOR Games

XOR games are a particular subclass characterized by two players with binary outputs. The winning conditions of these games are defined by the parity (XOR operation) of the players' outputs. The game operator associated with XOR games serves as a pivotal component in deriving quantum advantages over classical strategies. The authors prove that Kalai et al.'s compilation method not only maintains the classical game limits but also preserves the quantum bounds for XOR games. This entails that the quantum advantage facilitated by entangled states and quantum measurements is not diminished through the compilation process.

d-outcome CHSH Games

The exploration then extends to d-outcome CHSH games, which are generalizations of the well-known CHSH game to scenarios with multiple measurement outputs. The paper delineates the construction of the Bell operator for these games and explicates a sum-of-squares decomposition to illustrate the preservation of quantum bounds. The SATWAP inequality, which is pivotal in the self-testing of quantum systems, is examined as a specific instance of a d-outcome CHSH game that validates the theoretical claims.

Self-Testing and Cryptographic Implications

An intriguing component of the authors' analysis is the exploration of self-testing properties for compiled games. These results have significant implications for cryptographic protocols, particularly in the domain of device-independent quantum key distribution and randomness certification. The self-testing establishes a robust verification mechanism for quantum states and instruments used within these protocols, asserting that compiled nonlocal games can effectively certify quantum computational capabilities.

Future Research Directions

The paper concludes by identifying several open questions and potential avenues for further inquiry. Key among these is expanding the framework to include multi-player games and analyzing the interplay between quantum bound preservation and self-testing in more complex game configurations. The ongoing challenge lies in addressing the computational complexity of quantum strategies in compiled games and their cryptographic applications.

Conclusion

In summary, this paper substantiates the preservation of quantum bounds in compiled nonlocal games for XOR and d-outcome CHSH games. By reinforcing the connection between quantum homomorphic encryption and nonlocal games, the research provides a valuable resource for advancing quantum cryptographic protocols and enhancing the understanding of quantum advan

tage in computational scenarios.

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