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Testing Linearity against Non-signaling Strategies

Published:01 June 2020Publication History
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Abstract

Non-signaling strategies are collections of distributions with certain non-local correlations. They have been studied in physics as a strict generalization of quantum strategies to understand the power and limitations of nature’s apparent non-locality. Recently, they have received attention in theoretical computer science due to connections to Complexity and Cryptography.

We initiate the study of Property Testing against non-signaling strategies, focusing first on the classical problem of linearity testing (Blum, Luby, and Rubinfeld; JCSS 1993). We prove that any non-signaling strategy that passes the linearity test with high probability must be close to a quasi-distribution over linear functions.

Quasi-distributions generalize the notion of probability distributions over global objects (such as functions) by allowing negative probabilities, while at the same time requiring that “local views” follow standard distributions (with non-negative probabilities). Quasi-distributions arise naturally in the study of quantum mechanics as a tool to describe various non-local phenomena.

Our analysis of the linearity test relies on Fourier analytic techniques applied to quasi-distributions. Along the way, we also establish general equivalences between non-signaling strategies and quasi-distributions, which we believe will provide a useful perspective on the study of Property Testing against non-signaling strategies beyond linearity testing.

References

  1. Samson Abramsky and Adam Brandenburger. 2011. The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 11 (2011), 113036.Google ScholarGoogle ScholarCross RefCross Ref
  2. William Aiello, Sandeep N. Bhatt, Rafail Ostrovsky, and Sivaramakrishnan Rajagopalan. 2000. Fast verification of any remote procedure call: Short witness-indistinguishable one-round proofs for NP. In Proceedings of the 27th International Colloquium on Automata, Languages and Programming (ICALP’00). 463--474.Google ScholarGoogle ScholarCross RefCross Ref
  3. Sabri W. Al-Safi and Anthony J. Short. 2013. Simulating all nonsignaling correlations via classical or quantum theory with negative probabilities. Phys. Rev. Lett. 111 (2013), 170403. Issue 17.Google ScholarGoogle ScholarCross RefCross Ref
  4. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. 1998. Proof verification and the hardness of approximation problems. J. ACM 45, 3 (1998), 501--555.Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Sanjeev Arora and Shmuel Safra. 1998. Probabilistic checking of proofs: A new characterization of NP. J. ACM 45, 1 (1998), 70--122.Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Sanjeev Arora and Madhu Sudan. 2003. Improved low-degree testing and its applications. Combinatorica 23, 3 (2003), 365--426.Google ScholarGoogle ScholarCross RefCross Ref
  7. László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. 1991. Checking computations in polylogarithmic time. In Proceedings of the 23rd ACM Symposium on Theory of Computing (STOC’91). 21--32.Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. László Babai, Lance Fortnow, and Carsten Lund. 1991. Non-deterministic exponential time has two-prover interactive protocols. Comput. Complex. 1 (1991), 3--40.Google ScholarGoogle ScholarCross RefCross Ref
  9. Jonathan Barrett. 2007. Information processing in generalized probabilistic theories. Phys. Rev. A 75 (2007), 032304. Issue 3.Google ScholarGoogle ScholarCross RefCross Ref
  10. Jonathan Barrett, Lucien Hardy, and Adrian Kent. 2005. No signaling and quantum key distribution. Phys. Rev. Lett. 95 (2005), 010503. Issue 1.Google ScholarGoogle ScholarCross RefCross Ref
  11. Jonathan Barrett, Noah Linden, Serge Massar, Stefano Pironio, Sandu Popescu, and David Roberts. 2005. Nonlocal correlations as an information-theoretic resource. Phys. Rev. Lett. 71 (2005), 022101. Issue 2.Google ScholarGoogle Scholar
  12. Jonathan Barrett and Stefano Pironio. 2005. Popescu--Rohrlich correlations as a unit of nonlocality. Phys. Rev. Lett. 95 (2005), 140401. Issue 14.Google ScholarGoogle ScholarCross RefCross Ref
  13. Mihir Bellare, Don Coppersmith, Johan Håstad, Marcos A. Kiwi, and Madhu Sudan. 1996. Linearity testing in characteristic two. IEEE Trans. Inf. Theor. 42, 6 (1996), 1781--1795.Google ScholarGoogle ScholarCross RefCross Ref
  14. Michael Ben-Or, Don Coppersmith, Mike Luby, and Ronitt Rubinfeld. 2008. Non-abelian homomorphism testing, and distributions close to their self-convolutions. Rand. Struct. Algor. 32, 1 (2008), 49--70.Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. 2010. Optimal testing of Reed-Muller codes. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science (FOCS’10). 488--497.Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. 1993. Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci. 47, 3 (1993), 549--595.Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Gilles Brassard, Harry Buhrman, Noah Linden, André Allan Méthot, Alain Tapp, and Falk Unger. 2006. Limit on nonlocality in any world in which communication complexity is not trivial. Phys. Rev. Lett. 96 (2006), 250401. Issue 25.Google ScholarGoogle ScholarCross RefCross Ref
  18. Anne Broadbent and André Allan Méthot. 2006. On the power of non-local boxes. Theoret. Comput. Sci. 358, 1 (2006), 3--14.Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Harry Buhrman, Matthias Christandl, Falk Unger, Stephanie Wehner, and Andreas Winter. 2006. Implications of superstrong non-locality for cryptography. Proc. Roy. Soc. Lond. A: Math., Phys. Eng. Sci. 462, 2071 (2006), 1919--1932.Google ScholarGoogle ScholarCross RefCross Ref
  20. Nicolas J. Cerf, Nicolas Gisin, Serge Massar, and Sandu Popescu. 2005. Simulating maximal quantum entanglement without communication. Phys. Rev. Lett. 94 (2005), 220403. Issue 22.Google ScholarGoogle ScholarCross RefCross Ref
  21. Rui Chao and Ben W. Reichardt. 2017. Test to separate quantum theory from non-signaling theories. Retrieved from arXiv quant-ph/1706.02008.Google ScholarGoogle Scholar
  22. Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. 2017. Direct sum testing. SIAM J. Comput. 46 (2017), 1336--1369. Issue 4.Google ScholarGoogle ScholarCross RefCross Ref
  23. Paul A. M. Dirac. 1942. The physical interpretation of quantum mechanics. Proc. Roy. Soc. Lond. A: Math., Phys Eng. Sci. 180, 980 (1942), 1--40.Google ScholarGoogle Scholar
  24. Cynthia Dwork, Michael Langberg, Moni Naor, Kobbi Nissim, and Omer Reingold. 2004. Succinct NP Proofs and Spooky Interactions. Retrieved from www.openu.ac.il/home/mikel/papers/spooky.ps.Google ScholarGoogle Scholar
  25. Uriel Feige, Shafi Goldwasser, Laszlo Lovász, Shmuel Safra, and Mario Szegedy. 1996. Interactive proofs and the hardness of approximating cliques. J. ACM 43, 2 (1996), 268--292.Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Richard P. Feynman. 1987. Negative probability. In Quantum Implications: Essays in Honour of David Bohm, Basil J. Hiley and D. Peat (Eds.). 235--248.Google ScholarGoogle Scholar
  27. Oded Goldreich, Shafi Goldwasser, and Dana Ron. 1998. Property testing and its connection to learning and approximation. J. ACM 45, 4 (1998), 653--750.Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Thomas Holenstein. 2009. Parallel repetition: Simplification and the no-signaling case. Theor. Comput. 5, 1 (2009), 141--172.Google ScholarGoogle ScholarCross RefCross Ref
  29. Tsuyoshi Ito. 2010. Polynomial-space approximation of no-signaling provers. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP’10). 140--151.Google ScholarGoogle ScholarCross RefCross Ref
  30. Tsuyoshi Ito, Hirotada Kobayashi, and Keiji Matsumoto. 2009. Oracularization and two-prover one-round interactive proofs against nonlocal strategies. In Proceedings of the 24th IEEE Annual Conference on Comput. Complex. (CCC’09). 217--228.Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. Tsuyoshi Ito and Thomas Vidick. 2012. A multi-prover interactive proof for NEXP sound against entangled provers. In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science (FOCS’12). 243--252.Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Nick S. Jones and Lluís Masanes. 2005. Interconversion of nonlocal correlations. Phys. Rev. A 72 (2005), 052312. Issue 5.Google ScholarGoogle ScholarCross RefCross Ref
  33. Yael Kalai, Ran Raz, and Ron Rothblum. 2013. Delegation for bounded space. In Proceedings of the 45th ACM Symposium on the Theory of Computing (STOC’13). 565--574.Google ScholarGoogle Scholar
  34. Yael Tauman Kalai, Ran Raz, and Oded Regev. 2016. On the space complexity of linear programming with preprocessing. In Proceedings of the 7th Innovations in Theoretical Computer Science Conference (ITCS’16). 293--300.Google ScholarGoogle Scholar
  35. Yael Tauman Kalai, Ran Raz, and Ron D. Rothblum. 2014. How to delegate computations: The power of no-signaling proofs. In Proceedings of the 46th ACM Symposium on Theory of Computing (STOC’14). 485--494. Retrieved from https://eccc.weizmann.ac.il/report/2013/183/.Google ScholarGoogle Scholar
  36. Leonid A. Khalfin and Boris S. Tsirelson. 1985. Quantum and quasi-classical analogs of Bell inequalities. In Proceedings of the Symposium on the Foundations of Modern Physics (1985), 441--460.Google ScholarGoogle Scholar
  37. Noah Linden, Sandu Popescu, Anthony J. Short, and Andreas Winter. 2007. Quantum nonlocality and beyond: Limits from nonlocal computation. Phys. Rev. Lett. 99 (2007), 180502. Issue 18.Google ScholarGoogle ScholarCross RefCross Ref
  38. Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. 1992. Algebraic methods for interactive proof systems. J. ACM 39, 4 (1992), 859--868.Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Lluís Masanes, Antonio Acín, and Nicolas Gisin. 2006. General properties of nonsignaling theories. Phys. Rev. A 73 (2006), 012112. Issue 1.Google ScholarGoogle ScholarCross RefCross Ref
  40. Ryan O’Donnell. 2014. Analysis of Boolean Functions. Cambridge University Press.Google ScholarGoogle Scholar
  41. Sandu Popescu and Daniel Rohrlich. 1994. Quantum nonlocality as an axiom. Found. Phys. 24, 3 (1994), 379--385.Google ScholarGoogle ScholarCross RefCross Ref
  42. Sandu Popescu and Daniel Rohrlich. 1998. Causality and Nonlocality as Axioms for Quantum Mechanics. Springer Netherlands, 383--389.Google ScholarGoogle Scholar
  43. Prasad Raghavendra and David Steurer. 2009. Integrality gaps for strong SDP relaxations of UNIQUE GAMES. In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS’09). 575--585. Retrieved from http://people.eecs.berkeley.edu/ prasad/Files/cspgaps.pdf.Google ScholarGoogle ScholarCross RefCross Ref
  44. Peter Rastall. 1985. Locality, Bell’s theorem, and quantum mechanics. Found. Phys. 15, 9 (1985), 963--972.Google ScholarGoogle ScholarCross RefCross Ref
  45. Ran Raz and Shmuel Safra. 1997. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the 29th ACM Symposium on Theory of Computing (STOC’97). 475--484.Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. Ronitt Rubinfeld and Madhu Sudan. 1996. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25, 2 (1996), 252--271.Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Adi Shamir. 1992. IP = PSPACE. J. ACM 39, 4 (1992), 869--877.Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Hanif D. Sherali and Warren P. Adams. 1990. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Disc. Math. 3, 3 (1990), 411--430.Google ScholarGoogle ScholarDigital LibraryDigital Library
  49. Anthony J. Short, Nicolas Gisin, and Sandu Popescu. 2006. The physics of no-bit-commitment: Generalized quantum non-locality versus oblivious transfer. Quant. Inf. Proc. 5, 2 (2006), 131--138.Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Anthony J. Short, Sandu Popescu, and Nicolas Gisin. 2006. Entanglement swapping for generalized nonlocal correlations. Phys. Rev. A 73 (2006), 012101. Issue 1.Google ScholarGoogle ScholarCross RefCross Ref
  51. Amir Shpilka and Avi Wigderson. 2004. Derandomizing homomorphism testing in general groups. In Proceedings of the 36th ACM Symposium on the Theory of Computing (STOC’04). 427--435.Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. Wim van Dam. 2013. Implausible consequences of superstrong nonlocality. Nat. Comput. 12 (2013), 9--12. Issue 1.Google ScholarGoogle ScholarDigital LibraryDigital Library
  53. Thomas Vidick. 2014. Linearity testing with entangled provers. Retrieved from http://users.cms.caltech.edu/ vidick/linearity_test.pdf.Google ScholarGoogle Scholar
  54. Stefan Wolf and Jürg Wullschleger. 2005. Oblivious transfer and quantum non-locality. In Proceedings of the International Symposium on Information Theory (ISIT’05). 1745--1748.Google ScholarGoogle ScholarCross RefCross Ref

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