A three-player coherent state embezzlement game

Zhengfeng Ji1, Debbie Leung2, and Thomas Vidick3

1Centre for Quantum Software and Information, University of Technology Sydney, Australia
2University of Waterloo and the Perimeter Institute, Canada. Email: \textttwcleung@uwaterloo.ca
3California Institute of Technology, USA. Email: \textttvidick@cms.caltech.edu

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We introduce a three-player nonlocal game, with a finite number of classical questions and answers, such that the optimal success probability of $1$ in the game can only be achieved in the limit of strategies using arbitrarily high-dimensional entangled states. Precisely, there exists a constant $0 <c\leq 1$ such that to succeed with probability $1-\varepsilon $ in the game it is necessary to use an entangled state of at least $\Omega(\varepsilon ^{-c})$ qubits, and it is sufficient to use a state of at most $O(\varepsilon ^{-1})$ qubits.

The game is based on the coherent state exchange game of Leung et al.\ (CJTCS 2013). In our game, the task of the quantum verifier is delegated to a third player by a classical referee. Our results complement those of Slofstra (arXiv:1703.08618) and Dykema et al.\ (arXiv:1709.05032), who obtained two-player games with similar (though quantitatively weaker) properties based on the representation theory of finitely presented groups and $C^*$-algebras respectively.

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[1] P. Aravind. Quantum mysteries revisited again. American Journal of Physics, 72(10):1303–1307, 2004.

[2] A. Aspect, P. Grangier, and G. Roger. Experimental tests of realistic local theories via Bell's theorem. Physical review letters, 47(7):460, 1981.

[3] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195–200, 1964.

[4] J. Briët, H. Buhrman, and B. Toner. A generalized Grothendieck inequality and nonlocal correlations that require high entanglement. Communications in Mathematical Physics, 305(3):827–843, 2011.

[5] R. Chao, B. W. Reichardt, C. Sutherland, and T. Vidick. Test for a large amount of entanglement, using few measurements. Quantum, 2:92, 2018.

[6] R. Cleve, P. Hoyer, B. Toner, and J. Watrous. Consequences and limits of nonlocal strategies. In Computational Complexity, 2004. Proceedings. 19th IEEE Annual Conference on, pages 236–249. IEEE, 2004.

[7] R. Cleve, L. Liu, and V. I. Paulsen. Perfect embezzlement of entanglement. Journal of Mathematical Physics, 58(1):012204, 2017.

[8] A. Coladangelo. A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations. Quantum, 4:282, June 2020.

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

[10] A. Coladangelo and J. Stark. Separation of finite and infinite-dimensional quantum correlations, with infinite question or answer sets. arXiv preprint arXiv:1708.06522, 2017.

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

[12] K. Dykema, V. I. Paulsen, and J. Prakash. Non-closure of the set of quantum correlations via graphs. Communications in Mathematical Physics, 365:1125–1142, 2019.

[13] M. Fannes. A continuity property of the entropy density for spin lattice systems. Communications in Mathematical Physics, 31(4):291–294, 1973.

[14] H. Fu. Constant-sized correlations are sufficient to robustly self-test maximally entangled states with unbounded dimension. arXiv preprint arXiv:1911.01494, 2019.

[15] M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, et al. Significant-loophole-free test of Bell's theorem with entangled photons. Physical review letters, 115(25):250401, 2015.

[16] B. Hensen, H. Bernien, A. Dréau, A. Reiserer, N. Kalb, M. Blok, J. Ruitenberg, R. Vermeulen, R. Schouten, C. Abellán, et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526(7575):682–686, 2015.

[17] Z. Ji. Binary Constraint System Games and Locally Commutative Reductions. arXiv:1310.3794, 2013.

[18] Z. Ji. Classical verification of quantum proofs. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 885–898. ACM, 2016.

[19] Z. Ji. Compression of quantum multi-prover interactive proofs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 289–302, New York, NY, USA, 2017. ACM.

[20] D. Leung, B. Toner, and J. Watrous. Coherent state exchange in multi-prover quantum interactive proof systems. Chicago Journal of Theoretical Computer Science, 11:1–18, 2013.

[21] L. Mančinska and T. Vidick. Unbounded entanglement can be needed to achieve the optimal success probability. In International Colloquium on Automata, Languages, and Programming, pages 835–846. Springer, 2014.

[22] C. A. Miller and Y. Shi. Optimal Robust Self-Testing by Binary Nonlocal XOR Games. In S. Severini and F. Brandao, editors, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013), volume 22 of Leibniz International Proceedings in Informatics (LIPIcs), pages 254–262, Dagstuhl, Germany, 2013. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.

[23] D. Ostrev and T. Vidick. Entanglement of approximate quantum strategies in xor games. Quantum Information and Computation, 18(7&8):0617–0631, 2018.

[24] K. F. Pál and T. Vértesi. Maximal violation of a bipartite three-setting, two-outcome bell inequality using infinite-dimensional quantum systems. Physical Review A, 82(2):022116, 2010.

[25] O. Regev and T. Vidick. Quantum XOR games. ACM Transactions on Computation Theory (ToCT), 7(4):15, 2015.

[26] B. W. Reichardt, F. Unger, and U. Vazirani. Classical command of quantum systems. Nature, 496(7446):456–460, 2013.

[27] L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, et al. Strong loophole-free test of local realism. Physical review letters, 115(25):250402, 2015.

[28] W. Slofstra. Lower bounds on the entanglement needed to play XOR non-local games. Journal of Mathematical Physics, 52(10):102202, 2011.

[29] W. Slofstra. A group with at least subexponential hyperlinear profile. arXiv preprint arXiv:1806.05267, 2018.

[30] W. Slofstra. The set of quantum correlations is not closed. Forum of Mathematics, Pi, 7:e1, 2019.

[31] W. Slofstra. Tsirelson's problem and an embedding theorem for groups arising from non-local games. J. Amer. Math. Soc., 33:1–56, 2020.

[32] W. Slofstra and T. Vidick. Entanglement in non-local games and the hyperlinear profile of groups. Annales Henri Poincaré, 19:2979–3005, 2018.

[33] W. van Dam and P. Hayden. Universal entanglement transformations without communication. Physical Review A, 67(6):060302, 2003.

Cited by

[1] Andrea Coladangelo and Jalex Stark, "Unconditional separation of finite and infinite-dimensional quantum correlations", arXiv:1804.05116.

[2] Joseph Fitzsimons, Zhengfeng Ji, Thomas Vidick, and Henry Yuen, "Quantum proof systems for iterated exponential time, and beyond", arXiv:1805.12166.

[3] Oded Regev and Thomas Vidick, "Bounds on Dimension Reduction in the Nuclear Norm", arXiv:1901.09480.

[4] Andrea Coladangelo, "A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations", arXiv:1904.02350.

[5] William Slofstra, "A group with at least subexponential hyperlinear profile", arXiv:1806.05267.

[6] Richard Cleve, Benoit Collins, Li Liu, and Vern Paulsen, "Constant gap between conventional strategies and those based on C*-dynamics for self-embezzlement", arXiv:1811.12575.

[7] Louis Mathieu and Mehdi Mhalla, "Separating pseudo-telepathy games and two-local theories", arXiv:1806.08661.

[8] Andrea Coladangelo and Jalex Stark, "An inherently infinite-dimensional quantum correlation", Nature Communications 11, 3335 (2020).

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