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Noise-Tolerant Testing of High Entanglement of Formation

Authors Rotem Arnon-Friedman, Henry Yuen

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Rotem Arnon-Friedman
  • ETH Zürich, Switzerland,
Henry Yuen
  • UC Berkeley, USA,

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Rotem Arnon-Friedman and Henry Yuen. Noise-Tolerant Testing of High Entanglement of Formation. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 11:1-11:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


In this work we construct tests that allow a classical user to certify high dimensional entanglement in uncharacterized and possibly noisy quantum devices. We present a family of non-local games {G_n} that for all n certify states with entanglement of formation Omega(n). These tests can be derived from any bipartite non-local game with a classical-quantum gap. Furthermore, our tests are noise-tolerant in the sense that fault tolerant technologies are not needed to play the games; entanglement distributed over noisy channels can pass with high probability, making our tests relevant for realistic experimental settings. This is in contrast to, e.g., results on self-testing of high dimensional entanglement, which are only relevant when the noise rate goes to zero with the system's size n. As a corollary of our result, we supply a lower-bound on the entanglement cost of any state achieving a quantum advantage in a bipartite non-local game. Our proof techniques heavily rely on ideas from the work on classical and quantum parallel repetition theorems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • device independence
  • quantum games
  • entanglement testing
  • noise tolerance


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  1. Rotem Arnon-Friedman and Jean-Daniel Bancal. Device-independent certification of one-shot distillable entanglement. arXiv, 2017. Google Scholar
  2. Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. Simple and tight device-independent security proofs. arXiv preprint arXiv:1607.01797, 2016. Google Scholar
  3. Koenraad Audenaert, Frank Verstraete, and Bart De Moor. Variational characterizations of separability and entanglement of formation. Physical Review A, 64(5):052304, 2001. Google Scholar
  4. Jonathan Barrett, Lucien Hardy, and Adrian Kent. No signaling and quantum key distribution. Physical Review Letters, 95(1):010503, 2005. Google Scholar
  5. Mohammad Bavarian, Thomas Vidick, and Henry Yuen. Parallel repetition via fortification: analytic view and the quantum case. arXiv preprint arXiv:1603.05349, 2016. Google Scholar
  6. Mohammad Bavarian, Thomas Vidick, and Henry Yuen. Hardness amplification for entangled games via anchoring. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 303-316, 2017. Google Scholar
  7. John S Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1(3), 1964. Google Scholar
  8. Charles H Bennett, Herbert J Bernstein, Sandu Popescu, and Benjamin Schumacher. Concentrating partial entanglement by local operations. Physical Review A, 53(4):2046, 1996. Google Scholar
  9. Charles H Bennett, David P DiVincenzo, John A Smolin, and William K Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54(5):3824, 1996. Google Scholar
  10. Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, John M Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. arXiv preprint arXiv:1608.00263, 2016. Google Scholar
  11. Fernando GSL Brandao and Michał Horodecki. On Hastings' counterexamples to the minimum output entropy additivity conjecture. Open Systems &Information Dynamics, 17(01):31-52, 2010. Google Scholar
  12. Nicolas Brunner, Stefano Pironio, Antonio Acin, Nicolas Gisin, André Allan Méthot, and Valerio Scarani. Testing the dimension of Hilbert spaces. Physical review letters, 100(21):210503, 2008. Google Scholar
  13. Yu Cai, Jean-Daniel Bancal, Jacquiline Romero, and Valerio Scarani. A new device-independent dimension witness and its experimental implementation. Journal of Physics A: Mathematical and Theoretical, 49(30):305301, 2016. Google Scholar
  14. Rui Chao, Ben W Reichardt, Chris Sutherland, and Thomas Vidick. Test for a large amount of entanglement, using few measurements. arXiv preprint arXiv:1610.00771, 2016. Google Scholar
  15. Kai-Min Chung, Xiaodi Wu, and Henry Yuen. Parallel repetition for entangled k-player games via fast quantum search. In the 30th Conference on Computational Complexity (CCC), pages 512-536, 2015. Google Scholar
  16. 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. Google Scholar
  17. Andrea Coladangelo, Alex Grilo, Stacey Jeffery, and Thomas Vidick. Verifier-on-a-leash: new schemes for verifiable delegated quantum computation, with quasilinear resources. arXiv preprint arXiv:1708.07359, 2017. Google Scholar
  18. Andrea Coladangelo and Jalex Stark. Robust self-testing for linear constraint system games. arXiv preprint arXiv:1709.09267, 2017. Google Scholar
  19. Tim Coopmans. Robust self-testing of (almost) all pure two-qubit states. Master’s thesis, Universiteit van Amsterdam, 2017. Google Scholar
  20. Matthew Coudron and Anand Natarajan. The parallel-repeated magic square game is rigid. arXiv preprint arXiv:1609.06306, 2016. Google Scholar
  21. Nicolai Friis, Sridhar Bulusu, and Reinhold A Bertlmann. Geometry of two-qubit states with negative conditional entropy. Journal of Physics A: Mathematical and Theoretical, 50(12):125301, 2017. Google Scholar
  22. Alexandru Gheorghiu, Elham Kashefi, and Petros Wallden. Robustness and device independence of verifiable blind quantum computing. New Journal of Physics, 17(8):083040, 2015. Google Scholar
  23. Marissa Giustina, Marijn AM Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-Åke Larsson, Carlos Abellán, et al. Significant-loophole-free test of Bell’s theorem with entangled photons. Physical review letters, 115(25):250401, 2015. Google Scholar
  24. Michal Hajdušek, Carlos A Pérez-Delgado, and Joseph F Fitzsimons. Device-independent verifiable blind quantum computation. arXiv preprint arXiv:1502.02563, 2015. Google Scholar
  25. Patrick M Hayden, Michal Horodecki, and Barbara M Terhal. The asymptotic entanglement cost of preparing a quantum state. Journal of Physics A: Mathematical and General, 34(35):6891, 2001. Google Scholar
  26. Bas Hensen, H Bernien, AE Dréau, A Reiserer, N Kalb, MS Blok, J Ruitenberg, RFL Vermeulen, RN Schouten, C Abellán, et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526(7575):682-686, 2015. Google Scholar
  27. Thomas Holenstein. Parallel repetition: Simplification and the no-signaling case. Theory of Computing, 5(8):141-172, 2009. URL:
  28. Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Reviews of modern physics, 81(2):865, 2009. Google Scholar
  29. IBM. IBM quantum experience. URL:
  30. Rahul Jain, Attila Pereszlényi, and Penghui Yao. A parallel repetition theorem for entangled two-player one-round games under product distributions. In Proceedings of Conference on Computational Complexity (CCC), pages 209-216, 2014. Google Scholar
  31. Marius Junge, Carlos Palazuelos, David Pérez-García, Ignacio Villanueva, and Michael M Wolf. Unbounded violations of bipartite Bell inequalities via operator space theory. Communications in Mathematical Physics, 300(3):715-739, 2010. Google Scholar
  32. Yang Liu, Xiao Yuan, Ming-Han Li, Weijun Zhang, Qi Zhao, Jiaqiang Zhong, Yuan Cao, Yu-Huai Li, Luo-Kan Chen, Hao Li, et al. High speed self-testing quantum random number generation without detection loophole. In Frontiers in Optics, pages FTh2E-1. Optical Society of America, 2017. Google Scholar
  33. Matthew McKague. Self-testing in parallel. New Journal of Physics, 18(4):045013, 2016. Google Scholar
  34. Carl A Miller and Yaoyun Shi. Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 417-426. ACM, 2014. Google Scholar
  35. 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, pages 1003-1015. ACM, 2017. Google Scholar
  36. Károly F Pál and Tamás Vértesi. Quantum bounds on Bell inequalities. Physical Review A, 79(2):022120, 2009. Google Scholar
  37. Stefano Pironio, Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, and Valerio Scarani. Device-independent quantum key distribution secure against collective attacks. New Journal of Physics, 11(4):045021, 2009. Google Scholar
  38. Martin B Plenio and Shashank Virmani. An introduction to entanglement measures. arXiv preprint quant-ph/0504163, 2005. Google Scholar
  39. Sandu Popescu and Daniel Rohrlich. Thermodynamics and the measure of entanglement. Physical Review A, 56(5):R3319, 1997. Google Scholar
  40. Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM Journal on Computing, 40(6):1871-1891, 2011. Google Scholar
  41. Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763-803, 1998. Google Scholar
  42. Ben W Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496(7446):456-460, 2013. Google Scholar
  43. 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, et al. Strong loophole-free test of local realism. Physical review letters, 115(25):250402, 2015. Google Scholar
  44. Barbara M Terhal and Karl Gerd H Vollbrecht. Entanglement of formation for isotropic states. Physical Review Letters, 85(12):2625, 2000. Google Scholar
  45. Umesh Vazirani and Thomas Vidick. Fully device-independent quantum key distribution. Physical review letters, 113(14):140501, 2014. Google Scholar
  46. Frank Verstraete and Michael M Wolf. Entanglement versus Bell violations and their behavior under local filtering operations. Physical review letters, 89(17):170401, 2002. Google Scholar
  47. Tamás Vértesi and Nicolas Brunner. Disproving the Peres conjecture: Bell nonlocality from bipartite bound entanglement. arXiv preprint arXiv:1405.4502, 2014. Google Scholar
  48. Karl Gerd H Vollbrecht and Reinhard F Werner. Entanglement measures under symmetry. Physical Review A, 64(6):062307, 2001. Google Scholar
  49. William K Wootters. Entanglement of formation of an arbitrary state of two qubits. Physical Review Letters, 80(10):2245, 1998. Google Scholar
  50. William K Wootters. Entanglement of formation and concurrence. Quantum Information &Computation, 1(1):27-44, 2001. Google Scholar
  51. William K. Wootters. Entanglement of formation. In Prem Kumar, Giacomo M D'Ariano, and Osamu Hirota, editors, Quantum Communication, Computing and Measurement 2, pages 69-74. Springer, 2002. Google Scholar
  52. Henry Yuen. A parallel repetition theorem for all entangled games. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 77:1-77:13, 2016. Google Scholar
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