A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations
Computing and Mathematical Sciences, Caltech
Published: | 2020-06-18, volume 4, page 282 |
Eprint: | arXiv:1904.02350v3 |
Doi: | https://doi.org/10.22331/q-2020-06-18-282 |
Citation: | Quantum 4, 282 (2020). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We describe a two-player non-local game, with a fixed small number of questions and answers, such that an $\epsilon$-close to optimal strategy requires an entangled state of dimension $2^{\Omega(\epsilon^{-1/8})}$. Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick [17]. It reduces the number of players from three to two, as well as the question and answer set sizes. Moreover, it provides an (arguably) elementary proof of the non-closure of the set of quantum correlations, based on embezzlement and self-testing. In contrast, previous proofs [26,16,19] involved representation theoretic machinery for finitely-presented groups and $C^*$-algebras.
► BibTeX data
► References
[1] Antonio Acín, Serge Massar, and Stefano Pironio. Randomness versus nonlocality and entanglement. Physical Review Letters, 108(10):100402, 2012. https://doi.org/10.1103/PhysRevLett.108.100402.
https://doi.org/10.1103/PhysRevLett.108.100402
[2] Jop Briët, Harry Buhrman, and Ben Toner. A generalized grothendieck inequality and nonlocal correlations that require high entanglement. Communications in mathematical physics, 305(3):827–843, 2011. https://doi.org/10.1007/s00220-011-1280-3.
https://doi.org/10.1007/s00220-011-1280-3
[3] John S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195–200, 1964. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195.
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
[4] Nicolas Brunner, Miguel Navascués, and Tamás Vértesi. Dimension witnesses and quantum state discrimination. Physical review letters, 110(15):150501, 2013. https://doi.org/10.1103/PhysRevLett.110.150501.
https://doi.org/10.1103/PhysRevLett.110.150501
[5] Cédric Bamps and Stefano Pironio. Sum-of-squares decompositions for a family of clauser-horne-shimony-holt-like inequalities and their application to self-testing. Physical Review A, 91(5):052111, 2015. https://doi.org/10.1103/PhysRevA.91.052111.
https://doi.org/10.1103/PhysRevA.91.052111
[6] 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. https://doi.org/10.1103/PhysRevLett.100.210503.
https://doi.org/10.1103/PhysRevLett.100.210503
[7] Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. All pure bipartite entangled states can be self-tested. Nature communications, 8:15485, 2017. https://doi.org/10.1038/ncomms15485.
https://doi.org/10.1038/ncomms15485
[8] John F Clauser, Michael A Horne, Abner Shimony, and Richard A Holt. Proposed experiment to test local hidden-variable theories. Physical review letters, 23(15):880, 1969. https://doi.org/10.1103/PhysRevLett.23.880.
https://doi.org/10.1103/PhysRevLett.23.880
[9] Matthew Coudron and Anand Natarajan. The parallel-repeated magic square game is rigid. arXiv preprint arXiv:1609.06306, 2016.
arXiv:1609.06306
[10] Andrea Coladangelo. Parallel self-testing of (tilted) epr pairs via copies of (tilted) chsh and the magic square game. Quantum Information & Computation, 17(9-10):831–865, 2017. https://doi.org/10.26421/QIC17.9-10.
https://doi.org/10.26421/QIC17.9-10
[11] Andrea Coladangelo. Generalization of the clauser-horne-shimony-holt inequality self-testing maximally entangled states of any local dimension. Physical Review A, 98(5):052115, 2018. https://doi.org/10.1103/PhysRevA.98.052115.
https://doi.org/10.1103/PhysRevA.98.052115
[12] Rui Chao, Ben W Reichardt, Chris Sutherland, and Thomas Vidick. Test for a large amount of entanglement, using few measurements. Quantum, 2:92, 2018. https://doi.org/10.22331/q-2018-09-03-92.
https://doi.org/10.22331/q-2018-09-03-92
[13] Andrea Coladangelo and Jalex Stark. Robust self-testing for linear constraint system games. arXiv preprint arXiv:1709.09267, 2017.
arXiv:1709.09267
[14] Andrea Coladangelo and Jalex Stark. Separation of finite and infinite-dimensional quantum correlations, with infinite question or answer sets. arXiv preprint arXiv:1708.06522, 2017.
arXiv:1708.06522
[15] Andrea Coladangelo and Jalex Stark. Unconditional separation of finite and infinite-dimensional quantum correlations. arXiv preprint arXiv:1804.05116, 2018.
arXiv:1804.05116
[16] Ken Dykema, Vern I Paulsen, and Jitendra Prakash. Non-closure of the set of quantum correlations via graphs. Communications in Mathematical Physics, pages 1–18, 2017. https://doi.org/10.1007/s00220-019-03301-1.
https://doi.org/10.1007/s00220-019-03301-1
[17] Zhengfeng Ji, Debbie Leung, and Thomas Vidick. A three-player coherent state embezzlement game. arXiv preprint arXiv:1802.04926, 2018.
arXiv:1802.04926
[18] Debbie Leung, Ben Toner, and John Watrous. Coherent state exchange in multi-prover quantum interactive proof systems. Chicago Journal of Theoretical Computer Science, 11(2013):1, 2013. https://doi.org/10.4086/cjtcs.2013.011.
https://doi.org/10.4086/cjtcs.2013.011
[19] Magdalena Musat and Mikael Rørdam. Non-closure of quantum correlation matrices and factorizable channels that require infinite dimensional ancilla. arXiv preprint arXiv:1806.10242, 2018. https://doi.org/10.1007/s00220-019-03449-w.
https://doi.org/10.1007/s00220-019-03449-w
arXiv:1806.10242
[20] Laura Mančinska and Thomas Vidick. Unbounded entanglement in nonlocal games. International Colloquium on Automata, Languages, and Programming, pages 835–846, 2014. https://doi.org/10.1007/978-3-662-43948-7_69.
https://doi.org/10.1007/978-3-662-43948-7_69
[21] Dominic Mayers and Andrew Yao. Self testing quantum apparatus. Quantum Information & Computation, 4(4):273–286, 2004. https://doi.org/10.26421/QIC4.4.
https://doi.org/10.26421/QIC4.4
[22] Matthew McKague, Tzyh Haur Yang, and Valerio Scarani. Robust self-testing of the singlet. Journal of Physics A: Mathematical and Theoretical, 45(45):455304, 2012. https://doi.org/10.1088/1751-8113/45/45/455304.
https://doi.org/10.1088/1751-8113/45/45/455304
[23] 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. https://doi.org/10.1145/3055399.3055468.
https://doi.org/10.1145/3055399.3055468
[24] William Slofstra. Lower bounds on the entanglement needed to play xor non-local games. Journal of Mathematical Physics, 52(10):102202, 2011. https://doi.org/10.1063/1.3652924.
https://doi.org/10.1063/1.3652924
[25] William Slofstra. A group with at least subexponential hyperlinear profile. arXiv preprint arXiv:1806.05267, 2018.
arXiv:1806.05267
[26] William Slofstra. The set of quantum correlations is not closed. In Forum of Mathematics, Pi, volume 7. Cambridge University Press, 2019. https://doi.org/10.1017/fmp.2018.3.
https://doi.org/10.1017/fmp.2018.3
[27] William Slofstra. Tsirelson’s problem and an embedding theorem for groups arising from non-local games. Journal of the American Mathematical Society, 33(1):1–56, 2020. https://doi.org/10.1090/jams/929.
https://doi.org/10.1090/jams/929
[28] William Slofstra and Thomas Vidick. Entanglement in non-local games and the hyperlinear profile of groups. In Annales Henri Poincaré, volume 19, pages 2979–3005. Springer, 2018. https://doi.org/10.1007/s00023-018-0718-y.
https://doi.org/10.1007/s00023-018-0718-y
[29] Volkher B Scholz and Reinhard F Werner. Tsirelson's problem. arXiv preprint arXiv:0812.4305, 2008.
arXiv:0812.4305
[30] Wim van Dam and Patrick Hayden. Universal entanglement transformations without communication. Physical Review A, 67(6):060302, 2003. https://doi.org/10.1103/PhysRevA.67.060302.
https://doi.org/10.1103/PhysRevA.67.060302
[31] Tzyh Haur Yang and Miguel Navascués. Robust self-testing of unknown quantum systems into any entangled two-qubit states. Physical Review A, 87(5):050102, 2013. https://doi.org/10.1103/PhysRevA.87.050102.
https://doi.org/10.1103/PhysRevA.87.050102
Cited by
[1] Ivan Šupić and Joseph Bowles, "Self-testing of quantum systems: a review", Quantum 4, 337 (2020).
[2] Ivan Šupić, "Nonlocality strikes again", Quantum Views 4, 38 (2020).
[3] Andrea Coladangelo and Jalex Stark, "An inherently infinite-dimensional quantum correlation", Nature Communications 11 1, 3335 (2020).
[4] Matthias Christandl, Nicholas Gauguin Houghton-Larsen, and Laura Mancinska, "An Operational Environment for Quantum Self-Testing", Quantum 6, 699 (2022).
[5] Shubhayan Sarkar, "An Operational Notion of Classicality Based on Physical Principles", Foundations of Physics 53 2, 47 (2023).
[6] Connor Paddock, William Slofstra, Yuming Zhao, and Yangchen Zhou, "An Operator-Algebraic Formulation of Self-testing", Annales Henri Poincaré (2023).
[7] Zhengfeng Ji, Debbie Leung, and Thomas Vidick, "A three-player coherent state embezzlement game", Quantum 4, 349 (2020).
[8] Salman Beigi, "Separation of quantum, spatial quantum, and approximate quantum correlations", Quantum 5, 389 (2021).
[9] Shubhayan Sarkar, Debashis Saha, Jędrzej Kaniewski, and Remigiusz Augusiak, "Self-testing quantum systems of arbitrary local dimension with minimal number of measurements", npj Quantum Information 7 1, 151 (2021).
[10] Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, "MIP*=RE", arXiv:2001.04383, (2020).
[11] Patryk Lipka-Bartosik, Henrik Wilming, and Nelly H. Y. Ng, "Catalysis in Quantum Information Theory", arXiv:2306.00798, (2023).
[12] Rui Chao and Ben W. Reichardt, "Quantum dimension test using the uncertainty principle", arXiv:2002.12432, (2020).
The above citations are from Crossref's cited-by service (last updated successfully 2023-12-07 09:44:43) and SAO/NASA ADS (last updated successfully 2023-12-07 09:44:44). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
Pingback: Perspective in Quantum Views by Ivan Šupić "Nonlocality strikes again"