Hostname: page-component-5cf477f64f-2wr7h Total loading time: 0 Render date: 2025-04-07T12:26:13.532Z Has data issue: false hasContentIssue false
  • English
  • Français

THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  14 January 2019

WILLIAM SLOFSTRA
WILLIAM SLOFSTRA*
Affiliation:
Department of Pure Mathematics and Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.

MSC classification

Primary: 81P45: Quantum information, communication, networks
Secondary: 20F05: Generators, relations, and presentations
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 7 , 2019 , e1
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits noncommercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author 2019

References

Baumslag, G., ‘On the residual finiteness of generalised free products of nilpotent groups’, Trans. Amer. Math. Soc. 106(2) (1963), 193–209.Google Scholar
Bell, J. S., ‘On the Einstein Podolsky Rosen paradox’, Physics 1(3) (1964), 195–200.Google Scholar
Burgdorf, S., Laurent, M. and Piovesan, T., ‘On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings’, Electron. J. Linear Algebra 32 (2017), 15–40.Google Scholar
Capraro, V. and Lupini, M., Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture, Lecture Notes in Mathematics, 2136 (Springer, Switzerland, 2015).Google Scholar
Cleve, R., Liu, L. and Slofstra, W., ‘Perfect commuting-operator strategies for linear system games’, J. Math. Phys. 58 012202 (2017).Google Scholar
Cleve, R. and Mitta, R., ‘Characterization of binary constraint system games’, inAutomata, Languages, and Programming, Lecture Notes in Computer Science, 8572 (Springer, Heidelberg, 2014), 320–331.Google Scholar
Dykema, K. J. and Paulsen, V., ‘Synchronous correlation matrices and Connes’ embedding conjecture’, J. Math. Phys. 57(1) 015214 (2016).Google Scholar
Filonov, N. and Kachkovskiy, I., ‘A Hilbert–Schmidt analog of Huaxin Lin’s theorem’. Preprint, 2010, arXiv:1008.4002.Google Scholar
Fritz, T., ‘Tsirelson’s problem and Kirchberg’s conjecture’, Rev. Math. Phys. 24(05) 1250012 (2012).Google Scholar
Glebsky, L., ‘Almost commuting matrices with respect to normalized Hilbert–Schmidt norm’. Preprint, 2010, arXiv:1002.3082.Google Scholar
Ji, Z., ‘Binary constraint system games and locally commutative reductions’. Preprint, 2013, arXiv:1310.3794 [quant-ph].Google Scholar
Ji, Z., ‘Compression of quantum multi-prover interactive proofs’, inProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 (ACM, New York, 2017), 289–302.Google Scholar
Junge, M., Navascués, M., Palazuelos, C., Pérez-García, D., Scholz, V. B. and Werner, R. F., ‘Connes’ embedding problem and Tsirelson’s problem’, J. Math. Phys. 52(1) 012102 (2011).Google Scholar
Kharlampovich, O. G., ‘A finitely presented solvable group with unsolvable word problem’, Math. USSR Izvestija 19(1) (1982), 151–169.Google Scholar
Kharlampovich, O. G., Myasnikov, A. and Sapir, M., ‘Algorithmically complex residually finite groups’, Bull. Math. Sci. 7(2) (2017), 309–352.Google Scholar
Laurent, M. and Piovesan, T., ‘Conic Approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone’, SIAM J. Optim. 25(4) (2015), 2461–2493.Google Scholar
Leung, D., Toner, B. and Watrous, J., ‘Coherent state exchange in multi-prover quantum interactive proof systems’, Chic. J. Theoret. Comput. Sci. 19(1) (2013), 1–18.Google Scholar
Debbie Leung and Bingjie Wang, in preparation.Google Scholar
Loring, T. A., Lifting Solutions to Perturbing Problems in C ∗ -algebras, Fields Institute monographs (American Mathematical Society, Providence, RI, 1997).Google Scholar
Mal’cev, A. I., ‘On the faithful representations of infinite groups of matrices’, Amer. Math. Soc. Transl. Ser. 2 45 (1965), 1–18.Google Scholar
Manc̆inska, L. and Vidick, T., ‘Unbounded entanglement can be needed to achieve the optimal success probability’, inAutomata, Languages, and Programming, Lecture Notes in Computer Science, 8572 (Springer, Heidelberg, 2014), 835–846.Google Scholar
Mermin, N. David, ‘Simple unified form for the major no-hidden-variables theorems’, Phys. Rev. Lett. 65(27) (1990), 3373–3376.Google Scholar
Navascués, M., Pironio, S. and Acín, A., ‘A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations’, New J. Phys. 10(7) 073013 (2008).Google Scholar
Ozawa, N., ‘About the Connes embedding conjecture’, Jpn. J. Math. 8(1) (2013), 147–183.Google Scholar
Pál, K. F. and Vértesi, T., ‘Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems’, Phys. Rev. A 82(2) 022116 (2010).Google Scholar
Peres, A., ‘Incompatible results of quantum measurements’, Phys. Lett. A 151(3) (1990), 107–108.Google Scholar
Pestov, V. G., ‘Hyperlinear and sofic groups: a brief guide’, Bull. Symbolic Logic 14(4) (2008), 449–480.Google Scholar
Regev, O. and Vidick, T., ‘Quantum XOR games’, ACM Trans. Comput. Theory 7(4) (2015), 15:1–15:43.Google Scholar
Scholz, V. B. and Werner, R. F., ‘Tsirelson’s problem’. Preprint, 2008, arXiv:0812.4305.Google Scholar
Sikora, J. and Varvitsiotis, A., ‘Linear conic formulations for two-party correlations and values of nonlocal games’, Math. Program. 162(1–2) (2017), 431–463.Google Scholar
Slofstra, W., ‘Tsirelson’s problem and an embedding theorem for groups arising from non-local games’. Preprint, 2016, arXiv:1606.03140.Google Scholar
Tsirelson, B. S., Bell inequalities and operator algebras. Problem statement for website of open problems at TU Braunschweig, 2006, available at http://web.archive.org/web/20090414083019/http://www.imaph.tu-bs.de/qi/problems/33.html.Google Scholar
Wehner, S., Christandl, M. and Doherty, A. C., ‘Lower bound on the dimension of a quantum system given measured data’, Phys. Rev. A 78(6) 062112 (2008).Google Scholar