Geodesic
THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED
Forum of Mathematics, Pi, Tome 7 (2019)

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
@article{10_1017_fmp_2018_3,
     author = {WILLIAM SLOFSTRA},
     title = {THE {SET} {OF} {QUANTUM} {CORRELATIONS} {IS} {NOT} {CLOSED}},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {7},
     year = {2019},
     doi = {10.1017/fmp.2018.3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2018.3/}
}
TY  - JOUR
AU  - WILLIAM SLOFSTRA
TI  - THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED
JO  - Forum of Mathematics, Pi
PY  - 2019
VL  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2018.3/
DO  - 10.1017/fmp.2018.3
LA  - en
ID  - 10_1017_fmp_2018_3
ER  - 
%0 Journal Article
%A WILLIAM SLOFSTRA
%T THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED
%J Forum of Mathematics, Pi
%D 2019
%V 7
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2018.3/
%R 10.1017/fmp.2018.3
%G en
%F 10_1017_fmp_2018_3
WILLIAM SLOFSTRA. THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED. Forum of Mathematics, Pi, Tome 7 (2019). doi: 10.1017/fmp.2018.3

[1] Baumslag, G., ‘On the residual finiteness of generalised free products of nilpotent groups’, Trans. Amer. Math. Soc. 106(2) (1963), 193–209.Google Scholar

[2] Bell, J. S., ‘On the Einstein Podolsky Rosen paradox’, Physics 1(3) (1964), 195–200.Google Scholar

[3] 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

[4] 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

[5] Cleve, R., Liu, L. and Slofstra, W., ‘Perfect commuting-operator strategies for linear system games’, J. Math. Phys. 58 012202 (2017).Google Scholar

[6] 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

[7] Dykema, K. J. and Paulsen, V., ‘Synchronous correlation matrices and Connes’ embedding conjecture’, J. Math. Phys. 57(1) 015214 (2016).Google Scholar

[8] Filonov, N. and Kachkovskiy, I., ‘A Hilbert–Schmidt analog of Huaxin Lin’s theorem’. Preprint, 2010, arXiv:1008.4002.Google Scholar

[9] Fritz, T., ‘Tsirelson’s problem and Kirchberg’s conjecture’, Rev. Math. Phys. 24(05) 1250012 (2012).Google Scholar

[10] Glebsky, L., ‘Almost commuting matrices with respect to normalized Hilbert–Schmidt norm’. Preprint, 2010, arXiv:1002.3082.Google Scholar

[11] Ji, Z., ‘Binary constraint system games and locally commutative reductions’. Preprint, 2013, arXiv:1310.3794 [quant-ph].Google Scholar

[12] 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

[13] 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

[14] Kharlampovich, O. G., ‘A finitely presented solvable group with unsolvable word problem’, Math. USSR Izvestija 19(1) (1982), 151–169.Google Scholar

[15] Kharlampovich, O. G., Myasnikov, A. and Sapir, M., ‘Algorithmically complex residually finite groups’, Bull. Math. Sci. 7(2) (2017), 309–352.Google Scholar

[16] 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

[17] 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

[18] Debbie Leung and Bingjie Wang, in preparation.Google Scholar

[19] Loring, T. A., Lifting Solutions to Perturbing Problems in C ∗ -algebras, Fields Institute monographs (American Mathematical Society, Providence, RI, 1997).Google Scholar

[20] 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

[21] 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

[22] Mermin, N. David, ‘Simple unified form for the major no-hidden-variables theorems’, Phys. Rev. Lett. 65(27) (1990), 3373–3376.Google Scholar

[23] 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

[24] Ozawa, N., ‘About the Connes embedding conjecture’, Jpn. J. Math. 8(1) (2013), 147–183.Google Scholar

[25] 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

[26] Peres, A., ‘Incompatible results of quantum measurements’, Phys. Lett. A 151(3) (1990), 107–108.Google Scholar

[27] Pestov, V. G., ‘Hyperlinear and sofic groups: a brief guide’, Bull. Symbolic Logic 14(4) (2008), 449–480.Google Scholar

[28] Regev, O. and Vidick, T., ‘Quantum XOR games’, ACM Trans. Comput. Theory 7(4) (2015), 15:1–15:43.Google Scholar

[29] Scholz, V. B. and Werner, R. F., ‘Tsirelson’s problem’. Preprint, 2008, arXiv:0812.4305.Google Scholar

[30] 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

[31] Slofstra, W., ‘Tsirelson’s problem and an embedding theorem for groups arising from non-local games’. Preprint, 2016, arXiv:1606.03140.Google Scholar

[32] 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

[33] 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

Cité par Sources :