For any finite number of parts, measurements, and outcomes in a Bell scenario, we estimate the probability of random N-qudit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension, the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the local Hilbert space dimension goes to infinity.
REFERENCES
1.
J. S.
Bell
, Rev. Mod. Phys.
38
, 447
(1966
).2.
E. R.
Loubenets
, J. Math. Phys.
53
, 022201
(2012
).3.
E. R.
Loubenets
, Found. Phys.
47
, 1100
(2017
).4.
N.
Brunner
, D.
Cavalcanti
, S.
Pironio
, V.
Scarani
, and S.
Wehner
, Rev. Mod. Phys.
86
, 419
(2014
).5.
A.
Cabello
, “Pruebas algebraicas de imposibilidad de variables ocultas en Mecánica Cuántica
,” Ph.D. thesis, Universidad Complutense de Madrid
, 1996
.6.
T.
Vértesi
and N.
Brunner
, Nat. Commun.
5
, 5297
(2014
).7.
J. B. D.
Rosset
and N.
Gisin
, J. Phys. A: Math. Theor.
47
, 424022
(2014
).8.
D.
Avis
, S.
Moriyama
, and M.
Owari
, IEICE Trans. Fundam. Electron. Commun. Comput. Sci.
E92-A
, 1254
(2009
).9.
J. I.
de Vicente
, J. Phys. A: Math. Theor.
47
, 424017
(2014
).10.
J.
Barrett
, N.
Linden
, S.
Massar
, S.
Pironio
, S.
Popescu
, and D.
Roberts
, Phys. Rev. A
71
, 022101
(2005
).11.
A.
Aspect
, Phys. Lett. A
54
, 117
(1975
).12.
B.
Hensen
, H.
Bernien
, A. E.
Dreau
, A.
Reiserer
, N.
Kalb
, M. S.
Blok
, J.
Ruitenberg
, R. F. L.
Vermeulen
, R. N.
Schouten
, C.
Abellan
, W.
Amaya
, V.
Pruneri
, M. W.
Mitchell
, M.
Markham
, D. J.
Twitchen
, D.
Elkouss
, S.
Wehner
, T. H.
Taminiau
, and R.
Hanson
, Nature
526
, 682
(2015
).13.
B.
Hensen
, N.
Kalb
, M. S.
Blok
, A. E.
Dréau
, A.
Reiserer
, R. F. L.
Vermeulen
, R. N.
Schouten
, M.
Markham
, D. J.
Twitchen
, K.
Goodenough
, D.
Elkouss
, S.
Wehner
, T. H.
Taminiau
, and R.
Hanson
, Sci. Rep.
6
, 30289
(2016
).14.
S. J.
Freedman
and J. F.
Clauser
, Phys. Rev. Lett.
28
, 938
(1972
).15.
A.
Winter
, J. Phys. A: Math. Theor.
47
, 424031
(2014
).16.
C. E.
González-Guillén
, C. H.
Jiménez
, C.
Palazuelos
, and I.
Villanueva
, in 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015), Leibniz International Proceedings in Informatics (LIPIcs)
, edited by S.
Beigi
and R.
Koenig
(Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl
, Germany
, 2015
), Vol. 44, pp. 39
–47
.17.
C. E.
González-Guillén
, C.
Lancien
, C.
Palazuelos
, and I.
Villanueva
, Ann. Henri Poincaré
18
, 3793
(2017
).18.
C.
Palazuelos
and Z.
Yin
, Phys. Rev. A
92
, 052313
(2015
).19.
20.
N.
Alon
and J. H.
Spencer
, The Probabilistic Method
, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley-Interscience
, 2008
) 3rd edition.21.
J.
Briët
and T.
Vidick
, Commun. Math. Phys.
321
, 181
(2013
).22.
Y.-C.
Liang
, N.
Harrigan
, S. D.
Bartlett
, and T.
Rudolph
, Phys. Rev. Lett.
104
, 050401
(2010
).23.
R. C.
Drumond
and R. I.
Oliveira
, Phys. Rev. A
86
, 012117
(2012
).24.
R.
Horodecki
, P.
Horodecki
, M.
Horodecki
, and K.
Horodecki
, Rev. Mod. Phys.
81
, 865
(2009
).25.
S.
Pironio
, V.
Scarani
, and T.
Vidick
, New J. Phys.
18
, 100202
(2016
).26.
Notice that there is no loss of generality in supposing that every box has access to the same number m of inputs as well as to the same number v of outputs. This abstraction could be realized considering extra buttons that are never pushed and additional light bulbs that are never turned on.
27.
B. S.
Tsirelson
, Hadronic J. Suppl.
8
, 329
–345
(1993
).29.
A.
Schrijver
, Combinatorial Optimization: Polyhedra and Efficiency
, Algorithms and Combinatorics (Springer
, 2003
).30.
31.
We say that two Bell inequalities are equivalent if a behavior violates one of them if and only if it also violates the other.
32.
33.
A.
Cabello
, S.
Severini
, and A.
Winter
, Phys. Rev. Lett.
112
, 040401
(2014
).34.
R.
Rafael
, D.
Cristhiano
, J. L.-T.
Antonio
, T. C.
Marcelo
, and C.
Adan
, J. Phys. A: Math. Theor.
47
, 424021
(2014
).35.
D.
Collins
, N.
Gisin
, N.
Linden
, S.
Massar
, and S.
Popescu
, Phys. Rev. Lett.
88
, 040404
(2002
).36.
S.
López-Rosa
, Z.-P.
Xu
, and A.
Cabello
, Phys. Rev. A
94
, 062121
(2016
).37.
C.
Śliwa
, Phys. Lett. A
317
, 165
(2003
).38.
M.
Sadiq
, P.
Badzia̧g
, M.
Bourennane
, and A.
Cabello
, Phys. Rev. A
87
, 012128
(2013
).39.
M.
Froissart
, Il Nuovo Cimento B
64
, 241
(1981
).40.
D.
Collins
and N.
Gisin
, J. Phys. A: Math. Gen.
37
, 1775
(2004
).41.
N.
Brunner
and N.
Gisin
, Phys. Lett. A
372
, 3162
(2008
).42.
S.
Popescu
and D.
Rohrlich
, Phys. Lett. A
166
, 293
(1992
).43.
D.
Cavalcanti
, M. L.
Almeida
, V.
Scarani
, and A.
Acin
, Nat. Commun.
2
, 184
(2011
).44.
M.
Ledoux
, The Concentration of Easure Phenomenon
, 2nd ed. (American Mathematical Society
, USA
, 2001
).45.
D.
Haussler
and E.
Welzl
, Discrete Comput. Geom.
2
, 127
(1987
).46.
47.
D.
Pérez-García
, M. M.
Wolf
, C.
Palazuelos
, I.
Villanueva
, and M.
Junge
, Commun. Math. Phys.
279
, 455
(2008
).48.
An apparently very drastic behavior like is already enough for our purposes.
49.
M. P.
Müller
, E.
Adlam
, L.
Masanes
, and N.
Wiebe
, Commun. Math. Phys.
340
, 499
(2015
).50.
F. G. S. L.
Brandão
, A. W.
Harrow
, and M.
Horodecki
, Commun. Math. Phys.
346
, 397
(2016
).51.
S.
Popescu
, A.
Short
, and A.
Winter
, Nat. Phys.
2
, 754
(2006
).52.
B.
Amaral
, “The Exclusivity principle and the set o quantum distributions
,” Ph.D. thesis, Universidade Federal de Minas Gerais
, 2014
.53.
N.
Gisin
, Phys. Lett. A
154
, 201
(1991
).© 2018 Author(s).
2018
Author(s)
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