We provide a complete geometric description of the set of synchronous quantum correlations for the three-experiment two-outcome scenario. We show that these correlations form a closed set. Moreover, every correlation in this set can be realized using projection valued measures on a Hilbert space of dimension no more than 16.

1.
J. S.
Bell
, “
On the Einstein Podolsky Rosen paradox
,”
Phys. Phys. Fiz.
1
,
195
200
(
1964
).
2.
C.
Bennett
and
G.
Brassard
, “
Quantum cryptography: Public key distribution and coin tossing
,”
Theor. Comput. Sci.
560
(Pt. 1),
7
11
(
2014
).
3.
T.
Vidick
and
S.
Wehner
, “
More nonlocality with less entanglement
,”
Phys. Rev. A
83
,
052310
(
2011
).
4.
D.
Collins
and
N.
Gisin
, “
A relevant two qubit bell inequality inequivalent to the CHSH inequality
,”
J. Phys. A: Math. Gen.
37
,
1775
1787
(
2004
).
5.
M.
Froissart
, “
Constructive generalization of bell’s inequalities
,”
Il Nuovo Cimento B (1971-1996)
64
,
241
251
(
1981
).
6.
K. F.
Pal
and
T.
Vertesi
, “
Maximal violation of a bipartite three-setting, two-outcome bell inequality using infinite-dimensional quantum systems
,”
Phys. Rev. A
82
,
022116
(
2010
).
7.
K.
Dykema
,
V. I.
Paulsen
, and
J.
Prakash
, “
The delta game
,”
Quantum Inf. Comput.
18
(7,8),
599
616
(
2018
).
8.
W.
Slofstra
, “
The set of quantum correlations is not closed
,”
Forum Math. Pi
7
(
e1
),
41
(
2019
).
9.
K.
Dykema
,
V. I.
Paulsen
, and
J.
Prakash
, “
Non-closure of the set of quantum correlations via graphs
,”
Commun. Math. Phys.
365
,
1125
1142
(
2019
).
10.
M.
Junge
,
M.
Navascues
,
C.
Palazuelos
,
D.
Perez-Garcia
,
V. B.
Scholz
, and
R. F.
Werner
, “
Connes’ embedding problem and Tsirelson’s problem
,”
J. Math. Phys.
52
,
012102
(
2011
).
11.
T.
Fritz
, “
Tsirelson’s problem and Kirchberg’s conjecture
,”
Rev. Math. Phys.
24
,
1250012
(
2012
).
12.
N.
Ozawa
, “
About the Connes embedding conjecture: Algebraic approaches
,”
Jpn. J. Math.
8
,
147
183
(
2013
).
13.
K.
Dykema
and
V. I.
Paulsen
, “
Synchronous correlation matrices and Connes’ embedding conjecture
,”
J. Math. Phys.
57
,
015214
(
2016
).
14.
S.-J.
Kim
,
V. I.
Paulsen
, and
C.
Schafhauser
, “
A synchronous game for binary constraint systems
,”
J. Math. Phys.
59
,
032201
(
2018
).
15.
K.
Goh
,
J.
Kaniewski
,
E.
Wolfe
,
T.
Vértesi
,
X.
Wu
,
Y.
Cai
,
Y.
Liang
, and
V.
Scarani
, “
Geometry of the set of quantum correlations
,”
Phys. Rev. A
97
,
022104
(
2018
).
16.
L. P.
Thinh
,
A.
Varvitsiotis
, and
Y.
Cai
, “
Geometric structure of quantum correlators via semidefinite programming
,”
Phys. Rev. A
99
,
052108
(
2019
).
17.
J. F.
Clauser
,
M. A.
Horne
,
A.
Shimony
, and
R. A.
Holt
, “
Proposed experiment to test local hidden-variable theories
,”
Phys. Rev. Lett.
23
,
880
884
(
1969
).
18.
A.
Einstein
,
B.
Podolsky
, and
N.
Rosen
, “
Can quantum-mechanical description of physical reality be considered complete?
,”
Phys. Rev.
47
,
777
780
(
1935
).
19.
A.
Aspect
,
P.
Grangier
, and
G.
Roger
, “
Experimental tests of realistic local theories via bell’s theorem
,”
Phys. Rev. Lett.
47
,
460
463
(
1981
).
20.
V. I.
Paulsen
,
S.
Severini
,
D.
Stahlke
,
I. G.
Todorov
, and
A.
Winter
, “
Estimating quantum chromatic numbers
,”
J. Funct. Anal.
270
,
2188
2222
(
2016
).
21.
K. R.
Davidson
,
C*-algebras by Example
, Fields Institute Monographs Vol. 6 (
American Mathematical Society
,
Providence, RI
,
1996
), p.
xiv+309
.
22.
B.
Lackey
and
N.
Rodrigues
, “
Nonlocal games, synchronous correlations, and bell inequalities
,” arXiv:1707.06200 (
2017
).
23.
M.
Lupini
,
L.
Mancinska
,
V. I.
Paulsen
, et al., “
Perfect strategies for non-local games
,”
Math. Phys. Anal. Geom.
23
,
7
(
2020
).
24.
E.
Alhajjar
and
T.
Russell
, “
Maximally entangled correlation sets
,”
Houston J. Math.
(unpublished) (
2019
).
25.
B. S.
Cirel’son
, “
Quantum generalizations of bell’s inequality
,”
Lett. Math. Phys.
4
,
93
100
(
1980
).
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