The sheaf-theoretic structure of non-locality and contextuality

New Journal of Physics

Volumn 13, Issue , 2011, Pages

  Abramsky, Samson ^a   Brandenburger, Adam ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONTEXTUALITY; LINEAR SUBSPACE; LINEAR-ALGEBRAIC; MATHEMATICAL LANGUAGES; NO-GO THEOREM; NON-EXISTENCE; NONLOCALITIES; PROBABILITY TABLES; SHEAF THEORY; TENSOR PRODUCTS;

QUANTUM THEORY;

TENSORS;

EID: 83655197951     PISSN: 13672630     EISSN: None     Source Type: Journal    
DOI: 10.1088/1367-2630/13/11/113036     Document Type: Article

References (61)

1. Bell J S 1964 On the Einstein-Podolsky-Rosen paradox Physics 1 195-200
2. Kochen S and Specker E P 1967 The problem of hidden variables in quantum mechanics J. Math. Mech. 17 59-87
3. Bartosik H, Klepp J, Schmitzer C, Sponar S, Cabello A, Rauch H and Hasegawa Y 2009 Experimental test of quantum contextuality in neutron interferometry Phys. Rev. Lett. 103 40403
4. Kirchmair G, Zähringer F, Gerritsma R, Kleinmann M, Gühne O, Cabello A, Blatt R and Roos C F 2009 State-independent experimental test of quantum contextuality Nature 460 494-7
5. Heywood P and Redhead M L G 1983 Nonlocality and the Kochen-Specker paradox Found. Phys. 13 481-99
6. Stairs A 1983 Quantum logic, realism and value definiteness Phil. Sci. 50 578-602
7. Mermin N D 1990 Simple unified form for the major no-hidden-variables theorems Phys. Rev. Lett. 65 3373-6
8. Cabello A, Severini S and Winter A 2010 Non-contextuality of physical theories as an axiom, arXiv:1010.2163
9. MacLane S and Moerdijk I 1992 Sheaves in Geometry and Logic: a First Introduction to Topos Theory (Berlin: Springer)
10. Popescu S and Rohrlich D 1994 Quantum nonlocality as an axiom Found. Phys. 24 379-85
11. Hardy L 1993 Nonlocality for two particles without inequalities for almost all entangled states Phys. Rev. Lett. 71 1665-8
12. Greenberger D M, Horne M A and Zeilinger A 1989 Going beyond Bell's theorem Bell's Theorem, Quantum Theory and Conceptions of the Universe ed M Kafatos (Dordrecht: Kluwer) pp 69-72
13. Lal R 2011 private communication
14. Wigner E 1932 On the quantum correction for thermodynamic equilibrium Phys. Rev. 40 749
15. Dirac P A M 1942 The physical interpretation of quantum mechanics Proc. R. Soc. A 180 1-40
16. Moyal J E 1949 Quantum mechanics as a statistical theory Math. Proc. Camb. Phil. Soc. 45 99-124
17. Feynman R P 1987 Negative probability Quantum Implications: Essays in Honour of David Bohm ed B J Hiley and F D Peat (London: Routledge) pp 235-48
18. Ghirardi G C, Rimini A and Weber T 1980 A general argument against superluminal transmission through the quantum mechanical measurement process Lett. Nuovo Cimento (1971-85) 27 293-8
19. Abramsky S 2010 Relational Hidden Variables and non-locality Studia Logica at press (arXiv:1007.2754)
20. Fritz T 2009 Possibilistic Physics FQXi Essay http://www.fqxi.org/ community/forum/topic/569
21. Jacobs B 2010 Convexity, duality and effects Theoretical Computer Science-6th IFIP TC 1/WG 2.2 International Conference, TCS 2010 323 1-19
22. Cabello A, Estebaranz J M and García-Alcaine G 1996 Bell-Kochen-Specker theorem: a proof with 18 vectors Phys. Lett. A 212 183-7 (Pubitemid 126392816)
23. Fine A 1982 Hidden variables, joint probability and the Bell inequalities Phys. Rev. Lett. 48 291-5
24. Basoalto R M and Percival I C 2003 Bell Test and CHSH experiments with more than two settings J. Phys. A: Math. Gen. 36 7411
25. Pironio S 2005 Lifting Bell inequalities J. Math. Phys. 46 062112
26. Mansfield S and Fritz T 2011 Hardy's Non-locality Paradox and possibilistic conditions for non-locality arXiv:1105.1819v1
27. Feynman R P 1982 Simulating physics with computers Int. J. Theor. Phys. 21 467-88
28. Peres A 1990 Incompatible results of quantum measurements Phys. Lett. A 151 107-8
29. Mermin N D 1990 Quantum mysteries revisited Am. J. Phys. 58 731-4
30. Elitzur A C, Popescu S and Rohrlich D 1992 Quantum nonlocality for each pair in an ensemble Phys. Lett. A 162 25-8
31. Barrett J, Kent A and Pironio S 2006 Maximally nonlocal and monogamous quantum correlations Phys. Rev. Lett. 97 170409
32. Aolita L, Gallego R, Acín A, Chiuri A, Vallone G, Mataloni P and Cabello A 2011 Fully nonlocal quantum correlations arXiv:1105.3598
33. Montanari U 1974 Networks of constraints: fundamental properties and applications to picture processing Inf. Sci. 7 95-132
34. Kumar V 1992 Algorithms for constraint-satisfaction problems: a survey AI Mag. 13 32
35. Waegell M and Aravind P K 2011 Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres arXiv:1103.6058v1
36. Liang Y C, Spekkens R W and Wiseman H M 2010 Specker's parable of the over-protective seer: implications for contextuality, nonlocality and complementarity arXiv:1010.1273
37. Berge C and Duchet P 1984 Strongly perfect graphs Top. Perfect Graphs 21 57-61
38. Lovász L, Saks M and Schrijver A 1989 Orthogonal representations and connectivity of graphs Linear Algebra Appl. 114 439-54
39. Peres A 1991 Two simple proofs of the Kochen-Specker theorem J. Phys. A: Math. Gen. 24 L175
40. Bub J 1996 Schütte's tautology and the Kochen-Specker theorem Found. Phys. 26 787-06
41. Pavičić M, Merlet J P, McKay B and Megill N D 2005 Kochen-Specker vectors J. Phys. A: Math. Gen. 38 1577
42. Lovász L, Saks M and Schrijver A 2000 A correction: orthogonal representations and connectivity of graphs Linear Algebra Appl. 313 101-6
43. Lovász L 1979 On the Shannon capacity of a graph Inf. Theory IEEE Trans. 25 1-7
44. Dickson W M 1999 Quantum Chance and Non-Locality (Cambridge: Cambridge University Press)
45. Jarrett J P 1984 On the physical significance of the locality conditions in the Bell arguments Noûs 18 569-89
46. Shimony A 1986 Events and processes in the quantum world Quantum Concepts in Space and Time (Oxford: Clarendon) pp 182-203
47. Tsirelson B S 2006 Bell inequalities and operator algebras private communication
48. Scholz V B and Werner R F 2008 Tsirelson's problem arXiv:0812.4305
49. Greenberger D M, Horne M A, Shimony A and Zeilinger A 1990 Bell's theorem without inequalities Am. J. Phys. 58 1131
50. Barrett J 2007 Information processing in generalized probabilistic theories Phys. Rev. A 75 032304
51. Oles F J 1985 Type categories, functor categories and block structure Algebraic Semantics ed M Nivat and J C Reynolds (Cambridge: Cambridge University Press) pp 543-74
52. Tennent R 1986 Functor-category semantics of programming languages and logics Category Theory and Computer Programming (Lecture Notes in Computer Science vol 240) ed D Pitt, S Abramsky, A Poigné and D Rydeheard (Berlin: Springer) pp 206-24
53. Cattani G and Winskel G 1997 Presheaf models for concurrency Computer Science Logic (Berlin: Springer) pp 58-75 (Pubitemid 127097678)
54. Abramsky S, Barbosa R S and Mansfield S 2011 The cohomology of nonlocality and contextuality Electronic Proceedings in Theoretical Computer Science at press
55. Brandenburger A and Keisler H J 2011 What does a hidden variable look like? http://pages.stern. nyu.edu/~abranden/wdhv-03-03-11.pdf
56. Brandenburger A and Yanofsky N 2008 A classification of hidden-variable properties J. Phys. A: Math. Theor. 41 425302
57. Giry M 1982 A categorical approach to probability theory Categorical Aspects of Topology and Analysis (Lecture Notes in Mathematics vol 915) (Berlin: Springer) pp 68-85
58. Isham C J and Butterfield J 1998 Topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations Int. J. Theor. Phys. 37 2669-733 (Pubitemid 128359345)
59. Döring A and Isham C J 2008 A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory J. Math. Phys. 49 053516
60. Heunen C, Landsman N P and Spitters B 2009 A topos for algebraic quantum theory Commun. Math. Phys. 291 63-110
61. Sudarshan E C G and Rothman T 1993 A new interpretation of Bell's inequalities Int. J. Theor. Phys. 32 1077-86