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Linköping University Post Print Quantum contextuality for rational vectors Adan Cabello and Jan-Åke Larsson N.B.: When citing this work, cite the original article. Original Publication: Adan Cabello and Jan-Åke Larsson, Quantum contextuality for rational vectors, 2010, Physics Letters A, (375), 2, 99-99. http://dx.doi.org/10.1016/j.physleta.2010.10.061 Copyright: Elsevier Science B.V., Amsterdam. http://www.elsevier.com/ Postprint available at: Linköping University Electronic Press http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-63933 Quantum contextuality for rational vectors Adán Cabello∗ Departamento de Fı́sica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain Jan-Åke Larsson† arXiv:1010.3733v1 [quant-ph] 18 Oct 2010 Institutionen för Systemteknik, Linköpings Universitet, SE-581 83 Linköping, Sweden (Dated: October 20, 2010) The Kochen-Specker theorem states that noncontextual hidden variable models are inconsistent with the quantum predictions for every yes-no question on a qutrit, corresponding to every projector in three dimensions. It has been suggested [D. A. Meyer, Phys. Rev. Lett. 83, 3751 (1999)] that the inconsistency would disappear when we are restricted to projectors on unit vectors with rational components; that noncontextual hidden variables could reproduce the quantum predictions for rational vectors. Here we show that a qutrit state with rational components violates an inequality valid for noncontextual hidden-variable models [A. A. Klyachko et al., Phys. Rev. Lett. 101, 020403 (2008)] using rational projectors. This shows that the inconsistency remains even when using only rational vectors. PACS numbers: 03.65.Ta, 03.65.Ud The Kochen-Specker theorem from 1967 [1] states that the quantum predictions from a three-dimensional quantum system (a qutrit) are inconsistent with noncontextual hidden variables. The proof uses 117 directions in three dimensions, arranged in a pattern such that they cannot be colored in a particular manner, see [1] for details. Later proofs use less directions, but one common feature (in the three-dimensional versions) is that the set of unit vectors includes irrational components. It was noted in [2] that the Kochen-Specker proof needs these irrational vectors to be completed. Indeed, when using only the rational subset of vectors, the set is colorable in the manner required by quantum mechanics. It was also suggested [2] that, for this reason, the inconsistency between quantum mechanics and noncontextual hidden variables disappears, and that quantum mechanics can be imitated by noncontextual hidden variable models restricted to rational vectors. It has been recently shown [3] that the following inequality is a necessary and sufficient condition for qutrit noncontextual hidden variables, for measurements Ai with possible outcomes −1 and +1, such that Ai and Ai+1 (modulo 5) are compatible: 4 X hAi Ai+1 i ≥ −3. associated to the rational vectors hv0 | = (1, 0, 0) , (4a) hv1 | = (0, 1, 0) , 48 55 hv2 | = , , 0, − 73 73 1925 2052 1680 hv3 | = , , , 3277 3277 3277 140 171 , ,− hv4 | = 0, 221 221 (4b) (4c) (4d) (4e) we obtain a value of −3.941 for the left-hand side of (1), which deviates very little from the maximum violation at −3.944. Thus, even when using only rational vectors, the inconsistency is not nullified. The violation shows that the (physical content of) the Kochen-Specker theorem remains, namely, that the quantum-mechanical predictions cannot be reproduced by noncontextual hidden variables. (1) i=0 Using the rational qutrit state 354 357 158 , , ,− hψ| = 527 527 527 ∗ (2) and the observables Ai = 2|vi ihvi | − 11, [email protected] [email protected] [1] S. Kochen and E. P. Specker, J. Math. Mech. 17, 59 (1967). [2] D. A. Meyer, Phys. Rev. Lett. 83, 3751 (1999). [3] A. A. Klyachko, M. A. Can, S. Binicioğlu, and A. S. Shumovsky, Phys. Rev. Lett. 101, 020403 (2008). † (3)