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How orthogonalities set Kochen-Specker sets
Stockholm University, Faculty of Science, Department of Physics.
2011 (English)In: AIP conference proceedings vol. 1327, American Institute of Physics (AIP), 2011, 326-328 p.Conference paper, Published paper (Refereed)
Abstract [en]

We look at generalisations of sets of vectors proving the Kochen-Specker theorem in 3 and 4 dimensions. It has been shown that two such sets, although unitarily inequivalent, are part of a larger 3-parameter family of vectors that share the same orthogonality graph. We find that these sets are unusual, in that the vectors in all other Kochen-Specker sets investigated here are fully determined by orthogonality conditions and thus admit no free parameters.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2011. 326-328 p.
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-87992DOI: 10.1063/1.3567454OAI: oai:DiVA.org:su-87992DiVA: diva2:608650
Conference
AQT - The International Conference on Advances in Quantum Theory, 14–17 June 2010, Växjö, Sweden
Available from: 2013-02-28 Created: 2013-02-28 Last updated: 2014-09-03Bibliographically approved
In thesis
1. Geometry and foundations of quantum mechanics
Open this publication in new window or tab >>Geometry and foundations of quantum mechanics
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis explores three notions in the foundations of quantum mechanics: mutually unbiased bases (MUBs), symmetric informationally-complete positive operator valued measures (SICs) and contextuality. MUBs and SICs are sets of vectors corresponding to special measurements in quantum mechanics, but there is no proof of their existence in all dimensions. We look at the MUB constructions by Ivanović and Alltop in prime dimensions and highlight the important role played by the Weyl-Heisenberg and Clifford groups. We investigate how these MUBs are related, first invoking the third level of the Clifford hierarchy and then examining their geometrical features in probability simplices and Grassmannian spaces. There is a special connection between SICs and elliptic curves in dimension three, known as the Hesse configuration, which we discuss before looking for higher dimensional generalisations. Contextuality is introduced in relation to hidden variable models, where sets of vectors show the impossibility of assigning non-contextual outcomes to their corresponding measurements in advance. We remark on geometrical properties of these sets, which sometimes include MUBs and SICs, before constructing inequalities that can experimentally rule out non-contextual hidden variable models. Along the way, we look at affine planes, group theory and quantum computing.

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University, 2014. 100 p.
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-107132 (URN)978-91-7447-965-2 (ISBN)
Public defence
2014-10-03, FP41, AlbaNova universitetscentrum, Roslagstullsbacken 33, Stockholm, 13:15 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 6: Accepted.

 

Available from: 2014-09-11 Created: 2014-09-03 Last updated: 2014-09-15Bibliographically approved

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