Geometry and foundations of quantum mechanics
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
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.
Research subject Theoretical Physics
IdentifiersURN: urn:nbn:se:su:diva-107132ISBN: 978-91-7447-965-2OAI: oai:DiVA.org:su-107132DiVA: diva2:743171
2014-10-03, FP41, AlbaNova universitetscentrum, Roslagstullsbacken 33, Stockholm, 13:15 (English)
Sudberry, Anthony, Professor
Bengtsson, Ingemar, Professor
At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 6: Accepted.
List of papers