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Linear Dependencies in Weyl-Heisenberg Orbits
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
2013 (English)In: Quantum Information Processing, ISSN 1570-0755, E-ISSN 1573-1332, Vol. 12, no 11, 3449-3475 p.Article in journal (Refereed) Published
Abstract [en]

Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl-Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.

Place, publisher, year, edition, pages
2013. Vol. 12, no 11, 3449-3475 p.
Keyword [en]
SIC-POVMs, Weyl-Heisenberg group, Elliptic curves, Hesse configuration, Linear dependencies
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-101402DOI: 10.1007/s11128-013-0609-6ISI: 000325814300007OAI: oai:DiVA.org:su-101402DiVA: diva2:703531
Funder
Swedish Research Council, 621-2010-4060
Available from: 2014-03-07 Created: 2014-03-07 Last updated: 2017-12-05Bibliographically 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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