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Order 3 Symmetry in the Clifford Hierarchy
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
2014 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 47, no 45, 455302- p.Article in journal (Refereed) Published
Abstract [en]

We investigate the action of Weyl-Heisenberg and Clifford groups on sets of vectors that comprise mutually unbiased bases (MUBs). We consider two distinct MUB constructions, the standard and Alltop constructions, in Hilbert spaces of prime dimension. We show how the standard set of MUBs turns into the Alltop set under the action of an element at the third level of the Clifford hierarchy. We prove that when the dimension is a prime number equal to one modulo three each Alltop vector is invariant under an element of the Clifford group of order 3. The set of all Alltop vectors splits into three different orbits of the Clifford group, and forms a configuration together with the set of all subspaces invariant under an order 3 element of the Clifford group. There is a well-known conjecture that SIC vectors can be found in the eigenspace of order 3 Cliffords. This, combined with a connection between MUB and SIC vectors, suggests our work may provide a clue to the SIC existence problem in these dimensions. We identify Alltop vectors as so-called magic states which appear in the context of fault-tolerant quantum computing. The appearance of distinct Clifford orbits implies an inequivalence between some magic states.

Place, publisher, year, edition, pages
2014. Vol. 47, no 45, 455302- p.
Keyword [en]
Mutually unbiased bases, magic state distillation, Clifford hierarchy
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-107056DOI: 10.1088/1751-8113/47/45/455302ISI: 000344628200014OAI: oai:DiVA.org:su-107056DiVA: diva2:742841
Available from: 2014-09-02 Created: 2014-09-02 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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