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Orbits of mutually unbiased bases
Stockholms universitet, Naturvetenskapliga fakulteten, Fysikum.
2014 (Engelska)Ingår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 47, nr 13, s. 135303-Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We express Alltop's construction of mutually unbiased bases as orbits under the Weyl–Heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4. We reproduce Alltop's mutually unbiased bases using abelian subgroups of the Clifford group in prime dimensions, in direct analogy to the well-known construction of mutually unbiased bases using abelian subgroups of the Weyl–Heisenberg group. Finally, we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases.

Ort, förlag, år, upplaga, sidor
2014. Vol. 47, nr 13, s. 135303-
Nyckelord [en]
Mutually unbiased bases, Weyl-Heisenberg group, Clifford group
Nationell ämneskategori
Fysik
Forskningsämne
teoretisk fysik
Identifikatorer
URN: urn:nbn:se:su:diva-107055DOI: 10.1088/1751-8113/47/13/135303OAI: oai:DiVA.org:su-107055DiVA, id: diva2:742838
Tillgänglig från: 2014-09-02 Skapad: 2014-09-02 Senast uppdaterad: 2017-12-05Bibliografiskt granskad
Ingår i avhandling
1. Geometry and foundations of quantum mechanics
Öppna denna publikation i ny flik eller fönster >>Geometry and foundations of quantum mechanics
2014 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: Department of Physics, Stockholm University, 2014. s. 100
Nationell ämneskategori
Fysik
Forskningsämne
teoretisk fysik
Identifikatorer
urn:nbn:se:su:diva-107132 (URN)978-91-7447-965-2 (ISBN)
Disputation
2014-10-03, FP41, AlbaNova universitetscentrum, Roslagstullsbacken 33, Stockholm, 13:15 (Engelska)
Opponent
Handledare
Anmärkning

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 6: Accepted.

 

Tillgänglig från: 2014-09-11 Skapad: 2014-09-03 Senast uppdaterad: 2014-09-15Bibliografiskt granskad

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