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A Kochen-Specker inequality for a SIC
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics. Universidad de Sevilla, Spain.
2012 (English)In: Physics Letters A, ISSN 0375-9601, E-ISSN 1873-2429, Vol. 376, no 4, 374-376 p.Article in journal (Refereed) Published
Abstract [en]

Yu and Oh (eprint) [1] have given a state-independent proof of the Kochen–Specker theorem in three dimensions using only 13 rays. The proof consists of showing that a non-contextual hidden variable theory necessarily leads to an inequality that is violated by quantum mechanics. We give a similar proof making use of 21 rays that constitute aSIC (symmetric informationally-complete positive operator-valued measure) and a complete set of MUB (mutually unbiased bases). A theory-independent inequality is also presented using the same 21 rays, as required for experimental tests of contextuality.

Place, publisher, year, edition, pages
2012. Vol. 376, no 4, 374-376 p.
Keyword [en]
Kochen–Specker theorem, Non-contextuality, SIC and four MUB, Hesse configuration
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-81909DOI: 10.1016/j.physleta.2011.12.011OAI: oai:DiVA.org:su-81909DiVA: diva2:564493
Funder
Swedish Research Council
Available from: 2012-11-01 Created: 2012-11-01 Last updated: 2017-12-07Bibliographically approved
In thesis
1. Configurations in Quantum Information
Open this publication in new window or tab >>Configurations in Quantum Information
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Measurements play a central role in quantum information. This thesis looksat two types: contextual measurements and symmetric measurements. Contextualityoriginates from the Kochen-Specker theorem about hidden variablemodels and has recently undergone a subtle shift in its manifestation. Symmetricmeasurements are characterised by the regular polytopes they formin Bloch space (the vector space containing all density matrices) and are thesubject of several investigations into their existence in all dimensions.We often describe measurements by the vectors in Hilbert space ontowhich our operators project. In this sense, both contextual and symmetricmeasurements are connected to special sets of vectors. These vectors areoften special for another reason: they form congurations in a given incidencegeometry.In this thesis, we aim to show various connections between congurationsand measurements in quantum information. The congurations discussedhere would have been well-known to 19th and 20th century geometers andwe show they are relevant for advances in quantum theory today. Specically,the Hesse and Reye congurations provide proofs of measurement contextuality,both in its original form and its newer guise. The Hesse congurationalso ties together dierent types of symmetric measurements in dimension3called SICs and MUBswhile giving insights into the group theoreticalproperties of higher dimensional symmetric measurements.

Place, publisher, year, edition, pages
Department of Physics, Stockholm University, 2012. 47 p.
National Category
Other Physics Topics
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-82921 (URN)
Opponent
Supervisors
Available from: 2012-11-30 Created: 2012-11-30 Last updated: 2013-02-28Bibliographically approved
2. 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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