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Geometry and foundations of quantum mechanics
Stockholm University, Faculty of Science, Department of Physics.
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: urn:nbn:se:su:diva-107132ISBN: 978-91-7447-965-2 (print)OAI: oai:DiVA.org:su-107132DiVA: diva2:743171
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At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 6: Accepted.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
List of papers
1. A Kochen-Specker inequality for a SIC
Open this publication in new window or tab >>A Kochen-Specker inequality for a SIC
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.

Keyword
Kochen–Specker theorem, Non-contextuality, SIC and four MUB, Hesse configuration
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-81909 (URN)10.1016/j.physleta.2011.12.011 (DOI)
Funder
Swedish Research Council
Available from: 2012-11-01 Created: 2012-11-01 Last updated: 2017-12-07Bibliographically approved
2. Proposed experiments of qutrit state-independent contextuality and two-qutrit contextuality-based nonlocality
Open this publication in new window or tab >>Proposed experiments of qutrit state-independent contextuality and two-qutrit contextuality-based nonlocality
Show others...
2012 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 85, no 3, 032108- p.Article in journal (Refereed) Published
Abstract [en]

Recent experiments have demonstrated ququart state-independent quantum contextuality and qutrit state-dependent quantum contextuality. So far, the most basic form of quantum contextuality pointed out by Kochen and Specker, and Bell, has eluded experimental confirmation. Here we present an experimentally feasible test to observe qutrit state-independent quantum contextuality using single photons in a three-path setup. In addition, we show that if the same measurements are performed on two entangled qutrits, rather than sequentially on the same qutrit, then the noncontextual inequality becomes a Bell inequality. We show that this connection also applies to other recently introduced noncontextual inequalities.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-76268 (URN)10.1103/PhysRevA.85.032108 (DOI)000301104400009 ()
Note

5

Available from: 2012-05-21 Created: 2012-05-10 Last updated: 2017-12-07Bibliographically approved
3. How orthogonalities set Kochen-Specker sets
Open this publication in new window or tab >>How orthogonalities set Kochen-Specker sets
2011 (English)In: AIP conference proceedings vol. 1327, American Institute of Physics (AIP), 2011, 326-328 p.Conference paper, Published paper (Refereed)
Abstract [en]

We look at generalisations of sets of vectors proving the Kochen-Specker theorem in 3 and 4 dimensions. It has been shown that two such sets, although unitarily inequivalent, are part of a larger 3-parameter family of vectors that share the same orthogonality graph. We find that these sets are unusual, in that the vectors in all other Kochen-Specker sets investigated here are fully determined by orthogonality conditions and thus admit no free parameters.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2011
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-87992 (URN)10.1063/1.3567454 (DOI)
Conference
AQT - The International Conference on Advances in Quantum Theory, 14–17 June 2010, Växjö, Sweden
Available from: 2013-02-28 Created: 2013-02-28 Last updated: 2014-09-03Bibliographically approved
4. Linear Dependencies in Weyl-Heisenberg Orbits
Open this publication in new window or tab >>Linear Dependencies in Weyl-Heisenberg Orbits
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.

Keyword
SIC-POVMs, Weyl-Heisenberg group, Elliptic curves, Hesse configuration, Linear dependencies
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-101402 (URN)10.1007/s11128-013-0609-6 (DOI)000325814300007 ()
Funder
Swedish Research Council, 621-2010-4060
Available from: 2014-03-07 Created: 2014-03-07 Last updated: 2017-12-05Bibliographically approved
5. Orbits of mutually unbiased bases
Open this publication in new window or tab >>Orbits of mutually unbiased bases
2014 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 47, no 13, 135303- p.Article in journal (Refereed) 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.

Keyword
Mutually unbiased bases, Weyl-Heisenberg group, Clifford group
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-107055 (URN)10.1088/1751-8113/47/13/135303 (DOI)
Available from: 2014-09-02 Created: 2014-09-02 Last updated: 2017-12-05Bibliographically approved
6. Order 3 Symmetry in the Clifford Hierarchy
Open this publication in new window or tab >>Order 3 Symmetry in the Clifford Hierarchy
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.

Keyword
Mutually unbiased bases, magic state distillation, Clifford hierarchy
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-107056 (URN)10.1088/1751-8113/47/45/455302 (DOI)000344628200014 ()
Available from: 2014-09-02 Created: 2014-09-02 Last updated: 2017-12-05Bibliographically approved

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