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Mutually unbiased bases and Hadamard matrices of order six
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
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2007 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, no 5, 052106- p.Article in journal (Refereed) Published
Abstract [en]

We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

Place, publisher, year, edition, pages
2007. Vol. 48, no 5, 052106- p.
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:su:diva-24392DOI: 10.1063/1.2716990ISI: 000246892400006OAI: oai:DiVA.org:su-24392DiVA: diva2:197432
Available from: 2007-08-15 Created: 2007-08-15 Last updated: 2013-05-02Bibliographically approved
In thesis
1. Exploring the Set of Quantum States
Open this publication in new window or tab >>Exploring the Set of Quantum States
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Quantum mechanical properties of finite dimensional quantum systems are used within the field of quantum information. In this thesis the set of states (density matrices) for such systems is studied and described, largely in geometrical terms. The introductory part also acquaints the reader with relevant background about majorization, bistochastic matrices, mutually unbiased bases, Hadamard matrices and entanglement, with the aim to make the papers attached easier to read.

Paper I considers Peres' criterion for separability, for two qubit states. Paper II deals with the problem of how density matrices can be mixed from pure states, especially what probability distributions over pure states that are possible. In Paper III the set of bistochastic matrices–Birkhoff's polytope–and the subset of unistochastic matrices is studied, with a detailed description in dimensions 3 and 4. In Paper IV it is seen how the states of a complete set of mutually unbiased bases form a polytope in the set of density matrices, with certain combinatorial properties. A search for mutually unbiased bases in dimension 6 is presented in Paper VI, which includes a thorough discussion on 6 by 6 Hadamard matrices. Paper V presents a result about geodesics in the set of quantum states with respect to the curved Bures-Uhlmann geometry.

Place, publisher, year, edition, pages
Stockholm: Fysikum, 2007. 82 p.
Keyword
quantum mechanics, quantum states, density matrices, quantum information, geometry
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-6993 (URN)978-91-7155-475-8 (ISBN)
Public defence
2007-09-17, sal FB42, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:00
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Available from: 2007-08-15 Created: 2007-08-15Bibliographically approved

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