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Mutually unbiased bases and Hadamard matrices of order six
Stockholms Universitet, AlbaNova, Fysikum, Stockholm, Sweden.
Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński, Kraków, Poland.
Stockholms Universitet, AlbaNova, Fysikum, Stockholm, Sweden.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.ORCID iD: 0000-0002-1082-8325
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2007 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, no 5, p. 052106-1-052106-21Article 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
American Institute of Physics (AIP), 2007. Vol. 48, no 5, p. 052106-1-052106-21
Keywords [en]
Hadamard matrices, Hilbert spaces, discrete Fourier transforms
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-39667DOI: 10.1063/1.2716990Local ID: 50657OAI: oai:DiVA.org:liu-39667DiVA, id: diva2:260516
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2019-08-16Bibliographically approved

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Larsson, Jan-Åke

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