Mutually unbiased bases and Hadamard matrices of order six | Zendy

Ingemar Bengtsson | Zendy; Wojciech Bruzda | Zendy; Åsa Ericsson | Zendy; Jan-Åke Larsson | Zendy; Wojciech Tadej | Zendy; Karol Życzkowski | Zendy

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Mutually unbiased bases and Hadamard matrices of order six

Author(s) -

Ingemar Bengtsson,

Wojciech Bruzda,

Åsa Ericsson,

Jan-Åke Larsson,

Wojciech Tadej,

Karol Życzkowski

Publication year - 2007

Publication title -

journal of mathematical physics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.708

H-Index - 119

eISSN - 1089-7658

pISSN - 0022-2488

DOI - 10.1063/1.2716990

Subject(s) - mutually unbiased bases , hadamard transform , conjecture , mathematics , order (exchange) , complex hadamard matrix , set (abstract data type) , hadamard matrix , upper and lower bounds , hilbert space , combinatorics , discrete mathematics , pure mathematics , computer science , mathematical analysis , finance , economics , programming language

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.

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