2006/10/31 by Ingemar Bengtsson, Wojciech Bruzda, Asa Ericsson +5 · 2 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Combinatorics #Complex Hadamard matrix #Computer science #Conjecture #Discrete mathematics #Hadamard matrix #Hadamard transform #Hilbert space #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Mutually unbiased bases #Order (exchange) #Polynomial and algebraic computation #Pure mathematics #Set (abstract data type) #Upper and lower bounds #graph theory and CDMA systems #quant-ph
paper · pdf · doi:10.1063/1.2716990
published as J. Math. Phys. 48, 052106 (2007) · 33 pages, 3 figures. References added in v2
openalex publication_date 2007/05/01 · arxiv created 2007/06/01 · openalex created_date 2016/06/24 · arxiv updated 2022/03/16 · openalex updated_date 2026/08/01
We report on a search for mutually unbiased bases (MUBs) in 6 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 organise 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.