1,036 research outputs found
Three ways to look at mutually unbiased bases
This is a review of the problem of Mutually Unbiased Bases in finite
dimensional Hilbert spaces, real and complex. Also a geometric measure of
"mubness" is introduced, and applied to some recent calculations in six
dimensions (partly done by Bjorck and by Grassl). Although this does not yet
solve any problem, some appealing structures emerge.Comment: 18 pages. Talk at the Vaxjo Conference on Foundations of Probability
and Physics, June 200
Complex Hadamard matrices and Equiangular Tight Frames
In this paper we give a new construction of parametric families of complex
Hadamard matrices of square orders, and connect them to equiangular tight
frames. The results presented here generalize some of the recent ideas of
Bodmann et al. and extend the list of known equiangular tight frames. In
particular, a (36,21) frame coming from a nontrivial cube root signature matrix
is obtained for the first time.Comment: 6 pages, contribution to the 16th ILAS conference, Pisa, 201
On quaternary complex Hadamard matrices of small orders
One of the main goals of design theory is to classify, characterize and count
various combinatorial objects with some prescribed properties. In most cases,
however, one quickly encounters a combinatorial explosion and even if the
complete enumeration of the objects is possible, there is no apparent way how
to study them in details, store them efficiently, or generate a particular one
rapidly. In this paper we propose a novel method to deal with these
difficulties, and illustrate it by presenting the classification of quaternary
complex Hadamard matrices up to order 8. The obtained matrices are members of
only a handful of parametric families, and each inequivalent matrix, up to
transposition, can be identified through its fingerprint.Comment: 7 page
Trades in complex Hadamard matrices
A trade in a complex Hadamard matrix is a set of entries which can be changed
to obtain a different complex Hadamard matrix. We show that in a real Hadamard
matrix of order all trades contain at least entries. We call a trade
rectangular if it consists of a submatrix that can be multiplied by some scalar
to obtain another complex Hadamard matrix. We give a
characterisation of rectangular trades in complex Hadamard matrices of order
and show that they all contain at least entries. We conjecture that all
trades in complex Hadamard matrices contain at least entries.Comment: 9 pages, no figure
On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six
The complete classification of all 6x6 complex Hadamard matrices is an open
problem. The 3-parameter Karlsson family encapsulates all Hadamards that have
been parametrised explicitly. We prove that such matrices satisfy a non-trivial
constraint conjectured to hold for (almost) all 6x6 Hadamard matrices. Our
result imposes additional conditions in the linear programming approach to the
mutually unbiased bases problem recently proposed by Matolcsi et al.
Unfortunately running the linear programs we were unable to conclude that a
complete set of mutually unbiased bases cannot be constructed from Karlsson
Hadamards alone.Comment: As published versio
Temperley-Lieb R-matrices from generalized Hadamard matrices
New sets of rank n-representations of Temperley-Lieb algebra TL_N(q) are
constructed. They are characterized by two matrices obeying a generalization of
the complex Hadamard property. Partial classifications for the two matrices are
given, in particular when they reduce to Fourier or Butson matrices.Comment: 17 page
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