124 research outputs found
Equiangular lines, mutually unbiased bases, and spin models
We use difference sets to construct interesting sets of lines in complex
space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in
C^k when k-1 is a prime power. Using semiregular relative difference sets with
parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k.
We show how to construct these difference sets from commutative semifields and
that several known maximal sets of mutually unbiased bases can be obtained in
this way, resolving a conjecture about the monomiality of maximal sets. We also
relate mutually unbiased bases to spin models.Comment: 23 pages; no figures. Minor correction as pointed out in
arxiv.org:1104.337
Sets of Complex Unit Vectors with Two Angles and Distance-Regular Graphs
We study {0,\alpha}-sets, which are sets of unit vectors of in
which any two distinct vectors have angle 0 or \alpha. We investigate some
distance-regular graphs that provide new constructions of {0,\alpha}-sets using
a method by Godsil and Roy. We prove bounds for the sizes of {0,\alpha}-sets of
flat vectors, and characterize all the distance-regular graphs that yield
{0,\alpha}-sets meeting the bounds at equality.Comment: 15 page
Constructions of biangular tight frames and their relationships with equiangular tight frames
We study several interesting examples of Biangular Tight Frames (BTFs) -
basis-like sets of unit vectors admitting exactly two distinct frame angles
(ie, pairwise absolute inner products) - and examine their relationships with
Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one
frame angle.
We demonstrate a smooth parametrization BTFs, where the corresponding frame
angles transform smoothly with the parameter, which "passes through" an ETF
answers two questions regarding the rigidity of BTFs. We also develop a general
framework of so-called harmonic BTFs and Steiner BTFs - which includes the
equiangular cases, surprisingly, the development of this framework leads to a
connection with the famous open problem(s) regarding the existence of Mersenne
and Fermat primes. Finally, we construct a (chordally) biangular tight set of
subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.Comment: 19 page
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