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

On Two Ways to Look for Mutually Unbiased Bases

Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in C(d) to the search for d(d+1) vectors in C(d*d) satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.Comment: three pages to be published in Acta Polytechnica (Czech Technical University in Prague

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

A simple construction of complex equiangular lines

A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.Comment: 11 pages; minor revisions and comments added in section 1 describing a link to previously known results; correction to Theorem 1 and updates to reference

ON TWO WAYS TO LOOK FOR MUTUALLY UNBIASED BASES

Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in Cd to the search for d(d+1) vectors in Cd2 satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein