• Corpus ID: 119164203

All complex equiangular tight frames in dimension 3

@inproceedings{SzollHosi2014AllCE,
  title={All complex equiangular tight frames in dimension 3},
  author={Ferenc SzollHosi},
  year={2014}
}
In this paper we describe some new algebraic features of the Gram matrices of complex Equiangular Tight Frames (ETF). This lead on the one hand to the nonexistence of several low dimensional complex ETFs; and on the other hand to the full algebraic classification of all complex ETFs in C. We use computer aided methods, in particular, Gröbner basis computations to obtain these results. 2000 Mathematics Subject Classification. Primary 05B20, secondary 46L10. 
Packings in real projective spaces
TLDR
A computer-assisted proof of the optimality of a particular 6-packing in $\mathbb{R}\mathbf{P}^3$, a linear-time constant-factor approximation algorithm for packing in the so-called Gerzon range, and local optimality certificates for two infinite families of packings are provided.
Mutually Unbiased Equiangular Tight Frames
TLDR
A new method for constructing ETFs is introduced, showing that it is sometimes possible to construct multiple ETFs for the same space that are “mutually unbiased” in a way that is analogous to the quantum-information-theoretic concept of mutually unbiased bases.
Tables of the existence of equiangular tight frames
TLDR
Every known construction of ETFs is surveyed and existence for sufficiently small dimensions is tabulated.
A Delsarte-Style Proof of the Bukh–Cox Bound
TLDR
First, ideas from the Bukh–Cox proof are used to find a new proof of the Welch bound, and then ideas from Delsarte’s linear program are use to finding a newProof of the Buk-Cox bound.
Moment maps and Galois orbits for SIC-POVMs
The equations that define covariant SIC-POVMs are interpreted in terms of moment maps. Attention is focussed on orbits of a cyclic subgroup of a maximal torus and their images in the moment polytope.
Game of Sloanes: best known packings in complex projective space
It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield
J ul 2 01 7 TOWARD THE CLASSIFICATION OF BIANGULAR HARMONIC FRAMES
Equiangular tight frames (ETFs) and biangular tight frames (BTFs) sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively
Optimal measures for $p$-frame energies on spheres
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel
...
...

References

SHOWING 1-10 OF 28 REFERENCES
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
Construction, classification and parametrization of complex Hadamard matrices
The intended purpose of this work is to provide the reader with a comprehensive, state-of-the art presentation of the theory of complex Hadamard matrices, or at least report on the very recent
Complex Hadamard matrices of order 6: a four‐parameter family
In this paper, we construct a new, previously unknown four‐parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various
The monomial representations of the Clifford group
We show that the Clifford group--the normaliser of the Weyl-Heisenberg group--can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This
Orthogonal Maximal Abelian *-Subalgebras of the N×n Matrices and Cyclic N-Roots
It is proved that for n = 5, there is up to isomorphism only one pair of orthogonal maximal abelian-subalgebras (MASA's) in the n n-matrices. The same result holds trivially for n = 2 and n = 3, but
All mutually unbiased bases in dimensions two to five
TLDR
It is confirmed that the complete sets of (d + 1)MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.
QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY
This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental
...
...