# A sharpening of the Welch bounds
and the existence of real and complex spherical $t$--designs

## Shayne Waldron

## Abstract:

The Welch bounds for a finite set of unit vectors
are a family of inequalities indexed by $t=1,2,\ldots$,
which describe how ``evenly spread'' the vectors are.
They have important applications in signal analysis,
where sequences giving equality in the first Welch bound
are known as WBE sequences or as unit norm tight frames.
Here we consider sequences of vectors giving equality in the
higher order Welch bounds. These are seen to correspond
to tight frames for the complex symmetric $t$--tensors
(which we prove always exist).
We show that for $t>1$ the Welch bounds can be sharpened
for real vectors, and again, vectors giving equality alway exist.
We give a unified treatment of various conditions for equality
in both the real and complex cases. In particular, we give
an explicit description of the corresponding cubature rules
($t$--designs).
Our results set up a framework for the construction and classification
several configurations of vectors of recent interest.
These include MUBs (mutually unbiased bases), SICs
(complex equiangular lines), spherical half--designs,
projective $t$--designs and minimisers of the higher
order frame potential.
One interesting consequence is a construction of
sets of complex equiangular lines which were
previously unknown.

**Keywords:**
Welch bounds,
WBE sequences (Welch bound equality sequences),
finite tight frames,
symmetric tensors,
cubature rules for the sphere,
spherical $t$--designs,
projective $t$--designs,
MUBs (mutually unbiased bases),
SICs (symmetric informationally complete positive operator valued measures),
complex equiangular lines

**Math Review Classification:**
Primary 05B30, 42C15, 65D30, 94A12;
Secondary 42C15, 42C40

**Length:** 19 pages

**Last Updated:** 14 September 2016

## Availability: