A Sharpening of the Welch Bounds and the Existence of Real and Complex Spherical $t$ –Designs

The Welch bounds for a finite set of unit vectors are a family of inequalities indexed by t = 1, 2, ..., 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 Welch bound equality 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 always 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 mutually unbiased bases, 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.