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.

Many applications of finite tight frames (e.g., the use of SICs and mutually unbiased bases (MUBs) in quantum information theory and harmonic frames for the analysis of signals subject to erasures) depend only on the vectors up to projective unitary equivalence. It is well known that two finite sequences of vectors in inner product spaces are unitarily equivalent if and only if their respective inner products (Gramian matrices) are equal. Here we present a corresponding result for the projective unitary equivalence of two sequences of vectors (lines) in inner product spaces, i.e., that a finite number of (Bargmann) projective (unitary) invariants are equal. This result is equivalent to finding a rank-one matrix completion of a certain matrix. We give an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants and apply it to SICs, MUBs, and harmonic frames. We also extend our results to the projective similarity of vectors.

This article shows what are called the Welch (1974) bound equality (WBE) sequences by the signal processing community are precisely the isometric/equal norm-normalized/uniform tight frames which are currently being investigated for a number of applications, and in the real case are the spherical 2-designs of combinatorics. Recent applications include wavelet expansions, Grassmannian frames, frames robust to erasures, and quantum measurements. This is done by giving an elementary proof of a generalization of Welch's inequality to vectors which need not have equal energy, and then showing that equality occurs in this exactly when the vectors form a tight frame.