Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets

TL;DR

This paper constructs optimal complex Hilbert space frames that achieve the orthoplex bound, forming weighted complex projective 2-designs, using Singer difference sets, especially when the dimension-related parameters are prime powers.

Contribution

It introduces a novel construction of frames achieving the orthoplex bound in complex spaces using Singer sets, expanding the toolkit for quantum state tomography.

Findings

Constructed frames achieve the orthoplex bound in complex Hilbert spaces.

Frames form weighted complex projective 2-designs useful for quantum tomography.

Explicit constructions are provided for prime power dimensions.

Abstract

Equiangular tight frames are examples of Grassmannian line packings for a Hilbert space. More specifically, according to a bound by Welch, they are minimizers for the maximal magnitude occurring among the inner products of all pairs of vectors in a unit-norm frame. This paper is dedicated to packings in the regime in which the number of frame vectors precludes the existence of equiangular frames. The orthoplex bound then serves as an alternative to infer a geometric structure of optimal designs. We construct frames of unit-norm vectors in $K$-dimensional complex Hilbert spaces that achieve the orthoplex bound. When $K-1$ is a prime power, we obtain a tight frame with $K^2+1$ vectors and when $K$ is a prime power, with $K^2+K-1$ vectors. In addition, we show that these frames form weighted complex projective 2-designs that are useful additions to maximal equiangular tight frames and maximal sets of mutually unbiased bases in quantum state tomography. Our construction is based on Singer's family of difference sets and the related concept of relative difference sets.