Maximal Orthoplectic Fusion Frames from Mutually Unbiased Bases and Block Designs

TL;DR

This paper explores optimal subspace packings in Euclidean spaces using maximal mutually unbiased bases and combinatorial designs, achieving the orthoplex bound and connecting to Grassmannian 2-designs.

Contribution

It introduces a novel approach to constructing optimal subspace packings via maximal mutually unbiased bases and block designs, extending known results to higher dimensions.

Findings

Achieves orthoplex bound using maximal mutually unbiased bases.

Shows conversion between coordinate projections and Grassmannian 2-designs.

Provides explicit examples in specific dimensions.

Abstract

The construction of optimal line packings in real or complex Euclidean spaces has shown to be a tantalizingly difficult task, because it includes the problem of finding maximal sets of equiangular lines. In the regime where equiangular lines are not possible, some optimal packings are known, for example, those achieving the orthoplex bound related to maximal sets of mutually unbiased bases. In this paper, we investigate the packing of subspaces instead of lines and determine the implications of maximality in this context. We leverage the existence of real or complex maximal mutually unbiased bases with a combinatorial design strategy in order to find optimal subspace packings that achieve the orthoplex bound. We also show that maximal sets of mutually unbiased bases convert between coordinate projections associated with certain balanced incomplete block designs and Grassmannian 2-designs. Examples of maximal orthoplectic fusion frames already appeared in the works by Shor, Sloane and by Zauner. They are realized in dimensions that are a power of four in the real case or a power of two in the complex case.