Real Mutually Unbiased Bases

TL;DR

This paper investigates the maximum number of real mutually unbiased bases in various dimensions, providing bounds, discussing limitations of current methods, and offering a simplified proof for an upper limit.

Contribution

It offers new bounds on the number of real mutually unbiased bases and presents a simpler proof for the maximum in any dimension.

Findings

Most dimensions have at most 2 or 3 mutually unbiased bases

Limitations of existing methods in certain dimensions are discussed

A simpler proof shows at most d/2+1 real mutually unbiased bases in dimension d

Abstract

We tabulate bounds on the optimal number of mutually unbiased bases in R^d. For most dimensions d, it can be shown with relatively simple methods that either there are no real orthonormal bases that are mutually unbiased or the optimal number is at most either 2 or 3. We discuss the limitations of these methods when applied to all dimensions, shedding some light on the difficulty of obtaining tight bounds for the remaining dimensions that have the form d=16n^2, where n can be any number. We additionally give a simpler, alternative proof that there can be at most d/2+1 real mutually unbiased bases in dimension d instead of invoking the known results on extremal Euclidean line sets by Cameron and Seidel, Delsarte, and Calderbank et al.