Real mutually unbiased bases and representations of groups of odd order by real scaled Hadamard matrices of 2-power size

TL;DR

This paper establishes a connection between real mutually unbiased bases, Hadamard matrices, and group representations, demonstrating constructions for various dimensions and group orders, especially focusing on groups of odd order and powers of two.

Contribution

It introduces new methods to construct real mutually unbiased bases using scaled Hadamard matrices derived from group representations, particularly for groups of odd order and dimensions related to powers of two.

Findings

Constructed real mutually unbiased bases from cyclic groups of order a power of two.

Demonstrated that groups of odd order have faithful real representations yielding mutually unbiased bases.

Provided explicit examples for groups of order 3 and 5 in specific dimensions.

Abstract

We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let $q$ be a power of 2 and $r$ a positive integer. Then we can find a $q^{2r}\times q^{2r}$ real orthogonal matrix $D$, say, of multiplicative order $q^{2r-1}+1$, whose $q^{2r-1}+1$ powers $D$, \dots, $D^{q^{2r-1}+1}=I$ define $q^{2r-1}+1$ mutually unbiased bases in $\mathbb{R}^{q^{2r}}$. Thus the scaled matrices $q^rD$, \dots, $q^rD^{q^{2r-1}}$ are $q^{2r-1}$ different Hadamard matrices. When we take $q=2$, we achieve the maximum number of real mutually unbiased bases in dimension $2^{2r}$ using the elements of a cyclic group. We also prove the following. Let $G$ be an arbitrary finite group of odd order $2k+1$, where $k\geq 3$. Then $G$ has a real representation $R$, say, of degree $2^{2^{k-1}}$ such that the elements $R(\sigma)$, $\sigma\in G$, define $|G|$ mutually unbiased bases in $\mathbb{R}^{d}$, where $d= 2^{2^{k-1}}$. In addition, a group of order 5 defines five real mutually unbiased bases in $\mathbb{R}^{16}$ and a group of order 3 defines three real mutually unbiased bases in $\mathbb{R}^{4}$. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.