2014/10/15 by Rod Gow, Gow, Rod · 1 citation
Computer Science · Engineering · Mathematics · #05E10 #20C15 #20C35 #Coding theory and cryptography #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #Group Theory (math.GR) #Quantum Physics (quant-ph) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1410.4059
openalex publication_date 2014/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 q2r× q2r real orthogonal matrix D, say, of multiplicative order q2r-1+1, whose q2r-1+1 powers D, …, D^q2r-1+1=I define q2r-1+1 mutually unbiased bases in ℝ^q2r. Thus the scaled matrices qrD, …, qrD^q2r-1 are q2r-1 different Hadamard matrices. When we take q=2, we achieve the maximum number of real mutually unbiased bases in dimension 22r 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≥ 3. Then G has a real representation R, say, of degree 2^2k-1 such that the elements R(σ), σ∈ G, define |G| mutually unbiased bases in ℝd, where d= 2^2k-1. In addition, a group of order 5 defines five real mutually unbiased bases in ℝ16 and a group of order 3 defines three real mutually unbiased bases in ℝ4. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.