2016/04/16 by Jonathan Jedwab, Jedwab, Jonathan, Lily Yen +1
Computer Science · Engineering · Mathematics · #05B20 #15A30 #81P45 #94B60 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1604.04797
openalex publication_date 2016/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of b mutually unbiased bases (MUBs) in ℂd (for d > 1) comprises bd vectors in ℂd, partitioned into b orthogonal bases for ℂd such that the pairwise angle between all vectors from distinct bases is \arccos(1/√(d)). The largest number μ(d) of MUBs that can exist in ℂd is at most d+1, but constructions attaining this bound are known only when d is a prime power. A set of b MUBs in ℂd that cannot be enlarged, even by the first vector of a potential (b+1)-th MUB, is called strongly unextendible. Until now, only one infinite family of dimensions d containing b(d) strongly unextendible MUBs in ℂd satisfying b(d) < μ(d) was known, this family, due to Szántó, is asymptotically "large" in the sense that b(d)/μ(d) → 1 as d → ∞. However, the existence of 2m-1+1 strongly unextendible MUBs in ℂ2m for each integer m > 1 has been conjectured by Mandayam et al. We prove their conjecture for all even values of m, using only elementary linear algebra. The existence of this "small" new infinite family suggests, contrary to widespread belief, that μ(d) for non-prime-powers d might be significantly larger than the size of particular unextendible sets.