A Fourier analytic approach to the problem of mutually unbiased bases (Q2859351)

Two orthonormal bases \(A =\{e_1, \cdots, e_d\}\) and \(B =\{f_1,\cdots, f_d\}\) in \(\mathbb{C}^d\) are called unbiased if for every \(1 \leq j, k\leq d\), \(| \langle e_j,f_k\rangle| =1/\sqrt{d}\). A collection \(B_0, \cdots, B_m\) of orthonormal bases is said to be (pairwise) mutually unbiased if every two of them are unbiased. What is the maximal number of pairwise mutually unbiased bases (MUBs) in \(\mathbb{C}^d\)? See [\textit{T. Durt} et al., Int.\ J.\ Quantum Inf. 8, No. 4, 535--640 (2010; Zbl 1208.81052)] for more information. The author of the present paper describes how the problem of MUBs fits into a general scheme in additive combinatorics and then applies the method to prove a generalization of the fact that there are at most \(d + 1\) MUBs in \(\mathbb{C}^d\). He also indicates the limitations of the method by introducing the notion of pseudo-MUBs, and discusses the possible existence of such in the case \(d = 6\).