On the equivalence between real mutually unbiased bases and a certain class of association schemes

Mutually unbiased bases (MUBs) in complex vector spaces play several important roles in quantum information theory. At present, even the most elementary questions concerning the maximum number of such bases in a given dimension and their construction remain open. In an attempt to understand the complex case better, some authors have also considered real MUBs, mutually unbiased bases in real vector spaces. The main results of this paper establish an equivalence between sets of real mutually unbiased bases and 4-class cometric association schemes which are both QQ-bipartite and QQ-antipodal. We then explore the consequences of this equivalence, constructing new cometric association schemes and describing a potential method for the construction of sets of real MUBs.