Latin squares and mutually unbiased bases - Hubert de Guise

Latin squares have been studied as mathematical objects since Euler introduced the 36-ocer problem in 1782. More recently, Fisher (amongst others) described how Latin squares can be used in the design of experiments; Latin squares have found a host of other applications. Mutually unbiased bases (MUBs) have a much more recent history; Wootters showed they are an optimal choice of measurements to reconstruct the density matrix of a general quantum state. An example of MUBs are the generalized Pauli matrices } introduced by Patera and Zassenhaus (in 1987) in their study of gradings. In this presentation I will describe a connection between MUBs and orthogonal Latin squares. The common bridge linking the two are curves in discrete phase space: certain types of MUBs can be represented by a curve, and this curve can be used to generate - under the right conditions - a Latin square. Logical operations transforming sets of MUBs amongst themselves can transform one set of orthogonal Latin squares into another set.