The correspondence between mutually unbiased bases and mutually orthogonal extraordinary supersquares

Abstract

We study the connection between mutually unbiased bases and mutually orthogonal extraordinary supersquares, a wider class of squares which does not contain only the Latin squares. We show that there are four types of complete sets of mutually orthogonal extraordinary supersquares for the dimension d = 8. We introduce the concept of physical striation and show that this is equivalent to the extraordinary supersquare. The general algorithm for obtaining the mutually unbiased bases and the physical striations is constructed and it is shown that the complete set of mutually unbiased physical striations is equivalent to the complete set of mutually orthogonal extraordinary supersquares. We apply the algorithm to two examples: one for two-qubit systems ( d = 4) and one for three-qubit systems ( d = 8), by using the Type II complete sets of mutually orthogonal extraordinary supersquares of order 8.