Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of a special type of orthogonal decomposition with respect to the Killing form of the special linear and symplectic Lie algebras.
We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of
μ MUBs in $\K^n$ gives rise to a collection of
μ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also
†-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for
n≤5 an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for
n=6 there are at most three monomial MUBs.
We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for n×n unitary and orthogonal matrices are the same. They are both the set {0,1,…,n2−n−4,n2−n−2,n2−n} for n>4. We explicitly construct examples of orthogonal matrices with the numbers in the set. We apply our results to construct a necessary condition by which a multipartite unitary operation is a product operation. The latter is a fundamental problem in quantum information. We also construct an n×n orthogonal matrix of Schmidt rank n2−1 with many zero entries, and it solves an open problem in Muller-Hermes and Nechita [Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra Appl. 2018;557:174—187].