Mutually unbiased bases and orthogonal decompositions of 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 ⁿ gives rise to a collection of μ Cartan subalgebras of the special linear Lie algebra slₙ() that are pairwise orthogonal with respect to the Killing form, where = or =. In particular, a complete collection of MUBs in ⁿ gives rise to a so-called orthogonal decomposition (OD) of slₙ(). 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 {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 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.