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**Mutually unbiased bases and orthogonal decompositions of Lie algebras.**
*(English)*
Zbl 1152.81680

Summary: 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 \(\mu\) MUBs in \(K^n\) gives rise to a collection of \(\mu\) Cartan subalgebras of the special linear Lie algebra \(sl_n(K)\) that are pairwise orthogonal with respect to the Killing form, where \(K = \mathbb R\) or \(K = \mathbb C\). In particular, a complete collection of MUBs in \(\mathbb C^n\) gives rise to a so-called orthogonal decomposition (OD) of \(sl_n(\mathbb C)\). 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 relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for \(n \leq 5\) 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.