@article {IOPORT.05320403,
    author       = {Boykin, P.O. and Sitharam, M. and Tiep, P.H. and Wocjan, P.},
    title        = {Mutually unbiased bases and orthogonal decompositions of Lie algebras.},
    year         = {2007},
    journal      = {Quantum Information \& Computation},
    volume       = {7},
    number       = {4},
    issn         = {1533-7146},
    pages        = {371-382},
    publisher    = {Rinton Press, Paramus, NJ},
    abstract     = {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 ${\text{K}}^n$ gives rise to a collection of $\mu$ Cartan subalgebras of the special linear Lie algebra $sl_n({\text{K}})$ that are pairwise orthogonal with respect to the Killing form, where K = R or K = C. In particular, a complete collection of MUBs in ${\text{C}}^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n({\text{C}})$. The converse holds if the Cartan subalgebras in the OD are also $\dag$-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 {\it 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.},
    identifier   = {05320403},
}