id: 05320403
dt: j
an: 05320403
au: Boykin, P.O.; Sitharam, M.; Tiep, P.H.; Wocjan, P.
ti: Mutually unbiased bases and orthogonal decompositions of Lie algebras.
so: Quantum Inf. Comput. 7, No. 4, 371-382 (2007).
py: 2007
pu: Rinton Press, Paramus, NJ
la: EN
cc: 
ut: 
ci: 
li: 
ab: 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 $μ$ MUBs in ${\text{K}}^n$ gives rise to a collection of
    $μ$ 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.
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