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 mu MUBs in (n) gives rise to a collection of mu Cartan subalgebras of the special linear Lie algebra sl(n)() that are pail-wise 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 sl(n)(). The converse holds if the Cartan subalgebras in the OD are also dagger-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 [2] 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.
cartan subalgebras special linear lie algebra quantum information processing quantum computing