Mutually
unbiased bases and orthogonal decompositions of Lie algebras
(pp371-382)
P.
Oscar Boykin, Meera Sitharam, Pham Huu Tiep, and Pawel Wocjan
doi:
https://doi.org/10.26421/QIC7.4-6
Abstracts:
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
$\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in
$\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$.
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 \cite{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 $n\le 5$
an essentially unique complete collection of MUBs exists. We define \emph{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.
Key words:
Quantum Information Processing, Quantum
Computing, Special Linear Lie Algebra, Cartan Subalgebras |