Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras

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dc.creatorBoykin, P. Oscar
dc.creatorSitharam, Meera
dc.creatorTiep, Pham Huu
dc.creatorWocjan, Pawel
dc.date2005-06-10
dc.date.accessioned2026-07-07T06:12:58Z
dc.date.available2026-07-07T06:12:58Z
dc.descriptionWe 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 m MUBs in K^n gives rise to a collection of m 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 *-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 relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.
dc.description13 pages
dc.identifierhttps://arxiv.org/abs/quant-ph/0506089
dc.identifierhttp://arxiv.org/abs/quant-ph/0506089
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/92950
dc.subjectQuantum Physics
dc.titleMutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras
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