2005/06/10 by P. Oscar Boykin, Meera Sitharam, Boykin, P. Oscar +6 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Coding theory and cryptography #FOS: Physical sciences #Finite Group Theory Research #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0506089
13 pages
arxiv created 2005/06/10 · openalex publication_date 2005/06/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 m MUBs in Kn gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sln(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 Cn gives rise to a so-called orthogonal decomposition (OD) of sln(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 sln(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 sln(C) potentially allows the application of known results on (partial) ODs of sln(C) to MUBs.