Mutually unbiased bases and orthogonal decompositions of Lie algebras

P. Oscar Boykin, Meera Sitharam, Pham Huu Tiep, Pawel Wocjan

Research output: Contribution to journalArticlepeer-review

52 Scopus citations


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 n gives rise to a collection of μ Cartan subalgebras of the special linear Lie algebra sln() that are pairwise 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 sln(). 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 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.

Original languageEnglish (US)
Pages (from-to)371-382
Number of pages12
JournalQuantum Information and Computation
Issue number4
StatePublished - May 2007


  • Cartan subalgebras
  • Quantum computing
  • Quantum information processing
  • Special linear Lie algebra

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Computational Theory and Mathematics


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