Mutually unbiased bases and orthogonal decompositions of Lie algebras

  • Authors:

  • P. Oscar Boykin;Meera Sitharam;Pham Huu Tiep;Pawel Wocjan

  • Affiliations:

  • Dept. of Electrical and Computer Engineering, University of Florida, Gainesville, FL;Computer and Information Science and Engineering, University of Florida, Gainesville, FL;Department of Mathematics, University of Florida, Gainesville, FL;Institute for Quantum Information, California Institute of Technology, Pasadena, CA

  • Venue:
  • Quantum Information & Computation
  • Year:
  • 2007

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Abstract

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.