## 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 sl_{n}() 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 sl_{n}(). 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 language | English (US) |
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Pages (from-to) | 371-382 |

Number of pages | 12 |

Journal | Quantum Information and Computation |

Volume | 7 |

Issue number | 4 |

State | Published - May 2007 |

Externally published | Yes |

## Keywords

- 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