### 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) |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Quantum Information and Computation*,

*7*(4), 371-382.

**Mutually unbiased bases and orthogonal decompositions of Lie algebras.** / Boykin, P. Oscar; Sitharam, Meera; Tiep, Pham Huu; Wocjan, Pawel.

Research output: Contribution to journal › Article

*Quantum Information and Computation*, vol. 7, no. 4, pp. 371-382.

}

TY - JOUR

T1 - Mutually unbiased bases and orthogonal decompositions of Lie algebras

AU - Boykin, P. Oscar

AU - Sitharam, Meera

AU - Tiep, Pham Huu

AU - Wocjan, Pawel

PY - 2007/5

Y1 - 2007/5

N2 - 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.

AB - 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.

KW - Cartan subalgebras

KW - Quantum computing

KW - Quantum information processing

KW - Special linear Lie algebra

UR - http://www.scopus.com/inward/record.url?scp=34247249418&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247249418&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34247249418

VL - 7

SP - 371

EP - 382

JO - Quantum Information and Computation

JF - Quantum Information and Computation

SN - 1533-7146

IS - 4

ER -