Mutually unbiased bases and orthogonal decompositions of Lie algebras

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

Research output: Contribution to journalArticle

39 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)371-382
Number of pages12
JournalQuantum Information and Computation
Volume7
Issue number4
StatePublished - May 2007
Externally publishedYes

Fingerprint

Mutually Unbiased Bases
Orthogonal Decomposition
Algebra
Lie Algebra
algebra
Decomposition
decomposition
Cartan Subalgebra
Monomial
Closed
Orthogonal matrix
Unitary matrix
Converse
Pairwise
Open Problems
Correspondence
Imply

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

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

In: Quantum Information and Computation, Vol. 7, No. 4, 05.2007, p. 371-382.

Research output: Contribution to journalArticle

Boykin, P. Oscar ; Sitharam, Meera ; Tiep, Pham Huu ; Wocjan, Pawel. / Mutually unbiased bases and orthogonal decompositions of Lie algebras. In: Quantum Information and Computation. 2007 ; Vol. 7, No. 4. pp. 371-382.
@article{492b37bf90aa4cccb056ca7008bd8ed1,
title = "Mutually unbiased bases and orthogonal decompositions of Lie algebras",
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.",
keywords = "Cartan subalgebras, Quantum computing, Quantum information processing, Special linear Lie algebra",
author = "Boykin, {P. Oscar} and Meera Sitharam and Tiep, {Pham Huu} and Pawel Wocjan",
year = "2007",
month = "5",
language = "English (US)",
volume = "7",
pages = "371--382",
journal = "Quantum Information and Computation",
issn = "1533-7146",
publisher = "Rinton Press Inc.",
number = "4",

}

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 -