Balanced tripartite entanglement, the alternating group A4 and the Lie algebra sl(3, ℂ)⊕ u(1)

Michel Planat, P. Lévay, Metod Saniga

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographie group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W. Then, we describe a peculiar "condensation" of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, ℂ) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.

Original languageEnglish
Pages (from-to)39-51
Number of pages13
JournalReports on Mathematical Physics
Volume67
Issue number1
DOIs
Publication statusPublished - Feb 2011

Fingerprint

Alternating group
Qubit
Entanglement
Lie Algebra
algebra
Encoding
Genetic Code
coding
Particle Physics
Entangled State
Maximal Subgroup
genetic code
Group Algebra
Condensation
Finite Group
subgroups
entry
Generator
generators
condensation

Keywords

  • Entanglement
  • Lie algebras
  • Quantum computation

ASJC Scopus subject areas

  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

Balanced tripartite entanglement, the alternating group A4 and the Lie algebra sl(3, ℂ)⊕ u(1). / Planat, Michel; Lévay, P.; Saniga, Metod.

In: Reports on Mathematical Physics, Vol. 67, No. 1, 02.2011, p. 39-51.

Research output: Contribution to journalArticle

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