Entanglement of four qubit systems: A geometric atlas with polynomial compass I (the finite world)

Journal Article · Wed Jan 15 04:00:00 EST 2014 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4858336· OSTI ID:22251202

Holweck, Frédéric ^[1]; Luque, Jean-Gabriel ^[2]; Thibon, Jean-Yves ^[3]

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1. Laboratoire IRTES-M3M, Université de Technologie de Belfort-Montbéliard, 90010 Belfort Cedex (France)
2. Université de Rouen, Laboratoire d'Informatique, du Traitement de l'Information et des Systèmes (LITIS), Avenue de l'Université – BP 8, 6801 Saint-étienne-du-Rouvray Cedex (France)
3. Labroatoire d'Informatique Gaspard Monge Université Paris-Est Marne-la-Vallée, 77454 Marne-la-Vallée Cedex 2 (France)

We investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory, which allows us to interpret certain entangled states as algebraic varieties. More precisely we describe the nullcone, i.e., the set of states annihilated by all invariant polynomials, and also the so-called third secant variety, which can be interpreted as the generalization of GHZ-states for more than three qubits. All our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper. These 47 varieties correspond to 47 non-equivalent entanglement patterns, which reduce to 15 different classes if we allow permutations of the qubits.

• OSTI ID: 22251202
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 55; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
GEOMETRY
GHZ RANGE
POLYNOMIALS
QUANTUM ENTANGLEMENT
QUBITS