Convex hulls of varieties and entanglement measures based on the roof construction

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • Tobias J. Osborne

Externe Organisationen

  • Royal Holloway University of London
Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Seiten (von - bis)209-227
Seitenumfang19
FachzeitschriftQuantum Information and Computation
Jahrgang7
Ausgabenummer3
PublikationsstatusVeröffentlicht - März 2007
Extern publiziertJa

Abstract

In this paper we study the problem of calculating the convex hull of certain a ne algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call polynomial entanglement measures, can be represented as a ne algebraic varieties. We consider the evaluation of certain mixed-state extensions of these polynomial entanglement measures, namely convex and concave roofs. We show that the evaluation of a roof-based mixed-state extension is equivalent to calculating a hyperplane which is multiply tangent to the variety in a number of places equal to the number of terms in an optimal decomposition for the measure. In this way we provide an implicit representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.

ASJC Scopus Sachgebiete

Zitieren

Convex hulls of varieties and entanglement measures based on the roof construction. / Osborne, Tobias J.
in: Quantum Information and Computation, Jahrgang 7, Nr. 3, 03.2007, S. 209-227.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Osborne, TJ 2007, 'Convex hulls of varieties and entanglement measures based on the roof construction', Quantum Information and Computation, Jg. 7, Nr. 3, S. 209-227.
Osborne, T. J. (2007). Convex hulls of varieties and entanglement measures based on the roof construction. Quantum Information and Computation, 7(3), 209-227.
Osborne TJ. Convex hulls of varieties and entanglement measures based on the roof construction. Quantum Information and Computation. 2007 Mär;7(3):209-227.
Osborne, Tobias J. / Convex hulls of varieties and entanglement measures based on the roof construction. in: Quantum Information and Computation. 2007 ; Jahrgang 7, Nr. 3. S. 209-227.
Download
@article{6ad0ced16398472b90ebbeb6be0eb140,
title = "Convex hulls of varieties and entanglement measures based on the roof construction",
abstract = "In this paper we study the problem of calculating the convex hull of certain a ne algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call polynomial entanglement measures, can be represented as a ne algebraic varieties. We consider the evaluation of certain mixed-state extensions of these polynomial entanglement measures, namely convex and concave roofs. We show that the evaluation of a roof-based mixed-state extension is equivalent to calculating a hyperplane which is multiply tangent to the variety in a number of places equal to the number of terms in an optimal decomposition for the measure. In this way we provide an implicit representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.",
keywords = "Algebraic geometry, Entanglement measures",
author = "Osborne, {Tobias J.}",
note = "Copyright: Copyright 2007 Elsevier B.V., All rights reserved.",
year = "2007",
month = mar,
language = "English",
volume = "7",
pages = "209--227",
number = "3",

}

Download

TY - JOUR

T1 - Convex hulls of varieties and entanglement measures based on the roof construction

AU - Osborne, Tobias J.

N1 - Copyright: Copyright 2007 Elsevier B.V., All rights reserved.

PY - 2007/3

Y1 - 2007/3

N2 - In this paper we study the problem of calculating the convex hull of certain a ne algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call polynomial entanglement measures, can be represented as a ne algebraic varieties. We consider the evaluation of certain mixed-state extensions of these polynomial entanglement measures, namely convex and concave roofs. We show that the evaluation of a roof-based mixed-state extension is equivalent to calculating a hyperplane which is multiply tangent to the variety in a number of places equal to the number of terms in an optimal decomposition for the measure. In this way we provide an implicit representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.

AB - In this paper we study the problem of calculating the convex hull of certain a ne algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call polynomial entanglement measures, can be represented as a ne algebraic varieties. We consider the evaluation of certain mixed-state extensions of these polynomial entanglement measures, namely convex and concave roofs. We show that the evaluation of a roof-based mixed-state extension is equivalent to calculating a hyperplane which is multiply tangent to the variety in a number of places equal to the number of terms in an optimal decomposition for the measure. In this way we provide an implicit representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.

KW - Algebraic geometry

KW - Entanglement measures

UR - http://www.scopus.com/inward/record.url?scp=34247253919&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34247253919

VL - 7

SP - 209

EP - 227

JO - Quantum Information and Computation

JF - Quantum Information and Computation

SN - 1533-7146

IS - 3

ER -