Details
| Original language | English |
|---|---|
| Pages (from-to) | 209-227 |
| Number of pages | 19 |
| Journal | Quantum Information and Computation |
| Volume | 7 |
| Issue number | 3 |
| Publication status | Published - Mar 2007 |
| Externally published | Yes |
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
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Nuclear and High Energy Physics
- Mathematics(all)
- Mathematical Physics
- Physics and Astronomy(all)
- Computer Science(all)
- Computational Theory and Mathematics
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In: Quantum Information and Computation, Vol. 7, No. 3, 03.2007, p. 209-227.
Research output: Contribution to journal › Article › Research › peer review
}
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 -