dorsal/arxiv
View SchemaConvex Hulls of Varieties and Entanglement Measures Based on the Roof Construction
| Authors | Tobias J. Osborne |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0402055 |
| URL | https://arxiv.org/abs/quant-ph/0402055 |
| Journal | Quantum Information and Computation, vol. 7 (3), pp.209-227 (2007) |
Abstract
In this paper we study the problem of calculating the convex hull of certain affine 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 affine 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.
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"abstract": "In this paper we study the problem of calculating the convex hull of certain\naffine algebraic varieties. As we explain, the motivation for considering this\nproblem is that certain pure-state measures of quantum entanglement, which we\ncall polynomial entanglement measures, can be represented as affine algebraic\nvarieties. We consider the evaluation of certain mixed-state extensions of\nthese polynomial entanglement measures, namely convex and concave roofs. We\nshow that the evaluation of a roof-based mixed-state extension is equivalent to\ncalculating a hyperplane which is multiply tangent to the variety in a number\nof places equal to the number of terms in an optimal decomposition for the\nmeasure. In this way we provide an implicit representation of optimal\ndecompositions for mixed-state entanglement measures based on the roof\nconstruction.",
"arxiv_id": "quant-ph/0402055",
"authors": [
"Tobias J. Osborne"
],
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"quant-ph"
],
"journal_ref": "Quantum Information and Computation, vol. 7 (3), pp.209-227 (2007)",
"title": "Convex Hulls of Varieties and Entanglement Measures Based on the Roof Construction",
"url": "https://arxiv.org/abs/quant-ph/0402055"
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