Tangles of superpositions and the convex-roof extension

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Osterloh, Andreas
Siewert, Jens
Uhlmann, Armin
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Abstract
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We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, $SL(2,\CC)$ invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the {\em zero-polytope} is a convex set of density matrices with vanishing tangle whereas the {\em convex characteristic curve} readily provides a non-trivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero-polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.
Comment: 7 pages, 5 figures, revtex4
Keywords
Quantum Physics, Condensed Matter - Other Condensed Matter
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