Convex hulls of varieties and entanglement measures based on the roof construction
Tobias J. Osborne · Quantum Information and Computation · 2007
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 \emph{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 \emph{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 \emph{implicit} representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.