A family of norms with applications in quantum information theory II

We consider the problem of computing the family of operator norms recently introducedin [1]. We develop a family of semidefinite programs that can be used to exactly computethem in small dimensions and bound them in general. Some theoretical consequencesfollow from the duality theory of semidefinite programming, including a new constructiveproof that for all r there are non-positive partial transpose Werner states that are r-undistillable. Several examples are considered via a MATLAB implementation of thesemidefinite program, including the case of Werner states and randomly generated statesvia the Bures measure, and approximate distributions of the norms are provided. Weextend these norms to arbitrary convex mapping cones and explore their implicationswith positive partial transpose states.

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