A Family of Norms With Applications In Quantum Information Theory II

TL;DR

This paper introduces a family of operator norms relevant to quantum information theory, providing methods for their exact computation in small dimensions and bounds in general, with applications to Werner states and positive partial transpose states.

Contribution

Develops semidefinite programming techniques to compute and bound a new family of operator norms, with theoretical insights and practical implementations in quantum states.

Findings

Exact computation of norms for small dimensions

Bounded norms for general cases

Identification of non-positive partial transpose Werner states that are r-undistillable

Abstract

We consider the problem of computing the family of operator norms recently introduced in arXiv:0909.3907. We develop a family of semidefinite programs that can be used to exactly compute them in small dimensions and bound them in general. Some theoretical consequences follow from the duality theory of semidefinite programming, including a new constructive proof that there are non-positive partial transpose Werner states that are r-undistillable for arbitrary r. Several examples are considered via a MATLAB implementation of the semidefinite program, including the case of Werner states and randomly generated states via the Bures measure, and approximate distributions of the norms are provided. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states.