Abstract:

We consider the problem of computing the family of operator norms recently introduced in [1]. 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.

Authors:

• Nathaniel Johnston
• David W. Kribs

Download:

• Official publication from QIC
• Preprint from arXiv:1006:0898 [quant-ph]
• Local preprint [pdf]
• Slideshow presentation [pdf]

Status:

• Accepted for publication in Quantum Information & Computation.

Cite as:

• N. Johnston and D. W. Kribs, A Family of Norms With Applications In Quantum Information Theory II. Quantum Information & Computation 11 1 & 2, 104–123 (2011).

Supplementary material:

• QuantumSeDuMi.m – A MATLAB front-end for solving quantum semidefinite programs using the SeDuMi solver
• SchmidtOperatorNorm.m – A MATLAB script for computing the S(k)-norm of an operator

Related publications:

• A Family of Norms With Applications In Quantum Information Theory (prequel publication)