Abstract:

We answer in the affirmative a recently-posed question that asked if there exists an “untypical” convex mapping cone – i.e., one that does not arise from the transpose map and the cones of k-positive and k-superpositive maps. We explicitly construct such a cone based on atomic positive maps. Our general technique is to consider the smallest convex mapping cone generated by a single map, and we derive several results on such mapping cones. We use this technique to also present several other examples of untypical mapping cones, including a family of cones generated by spin factors. We also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality.

Authors:

• Nathaniel Johnston
• Łukasz Skowronek
• Erling Størmer

Download:

• Official publication from Linear Algebra and Its Applications
• Preprint from arXiv:1209.0437 [math.OA]
• Local preprint [pdf]

Cite as:

• N. Johnston, Ł. Skowronek, and E. Størmer. Generation of mapping cones from small sets. Linear Algebra and Its Applications, 438:3062–3075, 2013.