Generation of Mapping Cones from Small Sets

TL;DR

This paper constructs and characterizes unusual convex mapping cones in quantum information theory, demonstrating the existence of 'untypical' cones not derived from standard maps, using techniques involving atomic positive maps and spin factors.

Contribution

It introduces a method to generate and analyze untypical convex mapping cones, including explicit examples and a full characterization in the qubit case.

Findings

Existence of untypical convex mapping cones not arising from transpose or standard positive maps.

Construction of such cones using atomic positive maps and spin factors.

Complete characterization of single-element generated cones in the qubit case.

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.