Generation of Mapping Cones from Small Sets
arXiv:1209.0437 · doi:10.1016/j.laa.2012.12.007
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.
18 pages
References in corpus (17)
- Quantum entanglement
- Separability Criterion for Density Matrices
- Separability of Mixed States: Necessary and Sufficient Conditions
- Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
- Unextendible Product Bases and Bound Entanglement
- Optimization of entanglement witnesses
- General Entanglement Breaking Channels
- A Schmidt number for density matrices
- Entangling operations and their implementation using a small amount of entanglement
- Cones of positive maps and their duality relations
- Entanglement purification via separable superoperators
- Three-by-three bound entanglement with general unextendible product bases
- Numerical studies of entangled PPT states in composite quantum systems
- Low rank extremal PPT states and unextendible product bases
- Quantum dynamical semigroups and non-decomposable positive maps
- On extremal positive maps acting between type I factors
- Mapping Cones are Operator Systems