Practical construction of positive maps which are not completely positive

There are many more positive maps than completely positive maps

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool are real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps which are not completely positive is given.

在完全正映射中的插值：数值解法

Interpolation for completely positive maps: numerical solutions

本文提出了一种寻找矩阵代数之间完全正映射的技术，该映射在给定数据上取预定值，并通过半定规划方法和一个数值例子展示了一种凸优化方法。

The outer spectral radius and dynamics of completely positive maps

We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the $n^2$ by $n^2$ matrix $\sum{\overline{X_i}}\otimes{X_i}.$ We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely postive map, including the special case …

谱面阴影与实闭域上的完全正映射

Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields

本文提出新方法刻画非谱面阴影的凸半代数集，证明$n\geq5$的余正矩阵锥非谱面阴影，回答Scheiderer问题，并基于模型论观察建立保持性条件。