  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.
