  A linear map between real symmetric matrix spaces is positive if all positive
semidefinite matrices are mapped to positive semidefinite ones. A real
symmetric matrix is separable if it can be written as a summation of Kronecker
products of positive semidefinite matrices. This paper studies how to check if
a linear map is positive or not and how to check if a matrix is separable or
not. We propose numerical algorithms, based on Lasserre type semidefinite
relaxations, for solving such questions. To check the positivity of a linear
map, we construct a hierarchy of semidefinite relaxations for minimizing the
associated bi-quadratic forms over the unit spheres. We show that the
positivity can be detected by solving a finite number of such semidefinite
relaxations. To check the separability of a matrix, we construct a hierarchy of
semidefinite relaxations. If it is not separable, we can get a mathematical
certificate for that; if it is, we can get a decomposition for the
separability.
