We consider the problem of constructing quantum operations or channels
transforming a given set of quantum states. In the mathematical setting,
the problem reduces to finding a completely positive linear map, if it
exists, that maps a given set of density matrices to another given set
of density matrices. This problem in turns is equivalent to constructing
a positive semidefinite matrix satisfying certain linear constraints.
We show that the problem is never "weakly infeasible".
Furthermoren, we exploit the structure of the problem and develop
efficient methods in solving instances of high dimensions
that standard linear semidefinite programming solvers cannot handle.
We present numerical experiments based on
various "alternating projection type" methods.
(work with Vris Yuen-Lam Cheung, Dmitriy Drusvyatskiy, Chi-Kwong Li,
pdf file of talk