Interpolations of entanglement breaking channels and equivalent conditions for completely positive maps

We first investigate the interpolation problem between the strong entanglement breaking maps and countably decomposable separable states. Also, we study the interpolation relation between Φ(A)=B and Φ(|A|)=|B|, where Φ is a completely positive trace-preserving map, A and B are self-adjoint trace-class operators. At last, we mainly give two results for completely positive maps to extend the Choi theorem in the infinite dimensional Hilbert space.