We study quantum entanglement by Schmidt decomposition for some typical quantum algorithms. In the Shor's exponentially fast algorithm the quantum entanglement holds almost maximal, which is a major factor that a classical computer is not adequate to simulate quantum efficient algorithms.

The capacity of quantum channel with product input states was formulated by the quantum coding theorem. However, whether entangled input states can enhance the quantum channel is still open. It turns out that this problem is reduced to a special case of the more general problem whether the capacity of product quantum channel exhibits additivity. In the present study, we apply one of the quantum Arimoto-Blahut type algorithms to the latter problem. The results suggest that the additivity of product quantum channel capacity always holds and that entangled input states cannot enhance the quantum channel capacity.