Existence and construction of a quantum channel with given inputs and outputs

We derive that sufficient and necessary conditions for existence of a quantum channel φ and a generalized unitary operation ε sending Ai to Bi(1≤ i≤ k) for two given families {Ai}i=1k,{Bi}i=1k of matrices,respectively.As an application,a sufficient and necessary condition for existence of a unitary duality quantum computer with given input-output states is obtained.