The range of generalized quantum operations

A completely positive map Φ   is called a generalized quantum operation if tr(Φ(A))⩽tr(A)tr(Φ(A))⩽tr(A) for all positive operators A  . In this note, we mainly characterize the sets of {Φ(B):ΦΦ(B):Φ is a generalized quantum operation}, {Φ(B):Φ†Φ(B):Φ† is a generalized quantum operation}, and {Φ(B):ΦΦ(B):Φ and Φ†Φ† are generalized quantum operations}, where Φ†Φ† is a dual of Φ and B is a self-adjoint operator. In particular, the third set is relevant to Uhlmann's theorem in quantum information.