If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections | x ⟩ ⟩ ⟨ ⟨ x | ∈ ∈ B ( C | X | ) , then the action of an instrument I is given by a quantum channel I : B ( H 1 ) → → B ( H 2 ) ⊗ ⊗ B ( C | X | ) with