Kraus ' theorem (named after Karl Kraus) characterizes completely positive maps, which model quantum operations between quantum states. Informally, the theorem ensures that the action of any such quantum operation Φ Φ on a state ρ ρ can always be written as Φ Φ ( ρ ρ ) = ∑ ∑ k B k ρ ρ B k ∗ ∗ , for some set of operators { B k } k satisfying ∑ ∑ k B k ∗ ∗ B k ≤ ≤ 1 , where 1 is the identity operator.