Theorem. Let H and G be Hilbert spaces of dimension n and m respectively, and Φ Φ be a quantum operation between H and G . Then, there are matrices { B i } 1 ≤ ≤ i ≤ ≤ n m mapping H to G such that, for any state ρ ρ , Φ Φ ( ρ ρ ) = ∑ ∑ i B i ρ ρ B i ∗ ∗ . Conversely, any map Φ Φ of this form is a quantum operation provided ∑ ∑ k B k ∗ ∗ B k ≤ ≤ 1 .