Theorem. Let Φ Φ be a (not necessarily trace-preserving) quantum operation on a finite-dimensional Hilbert space H with two representing sequences of Kraus matrices { B i } i ≤ ≤ N and { C i } i ≤ ≤ N . Then there is a unitary operator matrix ( u i j ) i j such that C i = ∑ ∑ j u i j B j .