Abstract
We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map Φ there exists an auxiliary Hilbert space K in a pure state |ψψ| as well as a unitary operator U on system plus environment such that Φ equals tr_K(U((·)⊗|ψψ|)U^*). The main tool of our proof is Sz.-Nagy's dilation theorem applied to isometries defined on a subspace. In our construction, the environment consists of a system of dimension "Kraus rank of Φ" together with a qubit, the latter only acting as a catalyst. In contrast, the original proof of Hellwig & Kraus given in the 70s yields an auxiliary system of dimension "Kraus rank plus one". We conclude by providing an example which illustrates how the constructions differ from each other.