From Kraus Operators to the Stinespring Form of Quantum Maps: An Alternative Construction for Infinite Dimensions

We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map $\Phi$ there exists an auxiliary Hilbert space $\mathcal K$ in a pure state $|\psi\rangle\langle\psi|$ as well as a unitary operator $U$ on system plus environment such that $\Phi$ equals $\operatorname{tr}_{\mathcal K}(U((\cdot)\otimes|\psi\rangle\langle\psi|)U^*)$. 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 $\Phi$" 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.