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

TL;DR

This paper offers a new constructive proof for representing quantum channels in infinite dimensions, utilizing Sz.-Nagy's dilation theorem, and introduces a minimal environment construction compared to classical approaches.

Contribution

It provides an alternative, constructive method for the Stinespring dilation of quantum maps in infinite dimensions, reducing the environment size and using Sz.-Nagy's dilation theorem.

Findings

Constructive proof of Stinespring dilation in infinite dimensions

Environment size is minimized to Kraus rank plus a qubit

Illustrative example comparing new and classical constructions

Abstract

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.