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A Kernel Approach to the Stinespring and Kraus Representations

James Tian

TL;DR

The paper develops a self-contained kernel-based derivation of the Stinespring and Kraus representations for completely positive maps. It introduces two scalar positive-definite kernels: the first generates the dilation space and a unital *-representation, and the second exposes a canonical tensor-product structure from which Kraus operators arise. This approach avoids heavy C*-algebra machinery, clarifies the geometric meaning of dilation, and shows how the Kraus decomposition emerges from orthogonal components of the ancillary space. The resulting framework provides a transparent link between positive definite kernels, reproducing kernel Hilbert spaces, and the canonical forms of CP maps, with convergence statements in the strong operator topology.

Abstract

We give a self-contained derivation of the Stinespring and Kraus structure theorems for completely positive maps using only scalar positive-definite kernels.

A Kernel Approach to the Stinespring and Kraus Representations

TL;DR

The paper develops a self-contained kernel-based derivation of the Stinespring and Kraus representations for completely positive maps. It introduces two scalar positive-definite kernels: the first generates the dilation space and a unital *-representation, and the second exposes a canonical tensor-product structure from which Kraus operators arise. This approach avoids heavy C*-algebra machinery, clarifies the geometric meaning of dilation, and shows how the Kraus decomposition emerges from orthogonal components of the ancillary space. The resulting framework provides a transparent link between positive definite kernels, reproducing kernel Hilbert spaces, and the canonical forms of CP maps, with convergence statements in the strong operator topology.

Abstract

We give a self-contained derivation of the Stinespring and Kraus structure theorems for completely positive maps using only scalar positive-definite kernels.

Paper Structure

This paper contains 3 sections, 2 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. The Stinespring Dilation
  3. The Kraus Representation

Key Result

Theorem 2.1

Let $\varphi:\mathscr{B}\left(H\right)\rightarrow\mathscr{B}\left(H\right)$ be a CP map. There exists a minimal Stinespring dilation $\left(\pi,H_{K},V\right)$ such that where:

Theorems & Definitions (4)

  • Theorem 2.1: Stinespring dilation
  • proof
  • Theorem 3.1: canonical tensor structure
  • proof