A simple proof of the Jamiolkowski criterion for complete positivity of linear maps

TL;DR

This paper presents an accessible and straightforward proof of the Jamiolkowski criterion, enabling physicists to easily verify complete positivity of linear maps and systematically derive Kraus decompositions.

Contribution

It offers a simple, direct proof of the Jamiolkowski criterion and a systematic method for obtaining Kraus matrices, improving accessibility for physicists.

Findings

Provides a more accessible proof for physicists.

Enables systematic derivation of Kraus matrices.

Simplifies the verification of complete positivity.

Abstract

We give a simple direct proof of the Jamiolkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accesible for physicists than others found in the literature and provides a systematic method to give any set of Kraus matrices of its Kraus decomposition.