A simple proof of the Jamiolkowski criterion for complete positivity of linear maps of algebras of Hilbert-Schmidt operators

TL;DR

This paper extends a simple proof of the Jamiolkowski criterion to determine the complete positivity of linear maps between algebras of Hilbert-Schmidt operators, including infinite-dimensional cases.

Contribution

It generalizes a previous proof to cover all Hilbert-Schmidt operator algebras, broadening the criterion's applicability.

Findings

Provides a simplified proof applicable to infinite-dimensional cases

Extends the Jamiolkowski criterion to all Hilbert-Schmidt operator algebras

Facilitates easier verification of complete positivity in complex quantum systems

Abstract

We generalize a preceding simple proof of the Jamiolkowski criterion to check whether a given linear map between algebras of operators is completely positive or not. The generalization is performed to embrace all algebras of Hilbert-Schmidt class operators, thus possibly infinite-dimensional.