The Choi-Cholesky algorithm for completely positive maps

TL;DR

This paper introduces a Cholesky-based algorithm for constructing natural dilations of completely positive maps using the Choi-Jamio{}kowski correspondence, enabling canonical adjoint actions.

Contribution

It develops a Cholesky algorithm for bi-partite systems to explicitly construct dilations of CP maps, extending Kraus's representation theorem.

Findings

Provides a canonical construction of adjoint actions for CP maps.

Enables explicit dilation construction under separability and boundedness assumptions.

Utilizes the Choi-Jamio{}kowski correspondence for algorithm development.

Abstract

We establish explicit means via which natural dilations of completely positive (CP) maps can be constructed \`a la Kraus's IInd representation theorem. To obtain this, we rely on the Choi-Jamio{\l}kowski correspondence and develop a Cholesky algorithm for bi-partite systems. This enables a canonical construction of adjoint actions which recover the behaviour of the original CP-maps. Our results hold under separability assumptions and the requirement that the maps are completely bounded and preserve the subideal of finite rank operators.