The Choi-Cholesky algorithm for completely positive maps

Abstract

We establish explicit means via which natural dilations of completely positive (CP) maps can be constructed à la Kraus's IInd representation theorem. To obtain this, we rely on the Choi-Jamiołkowski 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.

Added Remarks 1.4 (statement of related works), 2.13 (coherence of recursive construction); Minor adjustments to text; added reference to code base