Completely positive map

Summary

A completely positive map between C*-algebras is a linear map that stays positive after tensoring with the identity on every matrix algebra $M_n(\C)$. These maps are the natural morphisms of operator algebras when one keeps only the order structure. The Stinespring dilation theorem describes them all as compressions of $*$-homomorphisms. In quantum information theory, completely positive trace-preserving maps are the quantum channels.