 Choi's theorem on completely positive maps

A finite-dimensional linear map is completely positive if and only if its Choi matrix is positive semidefinite. One direction applies the map tensored with the identity to a maximally entangled positive operator. For the converse, factor the positive Choi matrix into rank-one terms and reshape their vectors into matrices; the spectral Kraus decomposition reconstructs the map as a Kraus representation, which is completely positive.

Choi's theorem is an important result in the theory of completely positive (CP) maps in the context of operator algebras and quantum information theory. It provides a characterization of completely positive maps in terms of their action on matrices.