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.
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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.