Choi–Jamiołkowski isomorphism
In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels (described by complete positive maps) and quantum states (described by density matrices), this is introduced by Man-Duen Choi and Andrzej Jamiołkowski. It is also called channel-state duality by some authors in the quantum information area, but mathematically, this is a more general correspondence between positive operators and the complete positive superoperators. Definition. To study a quantum channel formula_1 from system formula_2 to formula_3, which is a trace-preserving complete positive map from operator spaces formula_4 to formula_5 , we introduce an auxiliary system formula_6 with the same dimension as system formula_2. Consider the maximally entangled state in the space of formula_9, since formula_1 is complete positive, formula_11 is a nonnegative operator. Conversely, for any nonnegative operator on formula_12, we can associate a complete positive map from formula_4to formula_5, this kind of correspondence is called Choi-Jamiołkowski isomorphism.