Choi–Jamiołkowski isomorphism


In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels and quantum states, this is introduced by M. D. Choi and A. 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 from system to, which is a trace-preserving complete positive map from operator spaces to, we introduce an auxiliary system with the same dimension as system. Consider the maximally entangled state
in the space of, since is complete positive, is a nonnegative operator. Conversely, for any nonnegative operator on, we can associate a complete positive map from to, this kind of correspondece is called Choi-Jamiolkowski isomorphism.