Channel-state duality

Channel-state duality refers to the correspondence between quantum channels and bipartite states, and it is extremely useful and fruitful in quantum information theory. It is often called the Jamio{}kowski isomorphism, the Jamio{}kowski-Choi isomorphism, or the Choi-Jamio{}kowski isomorphism. We trace the original roots of this duality from a historic perspective, clarify the somewhat misleading term ``isomorphism,'' reveal the underlying subtle nature, and highlight the seemingly similar but actually different features of the original correspondences in the manner of Pillis, Jamio{}kowski, and Choi that lead to the duality. We further illustrate some fundamental properties and diverse applications.