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 Jamiołkowski isomorphism, the Jamiołkowski-Choi isomorphism, or the Choi-Jamiołkowski 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, Jamiołkowski, and Choi that lead to the duality. We further illustrate some fundamental properties and diverse applications.