Abstract
In their interesting paper Jiang, Luo, and Fu [Phys. Rev. A 87, 022310 (2013)] present a nice historical review of what is usually referred to as the Choi-Jamio\l{}kowski isomorphism and discuss some of its subtle properties. Our aim is to point out that there is a stronger correspondence between quantum maps and bipartite states: an isometric isomorphism between linear maps and functionals. This correspondence stems from the famous Grothendieck memoirs, and it has been a known fact at least since the 1970s. In particular, some properties studied by Jiang et al. are simple reminiscences of this more general approach. Moreover, for the sake of a complete historical review, we would like to note that the positive partial transpose criterion studied in the 1980s by Choi was, in fact, rediscovered in quantum information by Peres [Phys. Rev. Lett. 77, 1413 (1996)] and M. Horodecki et al. [Phys. Lett. A 223, 1 (1996)].
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