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\l{}kowski isomorphism, the Jamio\l{}kowski-Choi isomorphism, or the Choi-Jamio\l{}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\l{}kowski, and Choi that lead to the duality. We further illustrate some fundamental properties and diverse applications.
Provided a maximally entangled state vector in
H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of
CP(H,K) (completely positive maps of density operators on
H tothose on
K) onto the density operators on
K⊗H, which extends to a unitary map. Here,
H and
K are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both
H and
K which produces density operators on either
L(H,K) (Hilbert-Schmidt inner product) or
K⊗H∗ instead of
K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on
K⊗H, but those natural in only
K are canonically associated with unitary maps
H→H∗ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.
In chapter 7 we will study quantum channels. We will see how the so-called Choi–Jamiołkowski isomorphism—which links quantum channels with quantum states—allows for semidefinite programming techniques to be applied to quantum channels. We will then see how we can estimate an uncharacterised channel, calculate the diamond norm (which characterises the operational distinguishability between channels) and finally how to use duality to uncover a link between a channel optimisation problem and quantum entropies.
We address various aspects of a widely used tool in quantum information theory: the Choi-Jamiolkowski isomorphism [A. Jamiolkowski, Rep. Math. Phys., 3, 275 (1972)]. We review different versions of the isomorphism, their properties and propose a unified description that combines them all. To this end, we identify the physical reason for the appearance of the (basis-dependent) operation of transposition in the isomorphism as used in Choi's theorem [M.-D. Choi, Lin. Alg. Appl., 10, 285 (1975)]. This requires a careful distinction between Jordan algebras and the different C*-algebras they arise from, which are distinguished by their order of composition. Physically, the latter encodes a choice of time orientation in the respective algebras, which relates to a number of recent results, including a characterisation of quantum from more general non-signalling bipartite correlations [M. Frembs and A. Döring, arXiv:2204.11471] and a classification of bipartite entanglement [M. Frembs, arXiv:2207.00024].
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)].