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.
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)].
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.
Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on
M⊗ˉN but on the tensor product of
M with the \emph{opposite} algebra of
N, or equivalently with a commutant; returning to
N itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is
∗-anti-isomorphic to itself. Naturality cannot be dropped: whenever
M is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.
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].