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arXiv:2610.27394

Choi-Jamiołkowski isomorphism(s) and channel/state duality, naturally!

Paul E. Lammert

quant-phSubmitted Oct 19, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

Provided a maximally entangled state vector in ${\mathcal{H}}\otimes{\mathcal{H}}$, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of ${\mathsf CP}({\mathcal{H}},{\mathcal{K}})$ (completely positive maps of density operators on ${\mathcal{H}}$ tothose on ${\mathcal{K}}$) onto the density operators on ${\mathcal{K}}\otimes{\mathcal{H}}$, which extends to a unitary map. Here, ${\mathcal{H}}$ and ${\mathcal{K}}$ 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 ${\mathcal{H}}$ and ${\mathcal{K}}$ which produces density operators on either ${\mathsf L}({\mathcal{H}},{\mathcal{K}})$ (Hilbert-Schmidt inner product) or ${\mathcal{K}}\otimes{\mathcal{H}}^*$ instead of ${\mathcal{K}}\otimes{\mathcal{H}}$. In particular, no entangled state input is required. There are no such binatural maps carried on ${\mathcal{K}}\otimes{\mathcal{H}}$, but those natural in only ${\mathcal{K}}$ are canonically associated with unitary maps ${\mathcal{H}}\rightarrow{\mathcal{H}}^*$ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.

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Provided a maximally entangled state vector in H⊗H{\mathcal{H}}\otimes{\mathcal{H}}H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of CP(H,K){\mathsf CP}({\mathcal{H}},{\mathcal{K}})CP(H,K) (completely positive maps of density operators on H{\mathcal{H}}H tothose on K{\mathcal{K}}K) onto the density operators on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, which extends to a unitary map. Here, H{\mathcal{H}}H and K{\mathcal{K}}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{\mathcal{H}}H and K{\mathcal{K}}K which produces density operators on either L(H,K){\mathsf L}({\mathcal{H}},{\mathcal{K}})L(H,K) (Hilbert-Schmidt inner product) or K⊗H∗{\mathcal{K}}\otimes{\mathcal{H}}^*K⊗H∗ instead of K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, but those natural in only K{\mathcal{K}}K are canonically associated with unitary maps H→H∗{\mathcal{H}}\rightarrow{\mathcal{H}}^*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\mathcal{M}\bar{\otimes}\mathcal{N}M⊗ˉ​N but on the tensor product of M\mathcal{M}M with the \emph{opposite} algebra of N\mathcal{N}N, or equivalently with a commutant; returning to N\mathcal{N}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\mathcal{M}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.
The complete positivity vs positivity correspondence in the Choi-Jamiołkowski-Kraus-Sudarshan quantum channel-state isomorphism depends on the choice of basis. Instead of the "canonical" basis, if we use, e.g., the Pauli spin matrices along with the identity as the basis for the space of bounded operators on the two-dimensional complex Hilbert space, this correspondence breaks down. A sufficient condition on the basis for validity of this correspondence is provided in the work of Paulsen and Shult~\cite{Paulsen}, which was later proven to be necessary by Kye~\cite{Kye}. A correspondence is also present between the space of super-maps and the tensor product of the spaces of the inputs and outputs of the same. In particular, a super-map is completely CP-preserving if and only if its Choi-type representation is completely positive (CP). This correspondence also depends on a specific choice of basis. In this work, we find the necessary and sufficient condition on a basis such that this correspondence holds true.
Abstract The Choi–Jamiołkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C ∗ -algebras they arise from, which are distinguished by their order of composition. A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism (Choi 1975 Lin. Alg. Appl. 10 285) further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, crucially depends on the dynamical structure in C ∗ -algebras. We explain in detail how this dependence gives rise to the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiołkowski (1972 Rep. Math. Phys. 3 275). We then overcome this subtle but pervasive issue in a number of ways: first, we prove a version of Choi’s theorem for Jamiołkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C ∗ -algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.
Abstract In this chapter we visit the ‘church of the larger Hilbert space’: we take a closer look at the relationship of pure and mixed quantum states via purification, and discuss a similar connection between completely positive and trace-preserving (CPTP) maps and unitary dynamics on a larger Hilbert space. We show that such CPTP maps, which represent quantum channels, are equivalent to operations admitting a Kraus decomposition, as well as to unitary dynamics on a larger Hilbert space with factoring initial condition, the famous Stinespring dilation of quantum operations. In this context we introduce the Choi-Jamiołkowski isomorphism, which expresses a duality between maps between two Hilbert spaces and states on tensor products of these Hilbert spaces, and we review several important examples of quantum channels such as the dephasing channel, amplitude-damping channel, and depolarizing channel. Finally, we examine operations that are not possible in quantum theory, focusing on the no-cloning theorem.
A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.
Abstract The Choi representation of completely positive trace preserving (CPTP) maps, i.e. quantum channels is often used in the context of quantum information and computation as it is easy to work with. It is a correspondence between CPTP maps and quantum states also termed as the Choi–Jamiołkowski isomorphism. It is especially useful if a parametrization of the set of CPTP maps is needed in order to consider a general map or optimize over the set of these. Here we provide a brief introduction to this topic, focusing on certain useful calculational techniques which are presented in full detail.