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arXiv:2608.13750 · Entropy

Choi--Jamiołkowski-type isomorphisms for von Neumann algebras

Marcin Marciniak, Michał Cholewiak

math.OAmath-phquant-phSubmitted Aug 13, 2026• Announced Aug 17, 2026• 0 citations
v1 · Aug 13, 2026v2 · Sep 15, 2026
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Abstract

The Choi--Jamiołkowski isomorphism identifies completely positive maps with bipartite states and underlies much of finite-dimensional quantum information theory. For systems with infinitely many degrees of freedom, modelled by von Neumann algebras of type III, neither traces nor density matrices are available, and the isomorphism has to be reformulated. We show that for arbitrary von Neumann algebras $\mathcal{M}$ and $\mathcal{N}$ there is a canonical order isomorphism between the space of normal completely bounded maps from $\mathcal{M}$ into the predual of $\mathcal{N}$ and the predual of the spatial tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$; under this identification complete positivity corresponds to positivity. Replacing the opposite algebra by $\mathcal{N}$ itself requires an anti-isomorphism of $\mathcal{N}$ with itself, and we prove that this condition is not only sufficient but also necessary, provided the identification is required to be natural in $\mathcal{M}$. Some such requirement is unavoidable, since for every $\mathcal{M}$ anti-isomorphic to itself an isomorphism exists for trivial reasons. Consequently no Choi--Jamiołkowski correspondence exists for the type III factors constructed by Connes. Along the way we show that the space of all normal maps, taken with the operator norm, is strictly too large for this purpose, and that complete positivity does not force complete boundedness in this setting.

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The Choi--Jamiołkowski isomorphism identifies completely positive maps with bipartite states and underlies much of finite-dimensional quantum information theory. For systems with infinitely many degrees of freedom, modelled by von Neumann algebras of type III, neither traces nor density matrices are available, and the isomorphism has to be reformulated. We show that for arbitrary von Neumann algebras M\mathcal{M}M and N\mathcal{N}N there is a canonical order isomorphism between the space of normal completely bounded maps from M\mathcal{M}M into the predual of N\mathcal{N}N and the predual of the spatial tensor product of M\mathcal{M}M with the \emph{opposite} algebra of N\mathcal{N}N; under this identification complete positivity corresponds to positivity. Replacing the opposite algebra by N\mathcal{N}N itself requires an anti-isomorphism of N\mathcal{N}N with itself, and we prove that this condition is not only sufficient but also necessary, provided the identification is required to be natural in M\mathcal{M}M. Some such requirement is unavoidable, since for every M\mathcal{M}M anti-isomorphic to itself an isomorphism exists for trivial reasons. Consequently no Choi--Jamiołkowski correspondence exists for the type III factors constructed by Connes. Along the way we show that the space of all normal maps, taken with the operator norm, is strictly too large for this purpose, and that complete positivity does not force complete boundedness in this setting.
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.
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–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.
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].
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.
Almost-idempotent quantum channels and approximate C∗C^*C∗-algebras
Let ΦΦΦ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that ΦΦΦ is ηηη-idempotent, namely, ∥Φ2−Φ∥cb≤η\|Φ^2-Φ\|_{\mathrm{cb}} \leη∥Φ2−Φ∥cb​≤η, and construct an associated ε\varepsilonε-C∗C^*C∗ algebra (of almost-invariant observables) for ε=O(η)\varepsilon=O(η)ε=O(η). This type of structure has the axioms of a unital C∗C^*C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε\varepsilonε. We prove that any finite-dimensional ε\varepsilonε-C∗C^*C∗ algebra AAA is O(ε)O(\varepsilon)O(ε)-isomorphic to some genuine C∗C^*C∗ algebra BBB. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When AAA comes from a finite-dimensional ηηη-idempotent UCP map ΦΦΦ, the O(η)O(η)O(η)-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel Φ∗Φ^*Φ∗ into a decoding channel, producing a state on BBB, and an encoding channel.