Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on $\mathcal{M}\bar{\otimes}\mathcal{N}$ but on the tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$, or equivalently with a commutant; returning to $\mathcal{N}$ 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 $\mathcal{M}$ 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.
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.
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.
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∗-algebras
Let Φ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that Φ is η-idempotent, namely, ∥Φ2−Φ∥cb≤η, and construct an associated ε-C∗ algebra (of almost-invariant observables) for ε=O(η). This type of structure has the axioms of a unital C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε. We prove that any finite-dimensional ε-C∗ algebra A is O(ε)-isomorphic to some genuine C∗ algebra B. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When A comes from a finite-dimensional η-idempotent UCP map Φ, the O(η)-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel Φ∗ into a decoding channel, producing a state on B, and an encoding channel.
Let O↦A(O) be a local Haag--Kastler net and let Ad(O)=A(O′)′ be its dual net. Normal unital completely positive maps on B(H) that fix A(O′) pointwise are exactly the channels with Kraus operators in Ad(O). Thus Haag duality at O is equivalent to equality with the A(O)-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators a∈M and b∈N, the small-parameter order defect has coefficient diamσ(−i[a,b]). Uniform optimization gives Γ(M,N)=2Δsa(M,N), and reversible inner-channel cb balls recover the same invariant through limε↓0Ωεrev(M,N)/ε2=Γ(M,N)/4. The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable G, the mean-input-energy-constrained coefficient ΓG,E has the same zero criterion and increases to Γ as E→∞, while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either 0 or 4; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.