Anti-Isomorphisms and the Naturality of Channel–State Duality

The Choi–Jamiołkowski correspondence identifies completely positive maps with positive bipartite operators in finite-dimensional quantum information theory. For general von Neumann algebras, the canonical bipartite object is the spatial tensor product with the opposite algebra. For arbitrary von Neumann algebras M and N, we define a canonical space CN(M) of normal maps M→N* from the predual of M⊗¯N∘ and prove that positivity of the functional corresponds exactly to complete positivity of the associated map. At the structural level, we prove that the contravariant predual functors M↦(M⊗¯N∘)* and M↦(M⊗¯N)* are naturally order-isomorphic if and only if N is ∗-anti-isomorphic to itself. It is enough to require naturality only with respect to the corner embeddings C→Mn(C). Consequently, for factors not anti-isomorphic to themselves, including the examples constructed by Connes, no such natural channel–state duality can exist. The obstruction is global and functorial rather than state-wise; individual symmetric normal bipartite states may still exist.