In open quantum systems theory, a **quantum channel** is formally defined as a completely positive (CP), trace-preserving linear map representing the evolution of density operators under environmental interaction and unobserved measurement outcomes. The foundational concepts include **Kraus operator representations**, which decompose these channels into an operator sum ($\mathcal{E}(\rho) = \sum_A M_A^\dagger M_A \rho$), and the principle that only unitary evolution remains invertible, while non-unitary maps represent irreversible decoherence where information flows irretrievably to inaccessible environments. This framework generalizes closed-system axioms by establishing the isomorphism between quantum operations (channels) and entangled states (**channel-state duality**), proving that any physical operation must be completely positive rather than merely positive when extended to larger systems, a requirement exemplified by the failure of the transpose map in composite systems.
In open quantum systems theory, a **quantum channel** is formally defined as a completely positive (CP), trace-preserving linear map representing the evolution of density operators under environmenta…