Optimal convex approximation of quantum channels based on $\alpha$-affinity
Determining the minimal distance between a target channel and a convex hull of predefined set of implementable channels is a fundamental problem in quantum resource theory, and provides key guidance for experimental implementations. In this work, we develop a unified analytical framework for optimal convex approximation of quantum channels based on the quantum $\alpha$-affinity measure. We construct a channel distance metric induced by the {\alpha}-affinity and the ChoiJamiolkowski isomorphism, which satisfies the required properties of a well-defined channel distance. Subsequently, we present an optimization framework for the convex approximation of quantum channels, and derive analytical solutions for the optimal convex approximation of single-qubit unitary channels over both the SU(2)-covariant and Pauli channel families, obtaining closed-form expressions for the optimal parameters and the minimal approximation distance. This framework is further applied to the amplitude-damping channel, yielding the explicit form of its optimal approximation and the associated minimal {\alpha}-affinity distance. In contrast to conventional approaches based on the diamond norm, our framework provides a systematic and analytically tractable approach to quantum channel approximation under realistic constraints.