Choi-level twirling of quantum channels: finite constructions and non-compact transformations

Twirling, i.e. averaging over symmetry actions, is a standard tool for reducing quantum states and channels to a symmetry-invariant form. We study channel twirling from the perspective of the channel-state duality and provide a constructive Choi-level description of the averaging map induced by arbitrary input/output representations. Our main technical result concerns the collective setting: for π^{textrm{in}}(U)=U^{otimes t_{textrm{in}}} and π^{textrm{out}}(U)=U^{otimes t_{textrm{out}}}, we introduce a partial-transpose reduction that removes the contragredient action and converts the mixed (walled Brauer) channel twirl into an ordinary Schur-Weyl twirl of the partially transposed Choi operator under U^{otimes\(t_{textrm{in}}+t_{textrm{out}}\)}, enabling explicit permutation-based formulas without constructing walled Brauer idempotents or mixed Schur transforms. Beyond compact symmetries, we extend channel twirling to reductive, generally non-unitary groups via Cartan decomposition and obtain an invariant-sector decomposition of the averaged Choi operator with weights determined solely by the Abelian Cartan component. Finally, we provide two finite realizations of channel averaging: a "dual" implementation as a convex mixture of unitary-1-design channels acting on invariant sectors, and a design-like reconstruction showing that weighted group t-designs induce channel t-designs for t=t_{textrm{in}}+t_{textrm{out}}.