Averaging of quantum channels via channel-state duality

Abstract: Twirling, uniform averaging over symmetry actions, is a standard tool for reducing the description of quantum states and channels to symmetry-invariant data. We develop a framework for averaging quantum channels based on channel-state duality that converts pre- and post-processing averages into a group twirl acting directly on the Choi operator. For arbitrary unitary representations on the input and output spaces, the twirled channel is obtained as an explicit projection onto the commutant of the induced representation on $\mathcal H_{\rm out}\otimes \mathcal H_{\rm in}$. In the collective setting, where the commutant is the walled Brauer algebra, we introduce a partial-transpose reduction that maps channel twirling to an ordinary Schur-Weyl twirl of the partially transposed Choi operator, enabling formulas in terms of permutation operators. We further extend the construction beyond compact symmetries to reductive non-unitary groups via Cartan decomposition, yielding a weighted sum of invariant-sector projections with weights determined by the Abelian component. Finally, we provide two finite realizations of channel averaging. The first one is a ``dual'' averaging protocol as a convex mixture of unitary-$1$-design channels on invariant sectors. The second one is a notion of channel $t$-designs induced by weighted group $t$-designs for $t=t_{\rm in}+t_{\rm out}$.