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

TL;DR

This paper develops a comprehensive framework for quantum channel twirling, including finite constructions and extensions to non-compact groups, with explicit formulas and new invariance decompositions.

Contribution

It introduces a Choi-level description of channel twirling, extends it to non-unitary groups, and provides finite realizations linking group designs to channel designs.

Findings

Explicit permutation formulas for channel twirling without complex idempotents

Extension of twirling to reductive, non-unitary groups via Cartan decomposition

Finite realizations of channel averaging using group t-designs

Abstract

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 $\pi^{\textrm{in}}(U)=U^{\otimes t_{\textrm{in}}}$ and $\pi^{\textrm{out}}(U)=U^{\otimes t_{\textrm{out}}}$, 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 $t=t_{\textrm{in}}+t_{\textrm{out}}$.