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Averaging of quantum channels via channel-state duality

Marcin Markiewicz, Łukasz Pawela, Zbigniew Puchała

TL;DR

This work develops a channel–state duality framework to twirl quantum channels under symmetry actions, recasting the problem as a group average on the Choi operator under the induced representation and identifying it with a projection onto the commutant via generalized Schur–Weyl duality. In the collective setting, a partial-transpose reduction converts the problem to a standard Schur–Weyl twirl with permutation-operator structure, significantly simplifying construction by avoiding explicit walled Brauer idempotents. The authors extend the approach to reductive non-unitary groups via Cartan decomposition, yielding invariant-sector decompositions with weights determined by the Abelian component, and present two finite realizations: a dual averaging protocol using unitary-1-designs on invariant sectors and a channel-t-design framework for collective actions. These results provide practical, finite, design-like methods for averaging channels with broad symmetry groups, with potential impact on symmetry reduction, randomized benchmarking, and quantum information processing under structured noise.

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 $t=t_{\rm in}+t_{\rm out}$.

Averaging of quantum channels via channel-state duality

TL;DR

This work develops a channel–state duality framework to twirl quantum channels under symmetry actions, recasting the problem as a group average on the Choi operator under the induced representation and identifying it with a projection onto the commutant via generalized Schur–Weyl duality. In the collective setting, a partial-transpose reduction converts the problem to a standard Schur–Weyl twirl with permutation-operator structure, significantly simplifying construction by avoiding explicit walled Brauer idempotents. The authors extend the approach to reductive non-unitary groups via Cartan decomposition, yielding invariant-sector decompositions with weights determined by the Abelian component, and present two finite realizations: a dual averaging protocol using unitary-1-designs on invariant sectors and a channel-t-design framework for collective actions. These results provide practical, finite, design-like methods for averaging channels with broad symmetry groups, with potential impact on symmetry reduction, randomized benchmarking, and quantum information processing under structured noise.

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 . 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--design channels on invariant sectors. The second one is a notion of channel -designs induced by weighted group -designs for .
Paper Structure (7 sections, 8 theorems, 50 equations)

This paper contains 7 sections, 8 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Averaging quantum channels over arbitrary unitary transformations
  3. Averaging quantum channels over non-unitary operations defined via Cartan decomposition
  4. Dual averaging of channels
  5. Channel $t$-designs
  6. Relation to designs for channels.
  7. Conclusions and outlook

Key Result

Theorem 1

Let $\Phi:\mathcal{B}(\mathcal{H}_{\rm in})\to\mathcal{B}(\mathcal{H}_{\rm out})$ be a quantum channel with Choi operator $J_\Phi=(\Phi\otimes {\rm 1 l}_{\rm{in}})(\rho_{\rm ME})\in \mathcal{B}(\mathcal{H}_{\rm out}\otimes \mathcal{H}_{\rm in})$. Given unitary representations $\pi^{\rm in}_{\rm U}$ Then the Choi operator of the twirled channel satisfies i.e. channel twirling is a state twirl on

Theorems & Definitions (10)

  • Theorem 1: Channel twirl in Choi form, Section \ref{['sec:unitary']}
  • Corollary 1: Collective unitary twirl reduces to ordinary Schur--Weyl twirl, Section \ref{['sec:unitary']}
  • Theorem 2: Reductive (Cartan) twirl and invariant-sector decomposition, Section \ref{['sec:nonunitary']}
  • Theorem 3: Dual averaging as a mixture of unitary-$1$-design channels, Section \ref{['sec:dual']}
  • Proposition 1: Channel $t$-design induced by a weighted group $t$-design, Section \ref{['sec:designs']}
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof