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arXiv:2512.23586 · Journal of Physics A Mathematical and Theoretical

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

Marcin Markiewicz, Łukasz Pawela, Zbigniew Puchała

quant-phSubmitted Dec 29, 2025• Announced Dec 30, 2025• 0 citations• Revised Feb 25, 2026 (v2)
v1 · Dec 29, 2025v2 · Feb 25, 2026
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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 $π^{\textrm{in}}(U)=U^{\otimes t_{\textrm{in}}}$ 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 $t=t_{\textrm{in}}+t_{\textrm{out}}$.

Journal reference
J. Phys. A: Math. Theor., 2026
DOI
10.1088/1751-8121/ae965b
Comments
15 pages

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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 πin(U)=U⊗tinπ^{\textrm{in}}(U)=U^{\otimes t_{\textrm{in}}}πin(U)=U⊗tin​ and πout(U)=U⊗toutπ^{\textrm{out}}(U)=U^{\otimes t_{\textrm{out}}}πout(U)=U⊗tout​, 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⊗(tin+tout)U^{\otimes(t_{\textrm{in}}+t_{\textrm{out}})}U⊗(tin​+tout​), 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-111-design channels acting on invariant sectors, and a design-like reconstruction showing that weighted group ttt-designs induce channel ttt-designs for t=tin+toutt=t_{\textrm{in}}+t_{\textrm{out}}t=tin​+tout​.
We construct finite averaging sets for twirling finite-dimensional quantum states over a broad class of symmetry transformations represented by reductive Lie groups. The averaging is defined through the Cartan decomposition of the group, with Haar-uniform integration over the compact components and an arbitrary normalized measure over the non-compact Abelian component. We show that the resulting twirling map can always be represented as a finite probabilistic mixture of unitary 1-design channels acting on irreducible sectors of the representation restricted to the maximal compact subgroup. Thus, the finite construction is determined by the compact-sector decomposition, while the non-compact Cartan component contributes only scalar sector weights. This distinction is important for implementation: no Schur transform for the full reductive group is required. Compared with standard t -design-based finite averaging, our construction generally uses fewer averaging operators, at the price of replacing local tensor-power sampling by sector-wise unitary averaging. This could be especially beneficial for simulations and quantum algorithms on higher-dimensional systems as reduction of the size of the averaging set relatively increases with dimension.
Abstract It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging . The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs . In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.
The Schur transform, which block-diagonalizes the tensor representation U⊗nU^{\otimes n}U⊗n of the unitary group Ud\mathbf{U}_dUd​ on nnn qudits, is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) to the case of mixed tensor U⊗n⊗Uˉ⊗mU^{\otimes n} \otimes \bar{U}^{\otimes m}U⊗n⊗Uˉ⊗m, where Uˉ\bar{U}Uˉ is the dual representation. This representation is the symmetry of unitary-equivariant channels, which find various applications in quantum majority vote, multiport-based teleportation, asymmetric state cloning, black-box unitary transformations, etc. The "mixed" Schur transform contains several natural extensions of the representation theory used in the Schur transform, in which the main ingredient is a duality between the mixed tensor representations and the walled Brauer algebra. Another element is an efficient implementation of a "dual" Clebsch-Gordan transform for Uˉ\bar{U}Uˉ. The overall circuit has complexity O~((n+m)d4)\widetilde{O} ((n+m)d^4)O((n+m)d4). Finally, we show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings and discuss other potential applications, including an extension of permutational quantum computing that includes partial transposes.
This thesis explores the interplay between representation theory and quantum information. Specifically, we focus on mixed Schur–Weyl duality, which considers the action of the unitary group on mixed tensors. This setting naturally arises in quantum information tasks involving unitary-equivariant channels, such as port-based teleportation, quantum majority vote, and universal transposition of unitary operators. A key contribution of this thesis is an explicit derivation of the action of the generators of the partially transposed permutation matrix algebra—the commutant of the mixed unitary action—in the Gelfand–Tsetlin basis. As another key result of this thesis, we develop efficient quantum circuits for the mixed quantum Schur transform, a novel primitive in quantum information. The key ingredient of our construction is new efficient circuits for the dual Clebsch–Gordan transform of the unitary group. A significant application of our findings is the construction of efficient quantum algorithms for port-based teleportation, a variant of quantum teleportation that eliminates the need for corrective operations. Another application is a symmetry reduction of semidefinite optimisation problems with unitary equivariance symmetry. Finally, we study the extendibility of quantum states possessing unitary, mixed unitary, or orthogonal symmetry on the complete graph. We obtain analytically the exact maximum values for projections onto the maximally entangled state and the antisymmetric state for each of the three symmetry classes. This thesis demonstrates the usefulness of mixed Schur–Weyl duality in quantum information and computing. We expect that our tools will help address other problems in other areas of quantum information processing, such as communication, cryptography, and simulation.
Twirled and swirled worlds describe generalised probabilistic theories (GPTs) restricted by symmetry-imposed superselection rules. We extend these constructions to a fully compositional setting for finite-dimensional GPTs and compact groups, treating states, effects, and multipartite transformations uniformly as processes. We distinguish system actions from process-space actions. We define actions on a theory as a whole by requiring the corresponding process-space actions to respect sequential and parallel composition. Averaging physical theory actions defines twirling, while averaging nonphysical theory actions defines swirling. We show that invariant subtheories might not satisfy process tomography, and establish conditions under which they do, involving invariant dynamically faithful states or internal classical reference frames. For locally tomographic parent theories, we prove that every theory symmetry admits a representation by linear system actions and that the invariant subtheory satisfies process tomography. An invariant multipartite process space fails local tomography if and only if some bipartition has sides sharing a non-trivial irreducible representation. We characterise when ancillary systems are necessary, which representations they must carry, and when entanglement is required for ancilla-free process tomography. We develop a framework for unifying twirled subtheories and extend incoherent simulations to compact groups. Finally, we recover bilocal classical theory as a twirled subtheory of classical theory and real quantum theory as a swirled subtheory of complex quantum theory, deriving their tomographic properties from their representation structure.
Abstract Averaging physical quantities over Lie groups appears in many contexts across the rapidly developing branches of physics like quantum information science or quantum optics. Such an averaging process can be always represented as averaging with respect to a finite number of elements of the group, called a finite averaging set . In the previous research such sets, known as t -designs, were constructed only for the case of averaging over unitary groups (hence the name unitary t -designs ). In this work we investigate the problem of constructing finite averaging sets for averaging over general non-compact matrix Lie groups, which is much more subtle task due to the fact that the the uniform invariant measure on the group manifold (the Haar measure) is infinite. We provide a general construction of such sets based on the Cartan decomposition of the group, which splits the group into its compact and non-compact components. The averaging over the compact part can be done in a uniform way, whereas the averaging over the non-compact one has to be endowed with a suppressing weight function, and can be approached using generalized Gauss quadratures. This leads us to the general form of finite averaging sets for semisimple matrix Lie groups in the product form of finite averaging sets with respect to the compact and non-compact parts. We provide an explicit calculation of such sets for the group S L ( 2 , C ) , although our construction can be applied to other cases. Possible applications of our results cover finding finite ensembles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.