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⊗tin and
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), 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=tin+tout.
We provide a generalization of the idea of unitary designs to cover finite averaging over much more general operations on quantum states. Namely, we construct finite averaging sets for averaging quantum states over arbitrary reductive Lie groups, on condition that the averaging is performed uniformly over the compact component of the group. Our construction comprises probabilistic mixtures of unitary 1-designs on specific operator subspaces. Provided construction is very general, competitive in the size of the averaging set when compared to other known constructions, and can be efficiently implemented in the quantum circuit model of computation.
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.
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.
Quantum simulation at fixed particle number admits two equivalent descriptions, a first-quantized (particle) representation and a second-quantized (occupation-number) representation. Their quantum resource costs differ sharply across computational tasks, so the ability to convert coherently between them is valuable. We construct an explicit unitary
Q, with inverse
Q†, that maps a first-quantized state to its fixed-
N occupation-number form while diagnosing the input's particle-exchange symmetry. The conversion is therefore symmetry-agnostic at the input yet fully resolved at the output, and it applies uniformly to bosonic, fermionic, and parastatistical sectors. At its foundation lies a structural identification that we place at the center of this work: the quantum Schur transform supplied by Schur-Weyl duality is the non-abelian Fourier transform of the commuting pair
(SN,U(d)), and the occupation-number representation is its weight basis, retaining only the labels shared by both factors, the irrep
λ and the
u(d) weight. This reduction is lossless for bosons and fermions, while a canonical Gelfand-Tsetlin promise renders it one-to-one for the remaining sectors. Algorithmically,
Q composes the strong Schur transform with reversible arithmetic that computes occupations as successive row-sum differences of the Gelfand-Tsetlin pattern, yielding gate complexity
poly(N,d,log(1/ε)). The converted state is prepared efficiently in quantum memory. Any classical algorithm that outputs it explicitly, however, pays a cost set by the sector dimension, which is polynomial of degree
N in
d at fixed
N and exponential in
N when
d=Θ(N). Finally, an efficient classical sampler for the induced occupation-number distribution would yield one for arbitrary quantum circuits, contrary to standard complexity assumptions.