Twirled and swirled worlds: a compositional approach
Abstract
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.
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BibTeX
@misc{soltani2026twirled,
title = {{Twirled and swirled worlds: a compositional approach}},
author = {Sina Soltani and Thomas D. Galley and Marco Erba and John H. Selby},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33294},
eprint = {2610.33294},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33294},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twirled and swirled worlds: a compositional approach
AU - Sina Soltani
AU - Thomas D. Galley
AU - Marco Erba
AU - John H. Selby
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33294
UR - https://arxiv.org/abs/2610.33294
AB - 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.
ER -