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.
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.
Tomographic locality is a principle commonly used in the program of finding axioms that pick out quantum theory within the landscape of possible theories. The principle asserts the sufficiency of local measurements for achieving a tomographic characterization of any bipartite state. In this work, we explore the meaning of the principle of tomographic locality by developing a simple scheme for generating a wide variety of theories that violate the principle. In this scheme, one starts with a tomographically local theory -- which can be classical, quantum or post-quantum -- and a physical symmetry, and one restricts the processes in the theory to all and only those that are covariant with respect to the collective action of that symmetry. We refer to the resulting theories as twirled worlds. We show that failures of tomographic locality are ubiquitous in twirled worlds. From the possibility of such failures in classical twirled worlds, we argue that the failure of tomographic locality (i.e., tomographic nonlocality) does not imply ontological holism. Our results also demonstrate the need for researchers seeking to axiomatize quantum theory to take a stand on the question of whether there are superselection rules that have a fundamental status.
Generalised Probabilistic Theories (GPTs) provide a unifying framework encompassing classical theories, quantum theories, as well as hypothetical alternatives. We investigate both the problem of extending a system with a finite set of transformations, and the problem of adding to a translation-invariant set of systems a finite set of entangled states and effects, plus all their images under the translation symmetry. We show that determining whether such extensions are consistent with the axioms of GPTs is undecidable. The source of the undecidability is that these finite extensions can generate infinitely many conditions which must be checked: iterating transformations may produce infinitely many new transformations; and similarly, entangled states and effects may generate infinitely many new states via the analogue of teleportation. Our results show that extending GPTs to include dynamics or entanglement encounters fundamental computability obstructions, which can only be circumvented by introducing additional physical or mathematical assumptions.
Quasiprobability representations are well-established tools in quantum information science, with applications ranging from the classical simulability of quantum computation to quantum process tomography, quantum error correction, and quantum sensing. While traditional quasiprobability representations typically employ real-valued distributions, recent developments highlight the usefulness of complex-valued ones -- most notably, via the family of Kirkwood--Dirac quasiprobability distributions. Building on the framework of Schmid et al. [Quantum 8, 1283 (2024)], we extend the analysis to encompass complex-valued quasiprobability representations that need not preserve the identity channel. Additionally, we also extend previous results to consider mappings towards infinite-dimensional spaces. We show that, for each system, every such representation can be expressed as the composition of two maps that are completely characterized by their action on states and on the identity (equivalently, on effects) for that system. Our results apply to all complex-valued quasiprobability representations of any finite-dimensional, tomographically-local generalized probabilistic theory, with finite-dimensional quantum theory serving as a paradigmatic example. In the quantum case, the maps' action on states and effects corresponds to choices of frames and dual frames for the representation. This work offers a unified mathematical framework for analyzing complex-valued quasiprobability representations in generalized probabilistic theories.
Entanglement is a central and subtle feature of quantum theory, whose structure and operational behavior can change dramatically when additional physical constraints, such as symmetries or superselection rules, are imposed. Such constraints can give rise to striking and counter-intuitive phenomena, including local broadcasting of entangled states and failures of entanglement monogamy. These effects naturally arise in tomographically nonlocal theories (like real quantum theory, twirled worlds, or fermionic quantum theory), where composite systems possess holistic degrees of freedom that are inaccessible to local measurements. In this work, we study entanglement in such theories within the framework of generalized probabilistic theories. We show that the failure of tomographic locality leads to two qualitatively distinct forms of entanglement, which we term tomographically-local entanglement and tomographically-nonlocal entanglement. We analyze the operational consequences of this distinction, proving that tomographically-nonlocal entanglement is useless for Bell nonlocality, steering, and teleportation, but sufficient for dense coding and perfectly secure data hiding. This framework clarifies the origin of several previously puzzling features of entanglement that arise when tomographic locality fails, as can happen even in quantum theory when one considers fermions or fundamental superselection rules.
We extend the process-matrix framework for defining indefinite causal orders to a broad class of generalised physical theories. We give a simple re-axiomatisation of spacetime contexts in terms of naturality over classical probes, which precisely captures non-contextual assignment together with compatibility with convexity and coarse-graining within local laboratories. Using this compositional perspective, we give a clean characterisation of theories in which abstract spacetime contexts admit a process-matrix representation, and we pin down the precise conditions (satisfied by classical theory, quantum theory, and a Boxworld-like theory) under which the supermap and process-matrix perspectives on higher-order processes coincide.
Process tomography, the experimental characterization of physical processes, is a central task in science and engineering. Here we investigate the axiomatic requirements that guarantee the in-principle feasibility of process tomography in general physical theories. Specifically, we explore the requirement that process tomography should be achievable with a finite number of auxiliary systems. We show that this requirement is satisfied in every theory equipped with universal extensions, that is, correlated states from which all other correlations can be generated with non-zero probability. We show that universal extensions exist in two cases: (1) theories permitting conclusive state teleportation, and (2) theories satisfying three properties of Causality, Pure Product States, and Purification. In the second case, the existence of universal extensions follows from a symmetry property of Purification, whereby all pure bipartite states with the same marginal on one system are interconvertible. Crucially, our results hold even in theories that do not satisfy Local Tomography, the property that the state of any composite system can be identified from the correlations of local measurements. Summarizing, the existence of universal extensions, without any additional requirement of Local Tomography, is a sufficient guarantee for the characterizability of physical processes using a finite number of auxiliary systems.