Real-amplitude quantum theory is the subtheory of quantum theory invariant under complex conjugation, and experiments in networks of independent sources have measured correlations above the real bound. We construct a theory on which the same conjugation fails complete positivity and still leaves every probability in the unit interval, and whose correlations are exactly those of its own conjugation-invariant subtheory in every causal structure, the bilocality scenario included. Its states are all the density matrices, its effects are the operators every partial transpose of which is again a quantum effect, and its operations are the completely PPT-preserving maps. The symmetrized subtheory simulates it once each source carries a reference frame rather than each system. One map therefore receives three different verdicts in three theories, so the symmetry alone marks no boundary at all. Quantum theory admits every effect its states permit, and no theory with that property can hide a symmetry this way, so what sustains the separation measured in the network experiments is the absence of a restriction rather than any feature of conjugation. What does have a boundary is the class of theories whose correlations coincide with those of their symmetrized subtheory. We show that sectorial closure, meaning invariance of the effects and the operations under the symmetry acting independently on each source, suffices for the absence of a gap under any finite group, and that it cannot be weakened on the effects. Fixing unrestricted states and conjugation makes the effects of this theory the largest the symmetry admits and its operations the largest sectorially closed ones, so it is not one construction among several. Locating where sectorial closure fails, for a candidate effect set built from a fixed bound entangled state, is a finite computation on a single ray of effects.
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35 pages, 1 figure, 2 tables. Ancillary files contain the verification script and the certificate, also deposited at https://doi.org/10.5281/zenodo.22648589
Real-amplitude quantum theory is the subtheory of quantum theory invariant under complex conjugation, and experiments in networks of independent sources have measured correlations above the real bound. We construct a theory on which the same conjugation fails complete positivity and still leaves every probability in the unit interval, and whose correlations are exactly those of its own conjugation-invariant subtheory in every causal structure, the bilocality scenario included. Its states are all the density matrices, its effects are the operators every partial transpose of which is again a quantum effect, and its operations are the completely PPT-preserving maps. The symmetrized subtheory simulates it once each source carries a reference frame rather than each system. One map therefore receives three different verdicts in three theories, so the symmetry alone marks no boundary at all. Quantum theory admits every effect its states permit, and no theory with that property can hide a symmetry this way, so what sustains the separation measured in the network experiments is the absence of a restriction rather than any feature of conjugation. What does have a boundary is the class of theories whose correlations coincide with those of their symmetrized subtheory. We show that sectorial closure, meaning invariance of the effects and the operations under the symmetry acting independently on each source, suffices for the absence of a gap under any finite group, and that it cannot be weakened on the effects. Fixing unrestricted states and conjugation makes the effects of this theory the largest the symmetry admits and its operations the largest sectorially closed ones, so it is not one construction among several. Locating where sectorial closure fails, for a candidate effect set built from a fixed bound entangled state, is a finite computation on a single ray of effects.
Whether the complex numbers of standard quantum theory are experimentally indispensable has remained open for decades. Real quantum theory (RQT), obtained by replacing complex amplitudes with real ones while retaining the usual Kronecker-product composition rule, reproduces all single-party and bipartite Bell correlations of quantum theory (QT), but its lack of local tomography suggested that the two theories might diverge in more general local experiments. This possibility appeared to be confirmed by Renou et al., who argued that a bilocal network experiment can falsify RQT without falsifying QT. Here we show that this conclusion relies on an experimentally untestable assumption. The key distinction is between product-state independence, which constrains the mathematical form of source states, and operational independence, which is defined entirely by the absence of observable cross-source correlations. We prove that, once source independence is imposed operationally, every finite network correlation achievable in QT is also achievable in RQT with the same locality structure of the measurements. We then extend this equivalence to arbitrary finite sequential multipartite protocols involving channels and measurements with prescribed locality structure. Thus, as long as no violation of QT is observed, RQT cannot be experimentally falsified. Our results restore the empirical indistinguishability of QT and RQT, while showing that they support markedly different pictures of the correlation structure underlying the same observed world.
Recent work by Renou et al. (2021) has led to some controversy concerning the question of whether quantum theory requires complex numbers for its formulation. We promote the view that the main result of that work is best understood not as a claim about the relative merits of different representations of quantum theory, but rather as a claim about the possibility of experimentally adjudicating between standard quantum theory and an alternative theory -- a foil theory -- known as real-amplitude quantum theory (RQT). In particular, the claim is that this adjudication can be achieved given only an assumption about the causal structure of the experiment. Here, we aim to shed some light on why this is possible, by reconceptualizing the comparison of the two theories as an instance of a broader class of such theory comparisons. By recasting RQT as the subtheory of quantum theory that arises by symmetrizing with respect to the collective action of a time-reversal symmetry, we can compare it to other subtheories that arise by symmetrization, but for different symmetries. If the symmetry has a unitary representation, the resulting foil theory is termed a twirled quantum world, and if it does not (as is the case in RQT), the resulting foil theory is termed a swirled quantum world. We show that, in contrast to RQT, there is no possibility of distinguishing any twirled quantum world from quantum theory given only an assumption about causal structure. We also define analogues of twirling and swirling for an arbitrary generalized probabilistic theory and identify certain necessary conditions on a causal structure for it to be able to support a causal compatibility gap between the theory and its symmetrized version. We draw out the implications of these analyses for the question of how a lack of a shared reference frame state features into the possibility of such a gap.
The role of complex quantities in quantum theory has been puzzling physicists since the beginnings. It is, thus, natural to ask whether, in order to describe our experiments, the mathematical structure of the complex Hilbert spaces it is built on is really necessary. Recently, it was shown that this structure is inevitable in network scenarios with independent sources. More precisely, Real Quantum Theory cannot explain the predictions of (Complex) Quantum Theory [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Here, we revisit the independence assumption underlying this work. We show that assuming partial independence is sufficient for showing the inadequacy of Real Quantum Theory. We derive a tradeoff between source independence and the Bell value achievable in Real Quantum Theory, which also lower bounds the source correlations required to explain previous experiments by means of real quantum systems. We further show that 1 bit of entanglement is necessary and sufficient for recovering the complex quantum correlations by means of Real Quantum Theory in the scenario from [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Finally, building on [McKague et al., Phys. Rev. Lett. 102, 2009PRLTAO0031-900710.1103/PhysRevLett.102.020505], we provide a construction to simulate any complex quantum setup with m independent sources by means of Real Quantum Theory by allowing the sources to share an m real-qubit entangled state in the first round of the experiment.
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.
A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an analogous exact representation on a doubled complex space selected by an antiunitary symplectic structure Theta^2 = -I. Within that sector, a Choi fixed-point condition characterizes when a complex channel admits quaternionic Kraus operators, while exact correction of a finite-dimensional right-quaternionic code is characterized by compression coefficients in the real center and admits an explicit recovery. For octonions, nonassociativity precludes a unique continuation; para-linear, categorical, sectorial, Jordan, Clifford-envelope, and Moufang models are compared by the operational structures they retain and the additional data required to define composites, channels, or recovery. Across these cases, an exact ambient representation does not erase the complex or symplectic structure that selects the physical theory.
Central thesis: Quantum theory contains stable closure without universal completion. Quantum field theory repeatedly resists the classical assumption that every physically meaningful structure must correspond to an independently complete object. A quantum state can be definite relative to one observable while remaining unresolved relative to another; separated alternatives can retain phase relations and later interfere; composite systems can possess globally definite relational structure without factorizing into independently complete subsystems; gauge-related field configurations can represent a single physical state; effective descriptions depend systematically on scale; and local field degrees of freedom need not correspond to independently realizable asymptotic particles. This paper investigates whether these apparently different structures admit a common representation as forms of partial closure. A closure is defined as a physically meaningful stabilization or equivalence relative to a specified context, probe family, observable algebra, quotient, scale, or asymptotic condition. It is partial when stable disclosure exists while additional physically relevant structure remains latent, relational, global, scale-dependent, or otherwise outside that closure. The framework distinguishes redundant fibers from latent physical fibers through a future-distinguishability criterion and introduces reclosure for later sensitivity to information not exhausted by an earlier closure. Interference becomes phase-sensitive reclosure; incompatible measurements instantiate noncommuting closure; entanglement instantiates nonfactorizable closure; contextuality becomes a failure of local closures to assemble into one global classical assignment; gauge symmetry becomes quotient closure; renormalization becomes scale-relative effective closure; confinement exemplifies asymptotic closure restriction; and Berry, Wilson, and flavor-cycle invariants exemplify closed relational cycles retaining invariant memory. Two restricted theory-space applications are developed: a one-generation hypercharge calculation in which Yukawa invariance and anomaly cancellation fix relative hypercharges up to overall normalization, and a minimal representation reconstruction under explicit low-dimensional and minimality assumptions. The resulting Typed Quantum Partial Closure Representation Theorem is a representation theorem rather than a derivation theorem. The framework is therefore presently a unifying reconstruction program whose decisive future test is whether closure principles can generate restrictions or predictions not already contained in the formalisms they reorganize. Keywords: quantum field theory; partial closure; reclosure; gauge symmetry; effective field theory; entanglement; contextuality; confinement; asymptotic states; anomaly cancellation; flavor mixing