Complex finite-dimensional quantum theory is distinguished within the Euclidean Jordan family by the matching between traceless observables and infinitesimal reversible dynamics. Existing characterizations typically impose this relation globally, or require a collision model capable of realizing all reversible dynamics. We ask how much of this dynamical structure is needed once the single-system state space is already assumed to be a simple Euclidean Jordan algebra and composition is available in the sense of Barnum–Graydon–Wilce. Let J be a nontrivial simple finite-dimensional Euclidean Jordan algebra admitting a BGW self-composite, and let Sₜ = exp(tΓ) be a unit-preserving reversible binary collision. Suppose equal product preparations are infinitesimally stationary and that the interaction induces a nonzero first-order reduced dynamics for at least one reference state. We prove that J ≅ Hₙ(C)ₛₐ for some n ≥ 2. The conclusion is rigid: after identifying J with Hₙ(C)ₛₐ, every productive stationary first-order footprint is a nonzero scalar multiple of the Hamiltonian commutator b ↦ i[b, ·], so the collision-induced map J₀ → Der(J) is automatically an isomorphism. On the standard complex matrix self-composite, the global stationary interaction itself is partial swap up to an overall rate. Conversely, partial swap supplies such a collision for every complex Hermitian EJA. The key step is that equal-state stationarity forces the global collision generator to commute with the connected diagonal automorphism action. The local infinitesimal footprint therefore determines a nonzero equivariant map J₀ → Der(J), where J₀ is the traceless subspace. Classification of simple EJAs then leaves only the complex Hermitian family, and one-dimensionality of the complex intertwiner space gives the rigidity statement. The result is a Jordan type-selection theorem, not a reconstruction from arbitrary operational theories: EJA structure and BGW composition are hypotheses. At this stage, however, neither local tomography, purification, collision universality, nor an a priori observable–generator correspondence is required. We also construct higher spin-factor collisions that are globally nontrivial and exactly stationary but have vanishing first-order local response, showing that the productivity hypothesis cannot simply be replaced by global nontriviality.
Complex finite-dimensional quantum theory is distinguished within the Euclidean Jordan family by the matching between traceless observables and infinitesimal reversible dynamics. Existing characterizations typically impose this relation globally, or require a collision model capable of realizing all reversible dynamics. We ask how much of this dynamical structure is needed once the single-system state space is already assumed to be a simple Euclidean Jordan algebra and composition is available in the sense of Barnum–Graydon–Wilce. Let J be a nontrivial simple finite-dimensional Euclidean Jordan algebra admitting a BGW self-composite, and let Sₜ = exp(tΓ) be a unit-preserving reversible binary collision. Suppose equal product preparations are infinitesimally stationary and that the interaction induces a nonzero first-order reduced dynamics for at least one reference state. We prove that J ≅ Hₙ(C)ₛₐ for some n ≥ 2. The conclusion is rigid: after identifying J with Hₙ(C)ₛₐ, every productive stationary first-order footprint is a nonzero scalar multiple of the Hamiltonian commutator b ↦ i[b, ·], so the collision-induced map J₀ → Der(J) is automatically an isomorphism. On the standard complex matrix self-composite, the global stationary interaction itself is partial swap up to an overall rate. Conversely, partial swap supplies such a collision for every complex Hermitian EJA. The key step is that equal-state stationarity forces the global collision generator to commute with the connected diagonal automorphism action. The local infinitesimal footprint therefore determines a nonzero equivariant map J₀ → Der(J), where J₀ is the traceless subspace. Classification of simple EJAs then leaves only the complex Hermitian family, and one-dimensionality of the complex intertwiner space gives the rigidity statement. The result is a Jordan type-selection theorem, not a reconstruction from arbitrary operational theories: EJA structure and BGW composition are hypotheses. At this stage, however, neither local tomography, purification, collision universality, nor an a priori observable–generator correspondence is required. We also construct higher spin-factor collisions that are globally nontrivial and exactly stationary but have vanishing first-order local response, showing that the productivity hypothesis cannot simply be replaced by global nontriviality.
Complex finite-dimensional quantum theory is distinguished within the Euclidean Jordan family by the matching between traceless observables and infinitesimal reversible dynamics. Existing characterizations typically impose this relation globally, or require a collision model capable of realizing all reversible dynamics. We ask how much of this dynamical structure is needed once the single-system state space is already assumed to be a simple Euclidean Jordan algebra and composition is available in the sense of Barnum–Graydon–Wilce. Let J be a nontrivial simple finite-dimensional Euclidean Jordan algebra admitting a BGW self-composite, and let Sₜ = exp(tΓ) be a unit-preserving reversible binary collision. Suppose equal product preparations are infinitesimally stationary and that the interaction induces a nonzero first-order reduced dynamics for at least one reference state. We prove that J ≅ Hₙ(C)ₛₐ for some n ≥ 2. The conclusion is rigid: after identifying J with Hₙ(C)ₛₐ, every productive stationary first-order footprint is a nonzero scalar multiple of the Hamiltonian commutator b ↦ i[b, ·], so the collision-induced map J₀ → Der(J) is automatically an isomorphism. On the standard complex matrix self-composite, the global stationary interaction itself is partial swap up to an overall rate. Conversely, partial swap supplies such a collision for every complex Hermitian EJA. The key step is that equal-state stationarity forces the global collision generator to commute with the connected diagonal automorphism action. The local infinitesimal footprint therefore determines a nonzero equivariant map J₀ → Der(J), where J₀ is the traceless subspace. Classification of simple EJAs then leaves only the complex Hermitian family, and one-dimensionality of the complex intertwiner space gives the rigidity statement. The result is a Jordan type-selection theorem, not a reconstruction from arbitrary operational theories: EJA structure and BGW composition are hypotheses. At this stage, however, neither local tomography, purification, collision universality, nor an a priori observable–generator correspondence is required. We also construct higher spin-factor collisions that are globally nontrivial and exactly stationary but have vanishing first-order local response, showing that the productivity hypothesis cannot simply be replaced by global nontriviality.
We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities.These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit.Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. {In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.}
This paper extends the results of Hyperbolic Degree Bounds Signaling Dimension in Generalized Probabilistic Theories (Zenodo DOI: 10.5281/zenodo.22946778), which established that the signaling dimension of every canonical finite-dimensional Euclidean Jordan algebra (EJA) equals its Jordan rank. We investigate the consequences of this single-system identity under recursive composition. Combining the injectivity of local effect-space tensor products in standard non-signaling composites with the no-hypersignaling (NH) principle, we prove that the normalized linear-dimension ratio η_lin(S) = dim(S)/sign.dim(S)² is supermultiplicative. Consequently, any local excess above unity is amplified under repeated composition. For canonical EJA composites satisfying the structural axioms of Barnum, Graydon, and Wilce, product Jordan frames and NH enforce multiplicativity of composite rank. Together with established EJA classification results, this yields the conditional recursive constraint dim(J) ≤ rank(J)². Within this framework, quaternionic matrix Jordan systems and spin factors V_n with n ≥ 4 are excluded, while real matrix systems remain subcritical and complex matrix systems exactly saturate the bound. We also obtain finite-copy exclusion criteria and identify exact NH violations in specified EJA composition rules, including doubled complex and quaternionic composites. The analysis does not assume local tomography. Its conclusions are conditional on NH and the stated composition axioms; they do not establish unique selection of complex quantum theory. This work provides a compositional continuation of the preceding hyperbolic-signaling result, connecting classical simulation requirements to structural constraints on Jordan probabilistic theories.
Quantum theory satisfies the mathematically powerful property of self-duality, meaning that its state and effect spaces are made isomorphic by an inner product. This is why both quantum states and effects can be represented by positive Hermitian operators. The seminal Koecher-Vinberg theorem shows that self-duality is a rather special property: any system that is self-dual and also homogeneous, meaning that the group of symmetries of its cone of unnormalised states acts transitively on the strictly positive states, must be isomorphic to a Euclidean Jordan algebra. EJAs have been classified and are known to be just a slight generalisation of quantum systems. This is why the Koecher-Vinberg theorem, and hence self-duality and homogeneity, have been used in several reconstructions of quantum theory from first principles. While homogeneity has an operational derivation from the ability to steer states, self-duality currently lacks such a clear operational motivation. In this paper we prove an alternative to the Koecher-Vinberg theorem, substituting for self-duality the more operational property of pure transitivity: symmetries of the normalised state space act transitively on the pure states. We show that any system that is homogeneous and satisfies pure transitivity must be self-dual, and hence isomorphic to an EJA. Together with various ways of singling out the complex matrix algebras from among EJAs, and known ways of ruling out classicality in these, this yields several new and concise reconstructions of quantum theory. For example, quantum systems, classical systems, and composites of these are the only ones that are homogeneous, satisfy pure transitivity, and allow locally tomographic composites; fully quantum systems are the only ones that are homogeneous, have continuous pure transitivity, and allow a correspondence between observables and generators of reversible transformations.
Solèr's theorem classifies an orthomodular generalized Hilbert space containing an infinite orthonormal sequence as a real, complex, or quaternionic Hilbert space. This classification does not by itself select the complex field. We add a separate finite-composition requirement: a system of dimension d over the selected field has the full self-adjoint matrix space as its unnormalized state/effect space; a composite of dimensions a and b has dimension ab over the same field; and independent product effects are tomographically complete. The resulting dimension identity holds for complex Hermitian matrices and fails, by explicit nonzero defects, for real symmetric and quaternionic Hermitian matrices whenever a and b are both greater than one. Complex Hilbert space is therefore rigid within this declared reconstruction class. The conclusion is conditional and representation-sensitive: complex quantum theory can be realified by adjoining an orthogonal complex-structure operator, and alternative composite rules fall outside the theorem. Applied to MTT, the result identifies exact missing premises rather than deriving them. The current q79 finite quantum model and finite complex polarization are compatible with the complex branch, but MTT has not yet sourced the required infinite orthogonality and standard locally tomographic composition from one physical upper construction.