Abstract: We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory. In this broader context, we analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inverses. We prove that each one is a strictly special instance of the latter for certain subcategories, providing a categorical foundation for Bayesian inversion as a generalization of reversing a process. We unify the categorical and $C^*$-algebraic notions of almost everywhere (a.e.) equivalence. As a consequence, we prove many results including a universal no-broadcasting theorem for S-positive categories, a generalized Fisher--Neyman factorization theorem for a.e. modular categories, a relationship between error correcting codes and disintegrations, and the relationship between Bayesian inversion and Umegaki's non-commutative sufficiency.

Abstract: Local master equations are a widespread tool to model open quantum systems, especially in the context of many-body systems. These equations, however, are believed to lead to thermodynamic anomalies and violation of the laws of thermodynamics. In contrast, here we rigorously prove that local master equations are consistent with thermodynamics and its laws without resorting to a microscopic model, as done in previous works. In particular, we consider a quantum system in contact with multiple baths and identify the relevant contributions to the total energy, heat currents and entropy production rate. We show that the second law of thermodynamics holds when one considers the proper expression we derive for the heat currents. We confirm the results for the quantum heat currents by using a heuristic argument that connects the quantum probability currents with the energy currents, using an analogous approach as in classical stochastic thermodynamics. We finally use our results to investigate the thermodynamic properties of a set of quantum rotors operating as thermal devices and show that a suitable design of three rotors can work as an absorption refrigerator or a thermal rectifier. For the machines considered here, we also perform an optimisation of the system parameters using an algorithm of reinforcement learning.

Abstract: We explore definite theoretical assertions about consciousness, starting from a non-reductive psycho-informational solution of David Chalmers's 'hard problem', based on the hypothesis that a fundamental property of 'information' is its experience by the supporting 'system'. The kind of information involved in consciousness needs to be quantum for multiple reasons, including its intrinsic privacy and its power of building up thoughts by entangling qualia states. As a result we reach a quantum-information-based panpsychism, with classical physics supervening on quantum physics, quantum physics supervening on quantum information, and quantum information supervening on consciousness. We then argue that the internally experienced quantum state, since it corresponds to a definite experience-not to a random choice-must be pure, and we call it ontic, in contrast with the state predictable from the outside (i.e. the state describing the knowledge of the experience from the point of view of an external observer) which we call epistemic and is generally mixed. Purity of the ontic state requires an evolution that is purity preserving, namely a so-called 'atomic' quantum operation. The latter is generally probabilistic, and its particular outcome is interpreted as the free will, which is unpredictable even in principle since quantum probability cannot be interpreted as lack of knowledge. The same purity of state and evolution allows solving the 'combination problem' of panpsychism. Quantum state evolution accounts for a short-term buffer of experience and contains itself quantum-to-classical and classical-to-quantum information transfers. Long term memory, on the other hand, is classical, and needs memorization and recall processes that are quantum-to-classical and classical-to-quantum, respectively...

Abstract: We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.

Abstract: In this contribution results from different disciplines of science were compared to show their intimate interweaving with each other having in common the golden ratio φ respectively its fifth power φ5. The research fields cover model calculations of statistical physics associated with phase transitions, the quantum probability of two particles, new physics of everything suggested by the information relativity theory (IRT) including explanations of cosmological relevance, the e-infinity theory, superconductivity, and the Tammes problem of the largest diameter of N non-overlapping circles on the surface of a sphere with its connection to viral morphology and crystallography. Finally, Fibonacci anyons proposed for topological quantum computation (TQC) were briefly described in comparison to the recently formulated reverse Fibonacci approach using the Janicko number sequence. An architecture applicable for a quantum computer is proposed consisting of 13-step twisted microtubules similar to tubulin microtubules of living matter. Most topics point to the omnipresence of the golden mean as the numerical dominator of our world.

Abstract: In a previous paper tests for entanglement for two-mode systems involving identical massive bosons were obtained. In the present paper we consider sufficiency tests for Einstein-Podolsky-Rosen (EPR) steering in such systems. We find that spin squeezing in any spin component, a Bloch vector test, the Hillery-Zubairy planar spin variance test, and squeezing in two-mode quadratures all show that the quantum state is EPR steerable. We also find a generalization of the Hillery-Zubairy planar spin variance test for EPR steering. The relation to previous correlation tests is discussed. This paper is based on a detailed classification of quantum states for bipartite systems. States for bipartite composite systems are categorized in quantum theory as either separable or entangled, but the states can also be divided differently into Bell local or Bell nonlocal states in terms of local hidden variable theory (LHVT). For the Bell local states there are three cases depending on whether both, one of or neither of the LHVT probabilities for each subsystem are also given by a quantum probability involving subsystem density operators. Cases where one or both are given by a quantum probability are known as local hidden states (LHSs) and such states are nonsteerable. The steerable states are the Bell local states where there is no LHS, or the Bell nonlocal states. The relationship between the quantum and hidden variable theory classification of states is discussed.

Abstract: In this article, the authors discuss the use of quantum probability, that is, the probability theory of quantum mechanics, for option pricing and for finance in general. The authors discuss the motivations for applying quantum probability to finance. The critical issues are replacing random variables with operators, self-reflexivity of markets, and the existence of incompatible observations. The authors outline quantum probability theory, quantum stochastic processes, and the pricing of options in a quantum context. TOPICS:Options, portfolio theory, portfolio construction Key Findings • Quantum probability theory is a probabilistic theory of observations. Observations can change the system and be incompatible. • Quantum probability offers a more empirically faithful handling of large events and of uncertainty. • A better theory of valuation is offered by quantum probability theory than classical probability theory.

Abstract: We present a unifying variational calculus derivation of Groverian geodesics for both quantum state vectors and quantum probability amplitudes. In the first case, we show that horizontal affinely parametrized geodesic paths on the Hilbert space of normalized vectors emerge from the minimization of the length specified by the Fubini–Study metric on the manifold of Hilbert space rays. In the second case, we demonstrate that geodesic paths for probability amplitudes arise by minimizing the length expressed in terms of the Fisher information. In both derivations, we find that geodesic equations are described by simple harmonic oscillators (SHOs). However, while in the first derivation the frequency of oscillations is proportional to the (constant) energy dispersion $$\Delta E$$ of the Hamiltonian system; in the second derivation the frequency of oscillations is proportional to the square root $$\sqrt{\mathcal {F}}$$ of the (constant) Fisher information. Interestingly, by setting these two frequencies equal to each other, we recover the well-known Anandan–Aharonov relation linking the squared speed of evolution of an Hamiltonian system with its energy dispersion. Finally, upon transitioning away from the quantum setting, we discuss the universality of the emergence of geodesic motion of SHO type in the presence of conserved quantities by analyzing two specific phenomena of gravitational and thermodynamical origin, respectively.

Abstract: This paper is devoted to justification of the application of quantum probability theory to problems of cognition, psychology, and decision making. Such applications are heavily based on quantum-like representation of interference of events that is formalized with complex probability amplitudes ("mental wave functions") and the Born rule for calculation of probability. In this paper, we present universal mathematical formalization of interference of events based on the calculus of intensities of interacting processes. Generally, this formalization leads to the nonlinear law of superposition of complex probability amplitudes with quantum linear superposition as a special important case. For intensities characterized by discrete occurrence of events, the formula for interference of intensities is transferred into the quantum-like formula for interference of probabilities. We illustrate the formalism by simple examples of possible applications of the calculus of intensities of processes to decision making and economics. We show that in special cases the classical (Kolmogorov) probabilistic model can give the same results as the quantum rule of summation of probabilities.

Abstract: We examine two quite different threads in Pitowsky’s approach to the measurement problem that are sometimes associated with his writings. One thread is an attempt to understand quantum mechanics as a probability theory of physical reality. This thread appears in almost all of Pitowsky’s papers (see for example 2003, 2007). We focus here on the ideas he developed jointly with Jeffrey Bub in their paper ‘Two Dogmas About Quantum Mechanics’ (2010) (See also: Bub (1977, 2007, 2016, 2020); Pitowsky (2003, 2007)). In this paper they propose an interpretation in which the quantum probabilities are objective chances determined by the physics of a genuinely indeterministic universe. The other thread is sometimes associated with Pitowsky’s earlier writings on quantum mechanics as a Bayesian theory of quantum probability (Pitowsky 2003) in which the quantum state seems to be a credence function tracking the experience of agents betting on the outcomes of measurements. An extreme form of this thread is the so-called Bayesian approach to quantum mechanics. We argue that in both threads the measurement problem is solved by implicitly adding structure to Hilbert space. In the Bub-Pitowsky approach we show that the claim that decoherence gives rise to an effective Boolean probability space requires adding structure to Hilbert space. With respect to the Bayesian approach to quantum mechanics, we show that it too requires adding structure to Hilbert space, and (moreover) it leads to an extreme form of idealism.

Abstract: This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. The theory is then illustrated with an experiment that exploits these ideas. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language, namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the free-forgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the "fixed points" of a morphism with its adjoint encode interesting information.

Abstract: The interference pattern in electron double-slit diffraction is a hallmark of quantum mechanics. A long standing question for stochastic electrodynamics (SED) is whether or not it is capable of reproducing such effects, as interference is a manifestation of quantum coherence. In this study, we use excited harmonic oscillators to directly test this quantum feature in SED. We use two counter-propagating dichromatic laser pulses to promote a ground-state harmonic oscillator to a squeezed Schr\"{o}dinger cat state. Upon recombination of the two well-separated wavepackets, an interference pattern emerges in the quantum probability distribution but is absent in the SED probability distribution. We thus give a counterexample that rejects SED as a valid alternative to quantum mechanics.

Abstract: According to a standard view, quantum mechanics (QM) is a contextual theory and quantum probability does not satisfy Kolmogorov’s axioms. We show, by considering the macroscopic contexts associated with measurement procedures and the microscopic contexts (μ-contexts) underlying them, that one can interpret quantum probability as epistemic, despite its non-Kolmogorovian structure. To attain this result we introduce a predicate language L(x), a classical probability measure on it and a family of classical probability measures on sets of μ-contexts, each element of the family corresponding to a (macroscopic) measurement procedure. By using only Kolmogorovian probability measures we can thus define mean conditional probabilities on the set of properties of any quantum system that admit an epistemic interpretation but are not bound to satisfy Kolmogorov’s axioms. The generalized probability measures associated with states in QM can then be seen as special cases of these mean probabilities, which explains how they can be non-classical and provides them with an epistemic interpretation. Moreover, the distinction between compatible and incompatible properties is explained in a natural way, and purely theoretical classical conditional probabilities coexist with empirically testable quantum conditional probabilities.

Abstract: The energy dependence of quantum complex-forming reaction probabilities is well known to involve sharp fluctuations, but little seems to be known about their amplitudes. We develop here, for triatomic reactions, an analytical approach of their statistical distribution. This approach shows that the fluctuation amplitudes depend essentially on the number of available quantum states in the reagent and product channels. Moreover, the more numerous the product states, the more efficiently the fluctuations of their populations compensate each other when they add up to give the reaction probability. The predictions of our approach appear to be in good quantitative agreement with quantum scattering calculations for the prototypical reaction H+ + H2.

Abstract: This paper is a new step towards getting rid of nonlocality from quantum physics. This is an attempt to structure the nonlocality mess. "Quantum nonlocality" is Janus faced. One its face is projection (Einstein-Luders) nonlocality and another Bell nonlocality. The first one is genuine quantum nonlocality, the second one is subquantum nonlocality. Recently it was shown that Bell "nonlocality" is a simple consequence of the complementarity principle. We now show that projection nonlocality has no connection with physical space. Projection state update is generalization of the well known operation of probability update used in classical inference. We elevate the role of interpretations of a quantum state. By using the individual (physical) interpretation, one can really get the illusion of a spooky action at a distance resulting from Luders' state update. The statistical interpretation combined with treating the quantum formalism as machinery for update of probability is known as the Vaxjo interpretation. Here one follows the standard scheme of probability update adjusted to the quantum calculus of probability. The latter is based on operating with states represented by vectors (or density operators). We present in parallel classical and quantum probability updates. From this presentation, it is clear that both classical and quantum "faster-than-light change of statistical correlation" take place in mental and not physical space.

Abstract: This paper gives a systematic account of various metrics on probability distributions (states) and on predicates. These metrics are described in a uniform manner using the validity relation between states and predicates. The standard adjunction between convex sets (of states) and effect modules (of predicates) is restricted to convex complete metric spaces and directed complete effect modules. This adjunction is used in two state-and-effect triangles, for classical (discrete) probability and for quantum probability.

Abstract: We present a unified field theory of wave and particle in quantum mechanics. This emerges from an investigation of three weaknesses in the de Broglie–Bohm theory: its reliance on the quantum probability formula to justify the particle-guidance equation; its insouciance regarding the absence of reciprocal action of the particle on the guiding wavefunction; and its lack of a unified model to represent its inseparable components. Following the author’s previous work, these problems are examined within an analytical framework by requiring that the wave–particle composite exhibits no observable differences with a quantum system. This scheme is implemented by appealing to symmetries (global gauge and spacetime translations) and imposing equality of the corresponding conserved Noether densities (matter, energy, and momentum) with their Schrodinger counterparts. In conjunction with the condition of time-reversal covariance, this implies the de Broglie–Bohm law for the particle where the quantum potential mediates the wave–particle interaction (we also show how the time-reversal assumption may be replaced by a statistical condition). The method clarifies the nature of the composite’s mass, and its energy and momentum conservation laws. Our principal result is the unification of the Schrodinger equation and the de Broglie–Bohm law in a single inhomogeneous equation whose solution amalgamates the wavefunction and a singular soliton model of the particle in a unified spacetime field. The wavefunction suffers no reaction from the particle since it is the homogeneous part of the unified field to whose source the particle contributes via the quantum potential. The theory is extended to many-body systems. We review de Broglie’s objections to the pilot-wave theory and suggest that our field-theoretic description provides a realization of his hitherto unfulfilled ‘double solution’ programme. A revised set of postulates for the de Broglie–Bohm theory is proposed in which the unified field is taken as the basic descriptive element of a physical system.

Abstract: It is demonstrated that the use of Kolmogorov’s probability theory to describe results of quantum probability for EPRB (Einstein-Podolsky-Rosen-Bohm) experiments requires extreme care when different subsets of measurement outcomes are considered. J. S. Bell and his followers have committed critical inaccuracies related to spin-gauge and probability measures of such subsets, because they use exclusively a single probability space for all data sets and sub-sets of data. It is also shown that Bell and followers use far too stringent epistemological requirements for the consequences of space-like separation. Their requirements reach way beyond Einstein’s separation principle and cannot be met by the major existing physical theories including relativity and even classical mechanics. For example, the independent free will does not empower the experimenters to choose multiple independent spin-gauges in the two EPRB wings. It is demonstrated that the suggestion of instantaneous influences at a distance (supposedly “derived” from experiments with entangled quantum entities) is a consequence of said inaccuracies and takes back rank as soon as the Kolmogorov probability measures are related to a consistent global spin-gauge and permitted to be different for different data subsets: Using statistical interpretations and different probability spaces for certain subsets of outcomes instead of probability amplitudes related to single quantum entities, permits physical explanations without a violation of Einstein’s separation principle.

Abstract: The existing computational cognitive models in cyber security risk management have major limitations in the analysis of cyber security behaviors. To address this issue, we introduce Hilbert space and quantum cognition to cyber security risk management. We compare some key axioms and definitions of classical cognition and quantum cognition. We provide examples on how some unique principles of Hilbert space and quantum cognition such as compatibility can help with the event representation, and capturing cognitive biases that are not possible with classical probability. We shed light on how the mathematical formalism of quantum probability can model the observed deviations from the classicality in human reasoning. We also shed light on how quantum cognition can contribute to the science and practice of cyber security. We argue that Hilbert space and quantum cognition can reconcile violations of classical probability theory in cyber security risk management with a formal theory, and examine whether it is possible to express formally some of the key heuristics in cyber security behaviors.

Abstract: We employ some techniques involving projections in a von Neumann algebra to establish some maximal inequalities such as the strong and weak symmetrization, Levy, Levy-Skorohod, and Ottaviani inequalities in the realm of the quantum probability spaces.

Abstract: The current paper is a brief review of the emerging field of quantum-like modelling in game theory. This paper aims to explore several quantum games, which are superior compared to their classical counterparts, which means either they give rise to superior Nash equilibria or they make the game fairer. For example, quantum Prisoners Dilemma generates Pareto superior outcomes as compared to defection outcome in the famous classical case. Again, a quantum-like version of cards game can make the game fairer, increasing the chance of winning of players who are disadvantaged in the classical case. This paper explores all the virtues of simple quantum games, also highlighting some findings of the authors as regards Prisoners Dilemma game.,As this is a general review paper, the authors have not demonstrated any specific mathematical method, rather explored the well-known quantum probability framework, used for designing quantum games. They have a short appendix which explores basic structure of Hilbert space representation of human decision-making.,Along with the review of the extant literature, the authors have also highlighted some new findings for quantum Prisoners Dilemma game. Specifically, they have shown in the earlier studies (which are referred to here) that a pure quantum entanglement set up is not needed for designing better games, even a weaker condition, which is classical entanglement is sufficient for producing Pareto improved outcomes.,Theoretical research, with findings and implications for future game designs, it has been argued that it is not always needed to have true quantum entanglement for superior Nash Equilibria.,The main purpose here is to raise awareness mainly in the social science community about the possible applications of quantum-like game theory paradigm. The findings related to Prisoners Dilemma game are, however, original.

Abstract: Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.

Abstract: The Geometrization of Quantum Mechanics proposed in this work is based on the postulate that the quantum probability density can $curve$ the classical spacetime. It is shown that the gravitational field produced by $smearing$ a point-mass $M_o$ at $ r = 0$ throughout all of space (in an spherically symmetric fashion) can be interpreted as the gravitational field generated by a self-gravitating anisotropic fluid droplet of mass density $ 4 \pi M_o r^2 \varphi^* ( r ) \varphi ( r ) $ and which is sourced by the $probability$ $cloud$ (associated with a spinless point-particle of mass $ M_o$) $permeating$ a $3$-spatial domain region $ {\cal D}_3 = \int 4 \pi r^2 dr $ at any time $ t $. Classically one may smear the point mass in any way we wish leading to arbitrary density configurations $ \rho (r ) $. However, Quantum Mechanically this is $not$ the case because the radial mass configuration $ M (r) $ must obey a key third order nonlinear differential equation (nonlinear extension of the Klein-Gordon equation) displayed in this work and which is the static spherically symmetric relativistic analog of the Newton-Schr\"{o}dinger equation. We conclude by extending our proposal to the Lagrange-Finsler and Hamilton-Cartan geometry of (co) tangent spaces and involving the relativistic version of Bohm's Quantum Potential. By further postulating that the quasi-probability Wigner distribution $W(x,p)$ $curves$ phase spaces, and by encompassing the Finsler-like geometry of the cotangent-bundle with phase space quantum mechanics, one can naturally incorporate the $noncommutative$ and non-local Moyal star product (there are also non-associative star products as well). To conclude, Phase Space is the arena where to implement the space-time-matter unification program. It is our belief this is the right platform where the quantization $of$ spacetime and the quantization $in$ spacetime will coalesce.

Abstract: A concept of quantum computing is proposed which naturally incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultaneously characterized by a quantum probability and a membership function. Along with the quantum amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum probability only, while the membership function can be computed through the membership amplitudes according to a chosen model. Two different versions are given here: the "product" obscure qubit in which the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations can be performed independently (i.e. quantum computation alongside truth). The measurement and entanglement of obscure qubits are briefly described.

Abstract: Violations of Bell inequalities in classical optics have been demonstrated in terms of field mean intensities and correlations, however, the quantum meaning of violations point to statistics and probabilities. We present a violation of Bell inequalities for classical fields in terms of probabilities, where we convert classical-field intensities into probabilities via the standard photon-counting equation. We find violation for both, entangled and separable field states. We conclude that any obtained quantum effect might be fully ascribed to the quantum nature of the detector rather than the field itself. Finally, we develop a new Bell-like criterion which is satisfied by factorized states and it is not by the entangled state.

Abstract: Itamar Pitowsky contends that quantum states are derived entities, bookkeeping devices for quantum probabilities, which he understands to reflect the odds rational agents would accept on the outcomes of quantum gambles. On his view, quantum probability is subjective, and so are quantum states. We disagree. We take quantum states, and the probabilities they encode, to be objective matters of physics. Our disagreement has both technical and conceptual aspects. We advocate an interpretation of Gleason’s theorem and its generalizations more nuanced—and less directly supportive of subjectivism—than Itamar’s. And we contend that taking quantum states to be physical makes available explanatory resources unavailable to subjectivists, explanatory resources that help make sense of quantum state preparation.

Abstract: This doctoral consortium presents an overview of my anticipated PhD dissertation which focuses on employing quantum Bayesian networks for social learning. The project, mainly, aims to expand the use of current quantum probabilistic models in human decision-making from two agents to multi-agent systems. First, I cultivate the classical Bayesian networks which are used to understand information diffusion through human interaction on online social networks (OSNs) by taking into account the relevance of multitude of social, psychological, behavioral and cognitive factors influencing the process of information transmission. Since quantum like models require quantum probability amplitudes, the complexity will be exponentially increased with increasing uncertainty in the complex system. Therefore, the research will be followed by a study on optimization of heuristics. Here, I suggest to use an belief entropy based heuristic approach. This research is an interdisciplinary research which is related with the branches of complex systems, quantum physics, network science, information theory, cognitive science and mathematics. Therefore, findings can contribute significantly to the areas related mainly with social learning behavior of people, and also to the aforementioned branches of complex systems. In addition, understanding the interactions in complex systems might be more viable via the findings of this research since probabilistic approaches are not only used for predictive purposes but also for explanatory aims.

Abstract: Quantum bits can be isolated to perform useful information-theoretic tasks, even though physical systems are fundamentally described by very high-dimensional operator algebras. This is because qubits can be consistently embedded into higher-dimensional Hilbert spaces. A similar embedding of classical probability distributions into quantum theory enables the emergence of classical physics via decoherence. Here, we ask which other probabilistic models can similarly be embedded into finite-dimensional quantum theory. We show that the embeddable models are exactly those that correspond to the Euclidean special Jordan algebras: quantum theory over the reals, the complex numbers, or the quaternions, and "spin factors" (qubits with more than three degrees of freedom), and direct sums thereof. Among those, only classical and standard quantum theory with superselection rules can arise from a physical decoherence map. Our results have significant consequences for some experimental tests of quantum theory, by clarifying how they could (or could not) falsify it. Furthermore, they imply that all unrestricted non-classical models must be contextual.