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Quantum bits can be isolated to perform useful information-theoretic tasks,even though physical systems are fundamentally described by veryhigh-dimensional operator algebras. This is because qubits can be consistentlyembedded into higher-dimensional Hilbert spaces. A similar embedding ofclassical probability distributions into quantum theory enables the emergenceof classical physics via decoherence. Here, we ask which other probabilisticmodels can similarly be embedded into finite-dimensional quantum theory. Weshow that the embeddable models are exactly those that correspond to theEuclidean special Jordan algebras: quantum theory over the reals, the complexnumbers, or the quaternions, and "spin factors" (qubits with more than threedegrees of freedom), and direct sums thereof. Among those, only classical andstandard quantum theory with superselection rules can arise from a physicaldecoherence map. Our results have significant consequences for someexperimental tests of quantum theory, by clarifying how they could (or couldnot) falsify it. Furthermore, they imply that all unrestricted non-classicalmodels must be contextual.

Characterization of the probabilistic models that can be embedded in quantum theory