  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.
