  Quantum theory is usually formulated in terms of abstract mathematical
postulates, involving Hilbert spaces, state vectors, and unitary operators. In
this work, we show that the full formalism of quantum theory can instead be
derived from five simple physical requirements, based on elementary assumptions
about preparation, transformations and measurements. This is more similar to
the usual formulation of special relativity, where two simple physical
requirements -- the principles of relativity and light speed invariance -- are
used to derive the mathematical structure of Minkowski space-time. Our
derivation provides insights into the physical origin of the structure of
quantum state spaces (including a group-theoretic explanation of the Bloch ball
and its three-dimensionality), and it suggests several natural possibilities to
construct consistent modifications of quantum theory.
