Summary: Lucien Hardy laid out a program for deducing state spaces from reasonable axioms, and thereby derived classical and quantum physics (QP). In a level \(N\) state space, each experiment results in one of \(N\) possible outcome states. The probability transition form, \(\langle S, T\rangle\), gives the probability that state \(S\) will transition to state \(T\). We drop Hardy’s composite systems axiom and simplicity axiom, and add a new axiom which asserts that any information, that distinguishes the states, must be derivable from the transition form. This axiom is not so much an assumption, but an acknowledgement of the central role of the transition form. Notably, the new set of axioms make no mention of mixed states or composite systems. Restricting to finite dimensions, we are lead to the three well-known sequences of projective spaces, \(\mathbb{P}^n\), \(n \geq 1\), for the reals, complexes and quaternions, with the standard probability transition pairing. The complex case is, of course, QP. Our formulation of state space leads naturally to its standard embedding as the rank 1 trace 1 Hermitian matrices. These spaces are all isotropic and compact-properties that are not assumed, but are consequences of the axioms: The above-mentioned information axiom, State space is connected, There is a Lie group (not assumed compact) acting transitively on state space, The measurements span a finite dimensional linear subspace of the real-valued functions on state space, The measurements separate the states, Every state is a level 1 state space, Level \(N\) state space has state subspaces of all levels less than \(N\). Every state space is (isomorphic to) a state subspace of a higher level state space, and The restriction of a state to a state subspace is a scale multiple of a state.