Quantum Theory from 5 reasonable axioms

State

Mathematical object that can be used to deter-mine the probability associated with the out-comes of any measurement that may be per-formed on a system prepared by the given preparation.

However, we do not need to measure all possible probability measurements to determine the state of a system.

• K (degrees of freedom): minimum number of probability measurements required to determine the state of a system, i.e., number of real parameters re-quired to specify the state.
• N (dimension): maximum number of states that can be reliably distinguished from one another in a single shot measurement.

Axioms

1. Probabilities
2. Simplicity: $K = K(N)$
3. Subspaces: If state of a system $A$ $\in M_{dim}$ subspace $\implies dim(A) = M$
4. Composite systems: $N = N_aN_b \text{ and } K = K_aK_b$
5. Continuity: $\exists$ a continuous reversible transformation on a system between anytwo pure states of that system