# 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

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