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.

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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.

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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