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Quantum Theory from 5 reasonable axioms
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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.
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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)K = K(N)
3.
Subspaces: If state of a system AA ∈Mdim\in M_{dim} subspace   ⟹  dim(A)=M\implies dim(A) = M
4.
Composite systems: N=NaNb and K=KaKbN = 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