  Conventional quantum mechanics with a complex Hilbert space and the Born Rule
is derived from five axioms describing properties of probability distributions
for the outcome of measurements. Axioms I,II,III are common to quantum
mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first
noted by Turing and von Neumann, in which the increase in entropy resulting
from a measurement is reduced by a suitable intermediate measurement. This is
shown to be impossible for local hidden variable theories. Axiom IV, together
with the first three, almost suffice to deduce the conventional rules but allow
some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom
V recognizes a property of the distribution of outcomes of random measurements
on qubits which holds only in the complex Hilbert space model. It is then shown
that the five axioms also imply the conventional rules for all dimensions.
