  Recently, [arXiv:0810.3134] is accepted and published. We derive an
inequality with two settings as tests for the existence of the Bloch sphere in
a spin-1/2 system. The probability theory of measurement outcome within the
formalism of von Neumann projective measurement violates the inequality.
Namely, we have to give up the existence of the Bloch sphere. Or, we have to
give up the probability theory of measurement outcome within the formalism of
von Neumann projective measurement. Hence it turns out that there is a
contradiction in the Hilbert space formalism of the quantum theory, viz., there
is no axiomatic system for the theory.
