  We discuss that there is a crucial contradiction within quantum mechanics. We
derive a proposition concerning a quantum expectation value under the
assumption of the existence of the directions in a spin-1/2 system. The quantum
predictions within the formalism of von Neumann's projective measurement cannot
coexist with the proposition concerning the existence of the directions.
Therefore, we have to give up either the existence of the directions or the
formalism of von Neumann's projective measurement. Hence there is a crucial
contradiction within the Hilbert space formalism of the quantum theory. This
implies that there is no axiomatic system for the quantum theory. This also
reveals that we need new physical theories in order to explain the handing of
raw experimental data. We discuss that this crucial contradiction makes the
quantum-theoretical formulation of Deutsch's algorithm questionable.
