Can Von Neumann’s Theory Meet Quantum Computation?

Recently, it is shown that there is a crucial contradiction within von Neumann’s theory [K. Nagata and T. Nakamura, Int. J. Theor. Phys. 49, 162 (2010)]. We derive a proposition concerning a quantum expected 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 von Neumann’s theory. We discuss that this crucial contradiction makes the theoretical formulation of Deutsch’s algorithm questionable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. Our discussion, here, improves previously published argumentations very much.