Von Neumann’s Theory, Projective Measurement, and Quantum Computation

Koji Nagata¹, Tadao Nakamura²

¹Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea.
²Department of Information and Computer Science, Keio University, Yokohama, Japan.

DOI: 10.4236/jamp.2015.37108                

Abstract

We discuss the fact that there is a crucial contradiction within Von Neumann’s theory. We derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in N spin-1/2 systems (1 ≤ N < +∞). This assumption intuitively depictures our physical world. However, the quantum predictions within the formalism of Von Neumann’s projective measurement violate the proposition with a magnitude that grows exponentially with the number of particles. We have to give up either the existence of the directions or the formalism of Von Neumann’s projective measurement. Therefore, Von Neumann’s theory cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm relies on Von Neumann’s theory. We investigate whether Von Neumann’s theory meets the Deutsch-Jozsa algorithm. We discuss the fact that the crucial contradiction makes the quantum-theoretical formulation of Deutsch-Jozsa algorithm questionable. Further, we discuss the fact that projective measurement theory does not meet an easy detector model for a single Pauli observable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. We propose a solution of the problem. Our solution is equivalent to changing Planck’s constant to a new constant . It may be said that a new type of the quantum theory early approaches Newton’s theory in the macroscopic scale than the old quantum theory does. We discuss how our solution is used in an implementation of Deutsch’s algorithm.

Keywords

Von Neumann’s Theory, Projective Measurement, and Quantum Computation

Conflicts of Interest

The authors declare no conflicts of interest.

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