Von Neumann’s Theory of Quantum Measurement

P. A. Moldauer

Kenjiro Yanagi

Rathindra Nath Sen

Von Neumann's theory of measurement in Quantum Mechanics was spelled out in the last chapter of his book, which was published in 1932. This book was highly mathematical for its time, and in 1939 London and Bauer provided a simplified account of the measurement theory part of it (London and Bauer, 1939). Von Neumann died in 1957. Thirty years after the publication of von Neumann's book, Wigner published a review containing his views on the shortcomings of von Neumann's theory, but omitting any discussion of its mathematical core, namely von Neumann's analysis of composite systems (Wigner, 1963). He also published a set of lecture notes entitled Interpretation of Quantum Mechanics (Wigner, 1983) in which some of his concerns were spelled out in greater detail, and an article addressed ‘to an audience of non-physicists’ (Wigner, 1964). Wigner's own contributions to measurement theory were discussed by Shimony in a talk at the Wigner centennial conference (Shimony, 2002). The English translation of von Neumann's book was published in 1955.1 The account that follows is based on these sources. We shall assume that the reader is acquainted with notions such as wave function collapse and the Heisenberg cut, but we shall not assume familiarity with the technicalities of von Neumann's theory. This chapter is organized accordingly. Section 8.1 explains what we mean by the term von Neumann's measurement theory and gives an overview of the subject. It is followed by Sections 8.2 and 8.3, in which the theory is spelled out in detail. Section 8.4 recounts Wigner's reservations.

Pier A. Mello

Koji Nagata, Tadao Nakamura

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 ($1 \leq N < +\infty$). 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 ($\hbar$) to a new constant ($\hbar/\sqrt{2}$). 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.

Howard M. Wiseman, Gerard J Milburn