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Journal of Applied Mathematics and Physics

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

Koji Nagata, Tadao Nakamura

Published Jan 1, 2015• 0 citations
Publisher

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 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.

DOI
10.4236/jamp.2015.37108

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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 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.
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 NNN spin-1/2 systems (1≤N<+∞1 \leq N < +\infty1≤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 (ℏ\hbarℏ) to a new constant (ℏ/2\hbar/\sqrt{2}ℏ/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.
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.
We discuss that there is a crucial contradiction within quantum mechanics. We derive a proposition concerning a quantum expecta-tion 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. PACS: 03.67.Lx, 03.65.Ta, 03.65.Ca 1
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.
Ensembles of quantum systems, such as liquid state NMR, have been proposed as possibilities for implementing quantum algorithms. For such implementations, projective measurement outcomes are replaced by expectation values. Although these are apparently deterministic, we point out that real implementations involve ensembles of nite size, giving probabilistic approximations to expectation values. The performance of such algorithms must be compared to their classical probabilistic counterparts. We discuss proposed scalable ensemble versions of the Deutsch-Jozsa algorithm in this context and show that their performance is worse than the classical probabilistic algorithm. Appears in Phys. Rev. A 68, 052301 (2003) \\Standard " Quantum Computing Implementations on a single quantum system amenable to projective measurements. L-qubit quantum system Pure state preparation Apply algorithm unitary Projective measurement on L0 qubit subsystem Classical computation j ii f : = Ûalg j ii x 2 f0; : : : ; 2L 0 1g with probability xj f 2 Possible solution I For useful algorithms the probability of returning x corresponding to trivial or incorrect solutions is small. I Typically a decomposition Ûalg = Ûm: : : Û1 into simple unitary operations facilitates algorithm implementation. 1 Expectation Value Quantum Computing Implementations using an ensemble of identical, non-interacting quantum systems. Ensemble of L-qubit quantum systems Mixed state preparation Apply algorithm unitary Expectation value (EV) measurements on L0 qubit subsystem