Von Neumann's Postulate Research Articles

Quantum Theory is an Information Theory

In this paper we review the general framework of operational probabilistic theories (OPT), along with the six axioms from which quantum theory can be derived. We argue that the OPT framework along with a relaxed version of five of the axioms, define a general information theory. We close the paper with considerations about the role of the observer in an OPT, and the interpretation of the von Neumann postulate and the Schrodinger-cat paradox.

On the Logical Origins of Quantum Mechanics Demonstrated By Using Clifford Algebra

Recently we have given proof of two theorems characterizing the Clifford algebra. By using such two theorems we have reformulated the well known von Neumann postulate on quantum measurements giving evidence of the algebraic manner in which quantum wave function collapse of quantum mechanics happens. In the present paper we introduce logic in Clifford algebra interpreting its idempotents as logical statements. Using the previously mentioned theorems we demonstrate that the two basic foundations of quantum mechanics, as the indeterminism and the quantum interference, do not arise from physics itself but from logic. We advance the principles that there are levels of our reality in which we lose our possibility of unconditionally define the truth. At this level of reality we cannot separate matter per se from the basic foundations of the logic that we use to describe it. This logical relativism does not characterize classical mechanics but quantum physics. According to Y. F. Orlov, at quantum level the truths of logical statements about dynamic variables become dynamic variables themselves.

On the Possibility That We Think in a Quantum Probabilistic Manner

My discussion is articulated under the neurological as well as the psychological profile. I insist in particular on the view that mental events arise in analogy with quantum probability fields. I review some results obtained on quantum cognition discussing in detail those that we obtained on quantum interference in mental states during perception-cognition in ambiguous figures. Frequently, I use the approach to quantum mechanics by Clifford algebra. I insist in particular on two recent results. The first is the justification that I obtain of the von Neumann postulate on quantum measurement and the second relates my Clifford demonstration on the logical origins of quantum mechanics and thus on the arising feature that quantum mechanics relates conceptual entities. The whole discussion aims me to support the conclusion that we think in a quantum probabilistic manner.

A Reformulation of von Neumann’s Postulates on Quantum Measurement by Using Two Theorems in Clifford Algebra

According to a procedure previously introduced from Y. Ilamed and N. Salingaros, we start giving proof of two existing Clifford algebras, the Si that has isomorphism with that one of Pauli matrices and the Ni,±1 where Ni stands for the dihedral Clifford algebra. The salient feature is that we show that the Ni,±1 may be obtained from the Si algebra when we attribute a numerical value (+1 or −1) to one of the basic elements (e1,e2,e3) of the Si. We utilize such result to advance a criterium under which the Si algebra has as counterpart the description of quantum systems that in standard quantum mechanics are considered in absence of observation and quantum measurement while the Ni,±1 attend when a quantum measurement is performed on such system with advent of wave function collapse. The physical content of the criterium is that the quantum measurement with wave function collapse induces the passage in the considered quantum system from the Si to Ni,+1 or to the Ni,−1 algebras, where each algebra has of course its proper rules of commutation. After a proper discussion on the difference between decoherence and wave function collapse, we re-examine the von Neumann postulate on quantum measurement, and we give a proper justification of such postulate by using the Si algebra. Soon after we study some applications of the above mentioned criterium to some cases of interest in standard quantum mechanics, analyzing in particular a two state quantum system, the case of time dependent interaction of such system with a measuring apparatus and finally the case of a quantum system plus measuring apparatus developed at the order n=4 of the considered Clifford algebras and of the corresponding density matrix in standard quantum mechanics. In each of such cases examined, we find that the passage from the algebra Si to Ni,±1, considered during the quantum measurement of the system, actually describes the collapse of the wave function. Therefore we conclude that the actual quantum measurement has as counterpart in the Clifford algebraic description, the passage from the Si to the Ni,±1 Clifford algebras, reaching in this manner the objective to reformulate von Neumann postulate on quantum measurement and proposing a self-consistent formulation of quantum theory.

VON NEUMANN AND LUDERS POSTULATES AND QUANTUM INFORMATION THEORY

This note is devoted to some foundational aspects of quantum mechanics (QM) related to quantum information (QI), especially quantum teleportation and "one way quantum computing." We emphasize the role of the projection postulate (determining post-measurement states) in QI and the difference between its Lüders and von Neumann versions. As is well-known, these postulates differ in the case of observables with degenerate spectra. Such observables are important in operations with entangled states: any measurement on one subsystem is represented by an observable with degenerate spectrum in the Hilbert space of a composite system. Some QI schemes (e.g. quantum teleportation and "one way quantum computing") are based on the use of Lüders postulate. The formal application of von Neumann postulate can block these QI schemes. In this note, we present a list of natural conditions under which von Neumann's description of measurements via refinement implies Lüders projection postulate.

EPR "PARADOX", PROJECTION POSTULATE, TIME SYNCHRONIZATION "NONLOCALITY"

We show that the projection postulate plays a crucial role in the discussion on the so called "quantum nonlocality," in particular in the EPR-argument. We stress that the original von Neumann projection postulate was crucially modified by extending it to observables with degenerate spectra (the Lüders postulate) and we show that this modification is highly questionable from a physical point of view, and it is the real source of "quantum nonlocality." The use of the original von Neumann postulate eliminates this problem: instead of "action at the distance"-nonlocality, we obtain a classical measurement nonlocality. The latter is related to the synchronization of two measurements (on the two parts of a composite system). It seems that EPR did mistake in their 1935-paper: if one uses correctly von Neumann projection postulate, no "elements of reality" can be assigned to entangled systems. Our analysis of the EPR and projection postulate makes clearer Bohr's considerations in his reply to Einstein.

The role of von Neumann and Lüders postulates in the Einstein, Podolsky, and Rosen considerations: Comparing measurements with degenerate and nondegenerate spectra

We show that the projection postulate plays a crucial role in the discussion on the so-called quantum nonlocality, in particular, in the Einstein, Podolsky, and Rosen argument. We stress that the original von Neumann projection postulate was crucially modified by extending it to observables with degenerate spectra (the Lüders postulate) and we show that this modification is highly questionable from a physical point of view and is the real source of quantum nonlocality. The use of the original von Neumann postulate eliminates this problem: instead of an action at a distance nonlocality we obtain a classical measurement nonlocality, which is related to the synchronization of two measurements (on the two parts of a composite system).

Einstein-Podolsky-Rosen paradox, Bell's inequality, and the projection postulate

Our aim is to understand the role of implicit assumptions which has been used by Einstein, Podolsky, and Rosen (EPR) in their famous article [Phys. Rev., 47, 777 (1935)] devoted to the so-called EPR paradox. We found that the projection postulate plays a crucial role in the EPR argument. It seems that EPR made a mistake in this paper — the projection postulate was applied not in its original form (as it has been formulated in von Neumann's book [Mathematical Foundations of Quantum mechanics, Princeton University Press (1955)] but in the form which was later formalized as Lüders' postulate [Ann. Phys., Lpz. 8, 322 (1951)]. Von Neumann's postulate was crucially modified by extending it to observables with degenerate spectra. This modification is the real source of “quantum nonlocality.” The use of the original von Neumann postulate eliminates this problem — instead of (an action at a distance)-nonlocality, we obtain a classical measurement nonlocality, which is related to the synchronization of two measurements (produced on the two parts of a composite system). If one uses correctly von Neumann's projection postulate, no “elements of reality” can be assigned to entangled systems.

Asymptotic von Neumann measurement strategy for solid-state qubits

A measurement on a macroscopic quantum system does in general not lead to a projection of the wavefunction in the basis of the detector as predicted by von-Neumann's postulate. Hence, it is a question of fundametal interest, how the preferred basis onto which the state is projected is selected out of the macroscopic Hilbert space of the system. Detector-dominated von-Neumann measurements are also desirable for both quantum computation and verification of quantum mechanics on a macroscopic scale. The connection of these questions to the predictions of the spin-boson modelis outlined. I propose a measurement strategy, which uses the entanglement of the qubit with a weakly damped harmonic oscillator. It is shown, that the degree of entanglement controls the degree of renormalization of the qubit and identify, that this is equivalent to the degree to which the measurement is detector-dominated. This measurement very rapidly decoheres the initial state, but the thermalization is slow. The implementation in Josephson quantum bits is described and it is shown that this strategy also has practical advantages for the experimental implementation.

Optical quantum nondemolition measurements and the Copenhagen interpretation.

We start from the reasonable assumption that a quantum measurement can be properly described only when the measurement apparatus itself is also treated quantum mechanically. Using idealized optical measurements as an example and introducing a small loss into the measurement apparatus, we show that the combined density matrix of the system and apparatus decoheres in a manner that the outcome of the measurement can be described classically. The loss can be arbitrarily small when the classical limit of the measurement apparatus is taken, i.e., the gain is made to approach infinity. Further, we show that the von Neumann postulate of the collapse of the wave function of the measured system into an eigenstate of the measured observable can be deduced from the analysis of two quantum nondemolition experiments in cascade and need not be introduced as a separate postulate. \textcopyright{} 1996 The American Physical Society.