Von Neumann Projection Postulate Research Articles

Gravitational quantum collapse in dilute systems

Penrose has suggested that large fluctuations of the gravitational energy of quantum systems, resulting from fluctuations of its density in space, may induce a quantum collapse mechanism [1], but he did not propose a precise dynamics for this process. We use the GBC (Gravitational Bohmian Collapse) model [2], which provides such a dynamics. The effects of collapse in dilute quantum systems are investigated, both in ordinary 3D space and in configuration space. We first discuss how a single result appears during a quantum measurement. The GBC model predicts a continuous but very fast evolution of the state vector that, at the end of the measurement, reproduces the von Neumann projection postulate. This ensures that the model remains compatible with the relativistic nosignaling constraint. In the absence of any measurement, we study the spontaneous effects of the GBC process, which depend on the quantum correlation function of observables with the spatial density operator. If the selected observable is the local current of the density fluid, we show that the collapse term leads to modifications of the Newton force, in a scalar or tensor form.

A classical counting process and non-Wiener stochastic differential equations in quantum optics

The von Neumann projection postulate under the registration of a spontaneously emitted photon is described through the stochastic counting process as a non-Wiener linear stochastic differential equation for a non-normalized atomic state vector.

Time correlators from deferred measurements

Repeated measurements as typically occurring in two- or multi-time correlators rely on von Neumann's projection postulate, telling how to restart the system after an intermediate measurement. We invoke the principle of deferred measurement to describe an alternative procedure where co-evolving quantum memories extract system information through entanglement, combined with a final readout of the memories described by Born's rule. The new approach to repeated quantum measurements respects the unitary evolution of quantum mechanics during intermediate times, unifies the treatment of strong and weak measurements, and reproduces the projected and (anti-) symmetrized correlators in the two limits. As an illustration, we apply our formalism to the calculation of the electron charge correlator in a mesoscopic physics setting, where single electron pulses assume the role of flying memory qubits. We propose an experimental setup which reduces the measurement of the time correlator to the measurement of currents and noise, exploiting the (pulsed) injection of electrons to cope with the challenge of performing short-time measurements.

Measurement problem: from De Broglie to theory of classical random fields interacting with threshold detectors

The quantum measurement problem as was formulated by von Neumann in 1933 can be solved by going beyond the operational quantum formalism. In our "prequantum model" quantum systems are symbolic representations of classical random fields. The Schrödinger's dynamics is a special form of the linear dynamics of classical fields. Measurements are described as interactions of classical fields with detectors. Discontinuity, the "collapse of the wave function", has the trivial origin: usage of threshold type detectors. The von Neumann projection postulate can be interpreted as the formal mathematical encoding of the absence of coincidence detection in measurement on a single quantum system, e.g., photon's polarization measurement. Our model, prequantum classical statistical field theory (PCSFT), in combination with measurements by threshold detectors satisfies the quantum restriction on coincidence detections: the second order coherence is less than one (opposite to all known semiclassical and classical feld models). The basic rule of quantum probability, the Born's rule, is derived from properties of prequantum random felds interacting with threshold type detectors. Comparison with De Broglie's views to quantum mechanics as theory of physical waves with singularities is presented.

Two Versions of the Projection Postulate: From EPR Argument to One‐Way Quantum Computing and Teleportation

Nowadays it is practically forgotten that for observables with degenerate spectra the original von Neumann projection postulate differs crucially from the version of the projection postulate which was later formalized by Lüders. The latter (and not that due to von Neumann) plays the crucial role in the basic constructions of quantum information theory. We start this paper with the presentation of the notions related to the projection postulate. Then we remind that the argument of Einstein‐Podolsky‐Rosen against completeness of QM was based on the version of the projection postulate which is nowadays called Lüders postulate. Then we recall that all basic measurements on composite systems are represented by observables with degenerate spectra. This implies that the difference in the formulation of the projection postulate (due to von Neumann and Lüders) should be taken into account seriously in the analysis of the basic constructions of quantum information theory. This paper is a review devoted to such an analysis.

Relation between “no broadcasting” for noncommuting states and “no local broadcasting” for quantum correlations

The no-broadcasting theorem, first established by Barnum et al. [Phys. Rev. Lett. 76, 2818 (1996)], states that a set of quantum states can be broadcast if and only if it constitutes a commuting family. Quite recently, Piani et al. [Phys. Rev. Lett. 100, 090502 (2008)] showed, by using an ingenious and sophisticated method, that the correlations in a single bipartite state can be locally broadcast if and only if the state is effectively a classical one (i.e., the correlations therein are classical). In this Brief Report, under the condition of nondegenerate spectrum, we provide an alternative and significantly simpler proof of the latter result based on the original no-broadcasting theorem and the monotonicity of the quantum relative entropy. This derivation motivates us to conjecture the equivalence between these two elegant yet formally different no-broadcasting theorems and indicates a subtle and fundamental issue concerning spectral degeneracy which also lies at the heart of the conflict between the von Neumann projection postulate and the Lueders ansatz for quantum measurements. This relation not only offers operational interpretations for commutativity and classicality but also illustrates the basic significance of noncommutativity in characterizing quantumness from the informational perspective.

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.

Stochastic Schrödinger Equations as Limit of Discrete Filtering

We consider an open model possessing a Markovian quantum stochastic limit and derive the limit stochastic Schrödinger equations for the wave function conditioned on indirect observations using only the von Neumann projection postulate. We show that the diffusion (Gaussian) situation is universal as a result of the central limit theorem with the quantum jump (Poissonian) situation being an exceptional case. It is shown that, starting from the correponding limiting open systems dynamics, the theory of quantum filtering leads to the same equations, therefore establishing consistency of the quantum stochastic approach for limiting Markovian models.

Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century

A history and drama of the development of quantum theory is outlined starting from the discovery of the Plank's constant exactly 100 years ago. It is shown that before the rise of quantum mechanics 75 years ago, the quantum theory had appeared first in the form of the statistics of quantum thermal noise and quantum spontaneous jumps which have never been explained by quantum mechanics. Moreover, the only reasonable probabilistic interpretation of quantum theory put forward by Max Born was in fact in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes; some of them, related to the great inventors of quantum theory such as Einstein and Schrodinger, are reconsidered in the paper. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for the resolution of this interpretational crisis by a divorce of the algebra of dynamical generators and a subalgebra of the observables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of past observables with the potential future. This rule together with self-compatibility of measurements ensuring the consitency of histories is called the nondemolition principle. The application of these rules in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, as well as to more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a quantum probabilistic solution in the form of dynamical filtering equations to the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting from Schrodinger and Bohr. The simplest Markovian quantum stochastic model for time-continuous measurements involves a boundary-value problem in second quantization for input offer waves in one extra dimension, and a reduction of the algebra of actual observables to an Abelian subalgebra for the output waves.

Quantum Mechanics of the Composite System and Its Subsystems

We revisit the EPR problem and make clear what is a correct comprehension of its problem. When one applies the quantum mechanics correctly, it will be shown that there is no paradox. According to these lines of thought, a quantum teleportation scheme without resort to the von Neumann projection postulate is presented.

A Set of Conditions for Teleportation without Resort to von Neumann's Projection Postulate

Since the pioneering works of Feynman 1) and Deutsch 2) using elements following the laws of quantum mechanics, interest has been growing in the area of so-called quantum computation. Among other things, a quantum teleportation scheme of the type proposed by Bennett et a1. 3) is of great interest from the measurement theoretical point of view. The original teleportation scheme of an unknown state iq'», which consists of the story of Alice and Bob, however, is insufficient due to the unclear treatment of the EPR 4) problem and the assumption of von Neumann's projection postulate. At the same time, the former unclear treatment follows from the latter assumption. This teleportation scheme, based on the existence of non­ local long range correlations between EPR pairs of particles relevant to the notion of wave function collapse, necessarily leads to epistemological discussion on the nature of physical reality. Unfortunately, tentative solutions to this problem always depend on what the authors decide to define as realism. 5 ) Therefore, such philosophical arguments concerning the conceptual aspects of EPR phenomenon are beyond the realm of physics. We shall show that a quantum teleportation can be achieved within the framework of present-day quantum mechanics without resorting to von Neumann's projection postulate. Today, the naive Copenhagen interpretation is criticized for various reasons. For example, in Machida and Namiki's theory of measurement, 5),6) which does not rely upon von Neumann's projection postulate, making use of the statistical operator as the description of the quantum state and introducing the state of the apparatus system, they showed that the resolution of the difficulty concerning the wave function collapse does not consist of a demonstration of the disappearance of other branch waves, but follows from the concept of decoherence. Their theory is widely applicable, even to incomplete measurements. 5) According to these lines of thought, one of the present authors (A. M.) attempted to analyze the process of decoherence. 7) In this paper, reviewing what is a correct understanding of the EPR problem, we intend to show how we can obtain a correct teleportation scheme without resorting to von Neumann's projection postulate. A reasonable explanation of the EPR problem was given by Machida. 8) In order

A Note on Van Fraassen's Modal Interpretation of Quantum Mechanics

Although there has been some discussion in the literature of Bas van Fraassen's modal interpretation of Quantum Mechanics, it has for the most part been concentrated on difficulties that van Fraassen's viewpoint shares with those of some other authors, including Kochen, Dieks, and Healey. van Fraassen's approach has, however, some problems of its own; in this note we want to focus on what seems to us to be one of the most serious of these. The difficulty concerns immediately repeated non-disturbing measurements of the same observable on a single system. As is well known, von Neumann's Projection Postulate guarantees that such measurements will always give the same outcome; likewise, in the approaches of the “modalists” mentioned above, such ‘consilience of repeated measurements’ is in one way or another built into the formalism. By contrast, we shall argue, van Fraassen's modal interpretation neither guarantees this result nor adequately explains why it is unnecessary to do so.

Dynamical quantum Zeno effect.

A purely dynamical explanation for the quantum Zeno effect is proposed. It is argued that a quantum system undergoes a quantum-Zeno-type dynamics as a consequence of a particular type of evolution involving a series of frequent spectral decompositions. The role of quantum measurements and of the ``collapse of the wave function'' is investigated and it is clarified that, provided a final observation is performed, the dynamical quantum Zeno effect can be obtained without making use of von Neumann's projection postulate. The meaning of infinitely frequent measurements is critically discussed and it is argued that it should be regarded as a mathematical idealization, impossible to realize from a physical point of view.

What is wave-function collapse by measurement?

On the basis of the many-Hilbert-space approach to quantum measurements, we reformulate the notion of wave-function collapse as a dephasing process, without making use of von Neumann's projection postulate. A “detector” is described as an open quantum system, characterized by thermodynamical properties, and endowed with a many-Hilbert-space structure: These features enable us to provoke decoherence and to perform quantum measurements. Our approach is compared to other Copenhagen-based theories in some detail.

Probability-density-functional description of quantum photodetection processes

A concept 'probability-density functional' combined with the path integral technique is shown to provide a new framework for quantum photodetection theory. It replaces the conventional picture of unitary evolution of the total density operator followed by von Neumann's projection postulate for the case in which the effects of nonunitary state reduction and continuous measurements must be taken into account. This paper shows that an expression of the probability-density functional which incorporates both these effects provides complete quantum-statistical information about photoelectron statistics. Simple procedures are given for calculating the ensemble average and generating the functional of a quantity distributed in time.

Quantum Measurements and Stochastic Processes

A nonlinear stochastic process is presented that, for each realization and for large times, reproduces L\uder's projection postulate. The corresponding density operator undergoes a linear evolution reproducing von Neumann's projection postulate. The violation of the Bell inequality, for instance, is described with the two apparatus acting independently on the composed system.

Correlation between Measurements in Quantum Theory

An attempt to extend the postulational basis of quantum theory by introducing correlations between the results of measurements, developed in this article, leads to negative joint probabilities for otherwise meaningful sets of measured values. Since, so far as can be seen, the attempt made is the only one compatible with the theory of random variables, and is incompatible with the structure of Hilbert space, we conclude that correlations are absent. This result, though it is tantamount to a denial of von Neumann's projection postulate and the “reduction of wave packets” on measurement, is nevertheless shown to be entirely satisfactory from the physical point of view.