Why “modal” interpretations of quantum mechanics don't solve the measurement problem

According to modal interpretations of quantum mechanics, an observable Q can possess a definite value even when the quantum state is not an eigenstate of Q. In this paper, I discuss some interpretive difficulties faced by modal theorists. First, expanding upon Albert and Loewer, I identify two reasons why real-life measurements are never ideal, and I discuss why these considerations bode ill for modal interpretations. Second, I show that modal interpretations provide a less satisfactory explanation of interference effects than is provided by pilot-wave interpretations.     

Measurement: It ain't over till it's over

Elby (1993) has raised certain problems that appear to be devastating for modal interpretations of quantum mechanics, but do not arise for Bohm's pilot wave theory. Here I show that the features Elby identifies as objectionable in my version of the modal interpretation have their counterpart in Bohm's theory. To the extent that Bohm's theory works as a no collapse solution to the measurement problem - and I think it does - so does my modal interpretation.     

Measurement and quantum indeterminateness

Albert and Loewer^[1] have recently clarified their earlier objection to the interactive interpretation presented in Healey^[2]. They now charge that this interpretation fails to solve a problem of which the measurement problem is but a special case. The general problem is to reconcile quantum mechanics with theprima facie determinateness of such dynamical properties as the positions of macroscopic objects. In response I defend both the preeminent significance of determinate measurement outcomes and the claim that the models of Healey^[3] go a long way toward securing their determinateness.1. They note ironically that the proponent of this interpretation must argue that since a natural measure of the set of states whose biorthonormal expansion has equal amplitudes is zero these states should be neglected. In fact these states werenot neglected in Healey^[2] (see pp. 98-100) where I argued that even though the resulting indeterminateness in any such states would be real, it would be rapidly removed by subsequent environmental interactions.     

Modal Interpretations and Relativity

A proof is given, at a greater level of generality than previous no-go theorems, of the impossibility of formulating a modal interpretation that exhibits serious Lorentz invariance at the fundamental level. Particular attention is given to modal interpretations of the type proposed by Bub.     

量子力学若干基本问题研究的新进展

首先回顾了量子力学诠释的各种研究的思想起源.以量子测量问题为核心,通过介绍和分析最近5年完成的相关实验（如用腔QED展示的量子退相干过程实验和超冷原子which-way实验等）,系统地评述了量子力学若干基本问题研究（包括宏观物体相干性,量子测量的冯*诺伊曼链问题）的新进展,并阐明了其作用与意义.指出了量子力学基本问题的探讨已经走出纯哲学思辨的范畴,开始了大规模的实验检验,而且其研究结果对信息科学有重要的潜在应用.通过基本观念的正确阐释,对量子测量理解方面的某些误解（如超光速现象）给予一定程度的澄清.最后,通过介绍霍夫特(G.′tHooft)最近在量子力学方面的工作,展望了量子力学今后的可能发展.     

Simple Realism and Canonically Conjugate Observables in Non-Relativistic Quantum Mechanics

In this paper we list some minimal requirements for a physically natural, straightforwardly realist interpretation of non-relativistic quantum mechanics. The goal is to characterize what one might call a simple realism of quantum systems, and of the observables associated with them.Simple realism as developed here is a generalized interpretation-scheme, one that abstracts important shared features of Einsteinian naive realism, the so-called modal interpretations, and the orthodox interpretation itself. Some such schemes run afoul of the classic no-go theorems, while others do not. The role of non-commuting observables plays a major role in this success or failure. In particular, we show that if a simple-realist interpretation attributes simultaneously definite values to canonically conjugate observables, then it necessarily falls prey to Kochen-Specker contradictions.This exercise provides some insight into why modal interpretations work, while more generally placing limits on the scope of simple realism itself. In particular, we find that within the framework of simple realism, the only consistent interpretation of the uncertainty relations is the orthodox one. What's more, we point out that similar conclusions are bound to hold for many other non-commuting observables as well.     

On Arthur Fine's solution to the measurement problem

Arthur Fine has recently proposed a novel solution to the measurement problem. Fine's innovation is to exploit the idea that a measurement is a selective interaction in which the apparatus responds to a probability distribution. I develop Fine's solution in some detail, responding to various potential objections along the way. In the end, however, I suggest that Fine needs to tell us more before we can declare the measurement problem solved.     

Resolving the measurement problem: reply to Allen Stairs

We defend an account of quantum measurement as a species of selective interactions, an account that gets beyond the insolubility theorem to resolve the quantum measurement problem. In the course of the defense we propose a novel analysis of inexact measurements and discuss the problem of how to treat the states and selective interactions of composite systems.     

完全补偿法测量电阻及误差分析

介绍了完全补偿法测量电阻的原理,给出了误差分析,并简述其优点．     

Entanglement, Joint Measurement, and State Reduction

Entanglement is perhaps the most important new feature of the quantum world. It is expressed in quantum theory by the joint measurement formula. We prove the formula for projection valued observables from a plausible assumption, which for spacelike separated measurements is a consequence of causality. State reduction is simply a way to express the joint measurement formula after one measurement has been made, and its result known.     

On the physical impossibility of ideal quantum measurements

Albert and Loewer have argued in this journal [4] that modal interpretations of quantum mechanics are ruled out if the abstract structure of Hilbert space is taken realistically. Their argument contains a dubious inference from a measure-zero set of non-ideal interactions. I look at possible ways to make this inference valid and I conclude that the evidence against the modal interpretation cannot be found in the Hilbert space alone. Instead an analysis of specific cases is required.I want to thank the audience at the 1995 Florence IUHPS Conference, and Harvey Brown, Nancy Cartwright, Jim Cushing, Marco Del Seta, Arthur Fine, Margaret Morrison, Fred Müller and Pieter Vermaas for comments and suggestions.     

CCD测量误差的研究

为了提高CCD的测量精度,对CCD的测量误差进行了深入分析。以单CCD和双CCD错排,对直径为5.000mm、8.000mm、12.000mm的三个标准杆件直径的测量数据为基础,对CCD测量误差的大小及其影响因素展开研究,重点对测量误差、像素间距间的关系进行了深入分析。对N个像素间距为H的CCD错排进行测量,其最大误差减小为H/N,当被测对象大小刚好是像素间距的整数倍时,测量值以很高的机率靠近真实值,检测零件尺寸时,在极值处分别设置一列CCD像素,调整零件极值刚好位于CCD两相邻像素的中间,能准确地判别零件的合格性。     

Coordinate Formalism on Abstract Hilbert Space: Kinematics of a Quantum Measurement

Coordinate form of tensor algebra on an abstract (infinite-dimensional) Hilbert space is presented. The developed formalism permits one to naturally include the improper states in the apparatus of quantum theory. In the formalism the observables are represented by the self-adjoint extensions of Hermitian operators. The unitary operators become linear isometries. The unitary evolution and the non-unitary collapse processes are interpreted as isometric functional transformations. Several experiments are analyzed in the new context.     

On the Measurement Problem for a Two-level Quantum System

A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrödinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the perturbed metric, the Born rule for probabilities of collapse is derived. The approach is applied to a two-level quantum system to obtain a simple geometric interpretation of quantum commutators, the uncertainty principle and Planck’s constant. In light of this, a lucid analysis of the double-slit experiment with collapse and an experiment on a pair of entangled particles is presented.     

Diffusion and survival in a medium with imperfect traps

This paper deals with independent particles diffusing on a line with traps at random positions. It is shown how the long-time decay of the survival probability is exhanced when particles do not necessarily disappear upon hitting a trap. The results are compared with predictions for a model where particles are either absorbed or reflected by traps.     

An approach to estimating and extrapolating model error based on inverse problem methods:towards accurate numerical weather prediction

Model error is one of the key factors restricting the accuracy of numerical weather prediction(NWP). Considering the continuous evolution of the atmosphere, the observed data(ignoring the measurement error) can be viewed as a series of solutions of an accurate model governing the actual atmosphere. Model error is represented as an unknown term in the accurate model, thus NWP can be considered as an inverse problem to uncover the unknown error term. The inverse problem models can absorb long periods of observed data to generate model error correction procedures. They thus resolve the deficiency and faultiness of the NWP schemes employing only the initial-time data. In this study we construct two inverse problem models to estimate and extrapolate the time-varying and spatial-varying model errors in both the historical and forecast periods by using recent observations and analogue phenomena of the atmosphere. Numerical experiment on Burgers’ equation has illustrated the substantial forecast improvement using inverse problem algorithms. The proposed inverse problem methods of suppressing NWP errors will be useful in future high accuracy applications of NWP.     

光学焦距测量误差分布的实验研究

为解决光学检测中的误差分布问题 ,提出了用实验方法验证其分布 ,而不是简单地依据经验估计其分布为均匀分布、三角分布或正态分布。通过人眼目视测量与CCD光电测试 ,分析比较大量的采样数据 ,通常人为操作因素及仪器机械精度是目视光学检测中的一个主要误差源 ,进而得到目视测焦距的误差分布 ,并进行拟合性检验     

Non-ideal measurements

We have previously objected to modal interpretations of quantum theory by showing that these interpretations do not assign outcomes to non-ideal measurements. Bub and Healey replied (in this journal) by offering alternative accounts of non-ideal measurements. In this paper we argue, first, that our account of non-ideal measurements is correct and, second, even if it is not correct, it is overwhelmingly likely that interactions satisfying our characterization of non-ideal measurements actually occur and that such interactions possess outcomes. A successful defense of the modal interpretation must assign outcomes to these interactions or show that they do not have outcomes or show that in fact they never occur. Bub and Healey show none of this.1. Modal interpretations are those which assign values to observables even when the quantum state is not an eigenstate of the observable,and in accord with some rule which involves the quantum state or the Hamiltonian that governs the evolution of the state. Healey rejects the name modal interpretation for his account preferring interactive interpretation. In previous papers we cited a number of problems with modal interpretations including the difficulty of adding to them a dynamics for their hidden variables and their failure to assign outcomes to nonideal measurements. It is the latter objection which is at issue in these papers. This objection applies to Kochen's, Dieks', and Healey's interpretations, which all use the polar decomposition theorem to determine which observables possess values, and to Van Fraassen's interpretation, which characterizes the observables that possess values in terms of the Hamiltonian governing the interaction. It does not apply to Bub's interpretation, which doesn't rely on this theorem.2. The analysis is slightly more complicated for disturbing measurements. Healey seems to consider it a defect that in our original discussion we restricted our discussion to non-disturbing measurements, but he would surely admit that the added complication is irrelevant to the issues at hand.3. There are only two honest ways of responding to the measurement problem. One is to break the eigenstate-eigenvalue link. This is the approach taken by modal interpretations, hiddenvariable theories like Bohm's, and our many-minds interpretation. The other is to modify the laws of evolution of state. This is what is done by orthodox quantum theory, which exempts measurements from the linear laws of state evolution and, more plausibly, by recent collapse theories like GRW, which modify the laws of state evoultion so that macroscopic interactions result in states which are close to being eigenstates of familiar observables.4. When the amplitudes are exactly equal (as in the usual EPR state), there is not a unique bi-orthonormal form. An advocate of the modal interpretation must argue that, since the measure of the set of such states is 0, it is very unlikely that they are actually encountered.5. For any observableA*, there is an initial state that the Hamiltonian evolves into, a state in whichA* is well defined. We mentioned this in Albert and Loewer (1991) and owe the point to Yakir Aharonhov.6. In (1990) and (1991) we showed how the many-minds interpretation and CRW handle inaccurate measurements (viz. interactions that conform to (2)). The analysis of these interactions within Bohm's theory is straightforward.7. The irony here will not be lost on the reader who recalls, from Note 4, that Healey must argue that, since the measure of the set of states whose bi-orthonormal form has equal amplitudes is 0, one can neglect such states.8. The Copenhagen interpretation is sometimes so understood; see for example Van Fraassen's (1992) discussion.