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Physical Review A

Classical limit of von Neumann measurement

M.S. Wang

Published Mar 1, 2004• 2 citations
Publisher

Abstract

We show that, in the context of von Neumann measurement theory, the condition of strong measurement is the classical limit of the measuring device, and vice versa. This means that a strong measurement can be performed only by using a measuring device prepared in a classical limit state, and vice versa. Since it is the strong measurement that conforms with the postulate that the outcome of a single measurement is one of the eigenvalues of the corresponding operator, the aforementioned postulate is valid only in the classical limit for a von Neumann measurement.

DOI
10.1103/physreva.69.034101

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We show that, in the context of von Neumann measurement theory, the condition of strong measurement is the classical limit of the measuring device, and vice versa. This means that a strong measurement can be performed only by using a measuring device prepared in a classical limit state, and vice versa. Since it is the strong measurement that conforms with the postulate that the outcome of a single measurement is one of the eigenvalues of the corresponding operator, the aforementioned postulate is valid only in the classical limit for a von Neumann measurement.
The one-time and two-time measurements are studied by employing the idea of Aharonov et al. that the strength of a measurement depends on the state of the measuring device in the context of von Neumann measurement theory. A measurement is classified as either a strong or a weak measurement depending on whether the interaction between the system and the measuring device is infinitely large or negligibly small. A single strong measurement yields one of the eigenvalues of the operator while a single weak measurement, in addition to the eigenvalues, has infinite other possible outcomes. Nevertheless, the expectation values of the strong and the weak one-time measurements are the same. However, the expectation values of the two-time measurements depend on the strength of the first measurement. For the strong case the result is consistent with the conventional result without having to introduce the state collapsing. For the weak case the result is the real part of the expectation value of the time-ordered two-time operator. The fact that this expectation value cannot be factorized indicates that the entanglement between the system and the measuring device introduced by a weak measurement cannot be neglected.
Notwithstanding it is well known that von Neumann's projection postulate is inapplicable to most realistic measurement procedures, it keeps haunting the foundations of quantum mechanics. In particular its applicability to EPR experiments is often assumed. In the present contribution this problem is considered from the point of view of a quantum mechanical theory of measurement, allowing a treatment of projection in EPR experiments as a special case of conditional preparation. The conditions are spelled out under which the postulate may be applicable.
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We provide an introduction to the theory of quantum measurements that is centered on the pivotal role played by John von Neumann's model. This introduction is accessible to students and researchers from outside the field of foundations of quantum mechanics and presented within a historical context. We first explain the origins and the meaning of the measurement problem in quantum theory, and why it is not present in classical physics. We perform a chronological review of the quantization of action and explain how this led to successive restrictions on what could be measured in atomic phenomena, until the consolidation of the orthodox interpretation of quantum mechanics. The clear separation between quantum system and classical apparatus that causes these restrictions is subverted in von Neumann's paradigmatic model of quantum measurements, a subject whose concepts we explain, while also providing the mathematical tools necessary to apply it to new problems. We show how this model was important in discussing the interpretations of quantum mechanics and how it is still relevant in modern applications. In particular, we explain in detail how it can be used to describe weak measurements and the surprising results they entail. We also discuss the limitations of von Neumann's model of measurements, and explain how they can be overcome with POVMs and Kraus operators. We provide the mathematical tools necessary to work with these generalized measurements and to derive master equations from them. Finally, we demonstrate how these can be applied in research problems by calculating the Quantum Zeno Effect.
Heisenberg was an ‘anti-realist’. Although Bohr was infamously obscure in many of his writings, it seems that he adopted a generally anti-realist interpretation, too. As their debate became more bitter, in early June 1927 Pauli was called in to mediate. With Pauli’s help, they forged an uneasy consensus, which became known as the Copenhagen interpretation. Einstein didn’t like it at all, setting the stage for a great debate about the quantum representation of reality. Although von Neumann’s formalism broadly conforms to the Copenhagen interpretation, he saw no need to introduce an arbitrary split between the classical and quantum worlds. But eliminating the split poses the problem of quantum measurement: when scaled to classical dimensions, a superposition of different measurement outcomes appears contrary to our experience, exemplified by the famous paradox of Schrödinger’s cat. Von Neumann was obliged to break the infinite regress by postulating the ‘collapse of the wavefunction’.
Many textbooks on Quantum Mechanics are not very precise as to the meaning of making a measurement: as a consequence, they frequently make assertions which are not based on a dynamical description of the measurement process. A model proposed by von Neumann allows a dynamical description of measurement in Quantum Mechanics, including the measuring instrument in the formalism.In this article we apply von Neumann’s model to illustrate the measurement of an observable by means of a measuring instrument and show how various results, which are sometimens postulated without a dynamical basis, actually emerge.We also investigate the more complex, intriguing and fundamental problem of two successive measurements in Quantum Mechanics, extending von Neumann’s model to two measuring instruments. We present a description which allows obtaining, in a unified way, various results that have been given in the literature.