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Physical Review A

Path summation and von Neumann–like quantum measurements

D. Sokolovski, R. Sala Mayato

Published Apr 11, 2005• 12 citations
Publisher

Abstract

We demonstrate how a general von Neumann--like measurement can be analyzed in terms of histories (paths) constructed for the measured variable $A$. The Schr\"odinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time, or continuous type. The cases of a particle described by Feynman paths in the coordinate space and a qubit in a two-dimensional Hilbert space are studied in some detail.

DOI
10.1103/physreva.71.042101

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We demonstrate how a general von Neumann--like measurement can be analyzed in terms of histories (paths) constructed for the measured variable AAA. The Schr"odinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time, or continuous type. The cases of a particle described by Feynman paths in the coordinate space and a qubit in a two-dimensional Hilbert space are studied in some detail.
We propose a general theoretical approach to quantum measurements based on the path (histories) summation technique. For a given dynamical variable A, the Schrödinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time or continuous type. Transformations between the histories obtained for non-commuting variables and construction of simultaneous histories for non-commuting observables are discussed. Important cases of a particle described by Feynman paths in the coordinate space and a qubit in a two dimensional Hilbert space are studied in some detail.
We analyze a quantum observable fff given by the time average of a dynamical variable F along a Feynman path. We show that fff can be described by a (generalized) wave function satisfying the Schr\"odinger-like equation and measured by a meter similar to that of von Neumann. A connection with operators is established. Examples include the particle's momentum, energy and the fractional traversal time.
We analyze the connection between the path integral and operator approaches to the quantum measurement problem. In general, an act of measurement is shown to destroy interference between components of the wave function related to the particle's histories. Classically, there exist a class of meters suitable for measuring the value of a given dynamical variable F. Quantally, different meters produce different results. The standard von Neumann measurement is one particular case. Rearranging Feynman paths according to the value of (time average of) F defines a different type of meter. The two methods disagree if the duration of the measurement is very short. Possible ways to measure the particle's momentum are studied in detail. The semiclassical limit of a measurement is analyzed.
We analyze von Neumann-like quantum measurements in terms of simultaneous virtual paths constructed for two noncommuting variables. The approach is applied to measurements of operator functions of conjugate variables and to the joint measurements of such variables. The limits of applicability of the restricted phase space path integral are studied. We demonstrate that, for a simple joint measurement, using entangled meter states allows one to manipulate the order in which the measurements are conducted. The effects of ``weakening'' a measurement by choosing unsharp meter states are also discussed.
It is convenient to describe a quantum system at all times by means of a “history operator” [Formula: see text], encoding measurements and unitary time evolution between measurements. These operators naturally arise when computing the probability of measurement sequences, and generalize the “sum over position histories” of the Feynman path-integral. As we argue in this work, this description has some computational advantages over the usual state vector description, and may help to clarify some issues regarding nonlocality of quantum correlations and collapse. A measurement on a system described by [Formula: see text] modifies the history operator, [Formula: see text], where [Formula: see text] is the projector corresponding to the measurement. We refer to this modification as “history operator collapse”. Thus, [Formula: see text] keeps track of the succession of measurements on a system, and contains all histories compatible with the results of these measurements. The collapse modifies the history content of [Formula: see text], and therefore modifies also the past (relative to the measurement), but never in a way to violate causality. Probabilities of outcomes are obtained as [Formula: see text]. A similar formula yields probabilities for intermediate measurements, and reproduces the result of the two-vector formalism in the case of the given initial and final states. We apply the history operator formalism to a few examples: entangler circuit, Mach–Zehnder interferometer, teleportation circuit and three-box experiment. Not surprisingly, the propagation of coordinate eigenstates [Formula: see text] is described by a history operator [Formula: see text] containing the Feynman path-integral.
The (consistent or decoherent) histories interpretation provides a consistent realistic ontology for quantum mechanics, based on two main ideas. First, a logic (system of reasoning) is employed which is compatible with the Hilbert-space structure of quantum mechanics as understood by von Neumann: quantum properties and their negations correspond to subspaces and their orthogonal complements. It employs a special (single framework) syntactical rule to construct meaningful quantum expressions, quite different from the quantum logic of Birkhoff and von Neumann. Second, quantum time development is treated as an inherently stochastic process under all circumstances, not just when measurements take place. The time-dependent Schrödinger equation provides probabilities, not a deterministic time development of the world. The resulting interpretive framework has no measurement problem and can be used to analyze in quantum terms what is going on before, after, and during physical preparation and measurement processes. In particular, appropriate measurements can reveal quantum properties possessed by the measured system before the measurement took place. There are no mysterious superluminal influences: quantum systems satisfy an appropriate form of Einstein locality. This ontology provides a satisfactory foundation for quantum information theory, since it supplies definite answers as to what the information is about. The formalism of classical (Shannon) information theory applies without change in suitable quantum contexts, and this suggests the way in which quantum information theory extends beyond its classical counterpart.