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Cambridge University Press eBooks

Indirect measurements

Vladimir B. Braginsky, Farid Ya Khalili, Kip S. Thorne

Published Sep 24, 1992• 0 citations
Publisher

Abstract

The two main types of quantum measurements von Neumann's postulate of the reduction of the wave function answers the question of what happens to the quantum object during the measurement. However, the reduction postulate does not answer the question of how the measuring device must be designed to realize the desired measurement. To answer this second question, one must understand the connection between the measuring device as an ordinary physical system, and the nature of the desired measurement (the quantity to be measured, the desired precision, and the choice of how the measurement is to perturb the quantum object). The Schrödinger equation cannot tell us the connection between the design of the measuring device and the nature of the measurement, because the Schrödinger equation neither governs nor describes the process of measurement. More specifically: The evolution of the wave function as described by the Schrödinger equation has two key features: it is a reversible evolution (from the final state in principle one can always evolve back to the initial one), and it is a deterministic evolution (the final state is determined uniquely by the initial one). By contrast, the reduction of the wave function in a measurement is irreversible (once the measurement is finished and the information has been extracted from it, it is impossible to return to the pre-measurement state), and it is nondeterministic (one cannot predict, before the measurement, the post-measurement state of the measured object).

DOI
10.1017/cbo9780511622748.006

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The two main types of quantum measurements von Neumann's postulate of the reduction of the wave function answers the question of what happens to the quantum object during the measurement. However, the reduction postulate does not answer the question of how the measuring device must be designed to realize the desired measurement. To answer this second question, one must understand the connection between the measuring device as an ordinary physical system, and the nature of the desired measurement (the quantity to be measured, the desired precision, and the choice of how the measurement is to perturb the quantum object). The Schrödinger equation cannot tell us the connection between the design of the measuring device and the nature of the measurement, because the Schrödinger equation neither governs nor describes the process of measurement. More specifically: The evolution of the wave function as described by the Schrödinger equation has two key features: it is a reversible evolution (from the final state in principle one can always evolve back to the initial one), and it is a deterministic evolution (the final state is determined uniquely by the initial one). By contrast, the reduction of the wave function in a measurement is irreversible (once the measurement is finished and the information has been extracted from it, it is impossible to return to the pre-measurement state), and it is nondeterministic (one cannot predict, before the measurement, the post-measurement state of the measured object).
It is well known, that the causal Schrödinger evolution of a quantum state is not compatible with the reduction postulate, even when decoherence is taken into account. The violation of the causal evolution, introduced by the standard reduction postulate distinguishes certain systems (as the measurement devices), whose states are very close to statistical mixtures (as the ones resulting from the process of decoherence). In these systems, this violation has not any observable effect. In arbitrary quantum systems, the transition from the initial density matrix to a diagonal matrix predicted by the standard reduction postulate, would lead to a complete breakdown of the Schrödinger evolution, and consequently would destroy all the predictive power of quantum mechanics. What we show here, is that there is a modified version of the postulate that allows to treat all the quantum mechanical systems on equal footing. The standard reduction postulate may be considered as a good approximation, useful for practical purposes, of this modified version which is consistent with the Schrödinger evolution and via decoherence with the experimental results. Within this approach, the physical role played by the reduction postulate is as a tool for the computation of relative probabilities and consequently for the determination of the probabilities of consistent histories.
Conceptual difficulties still remain in quantum mechanics, even if they had already been identified by its inventors. This does not mean that the theory is not successful! The reality is quite the opposite: in fact, independently of these difficulties, quantum mechanics is certainly one of the most successful theories of all science. One can even consider that its ability to adapt to new situations is one of its most remarkable features. It continues to give efficient and accurate predictions while new experiments are performed, even in situations that the founding fathers had no way to imagine. Actually, there are very few theories that have been verified with the same accuracy in so many situations. Nevertheless, it remains true that conceptual difficulties subsist, and their discussion is the object of this chapter. As we will see, most of them relate to the process of quantum measurement, in particular to the very nature of the random process that takes place on this occasion. We have seen that, in most cases, the wave function evolves gently, in a perfectly predictable and continuous way, according to the Schrödinger equation; in some cases only (when a measurement is performed), unpredictable changes take place, according to the Von Neumann postulate of state vector reduction. Obviously, having two different postulates for the evolution of the same mathematical object is very unusual in physics. The notion was a complete novelty when it was introduced, and still remains unique, but also the source of difficulties – in particular logical difficulties related to the compatibility between the two different postulates.
A quantum measurement process, when non-trivial, is not a closed evolution: the appearance of classical outcomes is usually interpreted as the evidence of some decoherence-like mechanism causing quantum superpositions to degrade into classical mixtures. Such mechanism is due to a net flow of information from the input system (measurement object), through the physical apparatus interacting with the object (measurement probe), into some environment, the latter representing all those degrees of freedom which are not directly accessible by the experimenter. For this reason, the phenomenon of state reduction induced by the measurement process generally entails an irreversible state change. The aim of our contribution is to answer the following questions : how much information a measurement is able to extract? "How much" irreversible is the state reduction due to a particular measurement process? In which way information gain and irreversibility are related?
The measurement process and the uncertainty relation In the unstarred sections of previous chapters, the back action of the measuring device on the measured object was analyzed by simply invoking the Heisenberg uncertainty relation for the perturbation and the measurement error. Is this simple method of analysis justified? The measurement process does not show up in the standard formulation of the uncertainty relations. Instead, the standard formulation, which follows from the fundamental postulates of quantum mechanics (section 2.1), couples the intrinsic rms uncertainties of observables as defined by the quantum state. Nevertheless, the examples of indirect measurements discussed above (the Heisenberg microscope and Doppler speed meter in chapter I, and the electron probe in chapter III) exhibit a tight coupling between the measurement process and the uncertainty relations. In particular, both the measurement error and the perturbation of the object can be traced to the uncertainty properties of the probe's initial state, and from the fact that this initial state is subject to the uncertainty relations, one can show that the measurement error and perturbation are also subject to it. Is it also possible, starting from the uncertainty relations as properties of the object's quantum state, to obtain the uncertainty relations for the measurement process? To answer this question, let us discuss the following thought experiment: We suppose that a generalized coordinate of some quantum object is measured, and that before the measurement the object was in a state with a very well defined momentum.
The measurement problem in quantum mechanics can be characterized as explaining the question of definite outcomes, where a measurement of a superposed quantum state only gives a single definite outcome. According to quantum mechanics, the wave function, ψ, defines the state of the system, while the Schrödinger equation is a linear superposition of states that evolves deterministically. When a measurement is made, the outcome always uncovers the physical system as being in a definite state, and the future evolution of the wave function is amended based on the state that was measured. This change is often referred to as the “collapse” of the wave function. The problem concerns the question of what and how the measurement affected the system and why this changed the wave function. This paper addresses the measurement problem by separating it into five separate questions that are resolved through linking measurements to an entropy transfer from the superposed state to the environment. The underlying premise is the wave function is descriptive and not prescriptive, and entropy—the metric of quantum uncertainty and uncertainty in general—is the fundamental property rather than the wave function.