In this chapter, I summarize the general way in which the measurement process can be cast (by making use of effects and amplitude operators). Then, I show that there are two main problems with the state measurement: (i) how to avoid the disruptive back action on the state of the measured system during detection and (ii) how to extract complete information from this state. In order to deal with them I first introduce quantum non-demolition (QND) measurement and examine the problem whether the entire probability distribution of the measured observable is not altered by a QND measurement, which would allow repeated QND measurements with different observables to extract the whole information from the measured system. However, I show that this is not the case. Then, I deal with protective measurement as such and show that a reversible measurement is in fact not a measurement. However, by taking advantage of statistical methods (and therefore by renouncing measurement of the state of a single system), we can indeed reconstruct the wave function but only by partially recovering the information contained in the state. Two further prices to pay are to admit the existence of negative quasi-probabilities due to the interference terms and to make use of unsharp observables for guaranteeing informational completeness. Introduction The aim of this chapter is a review of the developments related to the measurement of the state vector and the problems that have been raised in this context. Traditionally, the state vector or the wave function describing quantum systems has been considered as a formal tool for calculating the probabilities associated with certain events like measurement outcomes, but few scholars have tried to attribute to it an ontological status. It is well known how many difficulties are related to an attempt at assigning a kind of reality to the state of quantum systems.
In this chapter, I summarize the general way in which the measurement process can be cast (by making use of effects and amplitude operators). Then, I show that there are two main problems with the state measurement: (i) how to avoid the disruptive back action on the state of the measured system during detection and (ii) how to extract complete information from this state. In order to deal with them I first introduce quantum non-demolition (QND) measurement and examine the problem whether the entire probability distribution of the measured observable is not altered by a QND measurement, which would allow repeated QND measurements with different observables to extract the whole information from the measured system. However, I show that this is not the case. Then, I deal with protective measurement as such and show that a reversible measurement is in fact not a measurement. However, by taking advantage of statistical methods (and therefore by renouncing measurement of the state of a single system), we can indeed reconstruct the wave function but only by partially recovering the information contained in the state. Two further prices to pay are to admit the existence of negative quasi-probabilities due to the interference terms and to make use of unsharp observables for guaranteeing informational completeness. Introduction The aim of this chapter is a review of the developments related to the measurement of the state vector and the problems that have been raised in this context. Traditionally, the state vector or the wave function describing quantum systems has been considered as a formal tool for calculating the probabilities associated with certain events like measurement outcomes, but few scholars have tried to attribute to it an ontological status. It is well known how many difficulties are related to an attempt at assigning a kind of reality to the state of quantum systems.
In this book, I have argued that the wave function in quantum mechanics is real, and it represents the state of random discontinuous motion (RDM) of particles in three-dimensional space. Moreover, this picture of quantum ontology is complete in accounting for our definite experience, but it requires that the quantum dynamics be revised to include a stochastic, nonlinear evolution term resulting from the RDM of particles. Obviously I took a road less traveled by researchers in the foundations of quantum mechanics. In order to convince more readers that this road is deserved to be taken, I will review the main results of this book and discuss them in a broader context. The starting point of my road is protective measurements (Chapter 1). In 1993, this new method of measurement in quantum mechanics was discovered (Aharonov and Vaidman, 1993; Aharonov, Anandan, and Vaidman, 1993). Distinct from conventional projective measurements, protective measurement is a method for measuring the expectation value of an observable on a single quantum system. By a series of protective measurements, one can even measure the wave function of a single quantum system. Therefore, besides the Born rule for projective measurements, protective measurements provide another, more direct connection between the wave function and results of measurements, which is not probabilistic but definite. It can be expected that a definite connection between the wave function and results of measurements is more important for understanding the meaning of the wave function and searching for the ontology of quantum mechanics. However, it seems that this connection provided by protective measurements is still less well known today, and its significance has not been fully realized by most researchers, either. The first stop on my road is the reality of the wave function (Chapter 4). It is correct to say that protective measurements alone do not imply the reality of the wave function. An additional connection between the state of reality and results of measurements is needed.
Introduction One of the crucial issues about protective measurements is the extent to which they provide (additional) grounds for some realist understanding of the wave function. This issue is notoriously subject to debate, which is not the aim of this chapter. Rather, this chapter aims at clarifying the further issue of the ontological picture corresponding to the realist understanding of the wave function possibly suggested by protective measurements. Oddly enough, proponents of a realist reading of protective measurements remain in general rather vague about ontology (see for instance Aharonov et al. (1993, 4624), who assimilate the wave function to an “extended object” without further details, or the brief and somewhat unprecise discussion in Dickson (1995, 135–136) of the interpretation of the wave function within Bohmian mechanics (BM) and within the theoretical framework elaborated by Ghirardi, Rimini and Weber (GRW); in this volume, Gao does propose a more elaborated ontological model for quantum mechanics (QM) based on protective measurements). Despite this vagueness (or maybe because of it), it seems that the most obvious ontological intuition resulting from protective measurements points towards some form of straightforward realism about the wave function understood in the sense of a real, physical field: indeed, protective measurements are claimed to allow measuring expectation values of observables on a single quantum system, thereby providing the possibility of reconstructing (i.e. “measuring” in some sense) the wave function of a single quantum system. Aharonov et al. (1996, 125) clearly express this intuition: “We can observe the expectations values of operators, and we can observe the density and the current of the Schrödinger wave. We can “see” in some sense the Schrödinger wave. This leads us to believe that it has physical reality.” Therefore, this kind of intuition seems in line with a realist conception of the wave function, according to which the wave function is considered as a real substantial, material in some sense) entity on its own (this view is often called “wave function realism”).
Protective measurement, in the language of standard quantum mechanics, is a method to measure the expectation value of an observable on a single quantum system (Aharonov and Vaidman, 1993; Aharonov, Anandan and Vaidman, 1993). For a conventional impulsive measurement, the state of the measured system is strongly entangled with the state of the measuring device during the measurement, and the measurement result is one of the eigenvalues of the measured observable. By contrast, during a protective measurement, the measured state is protected by an appropriate procedure so that it neither changes nor becomes entangled with the state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables on a single quantum system, and in particular, the wave function of the system can also be measured as expectation values of certain observables. It is expected that protective measurements can be performed in the near future with the rapid development of weak measurement technologies (e.g. Kocsis et al., 2011; Lundeen et al., 2011). In this chapter, we will give a clear introduction to protective measurement in quantum mechanics. Standard quantum mechanics and impulsive measurement The standard formulation of quantum mechanics, which was first developed by Dirac (1930) and von Neumann (1955), is based on the following basic principles. 1 Physical states The state of a physical system is represented by a normalized wave function or unit vector | ψ ( t )〉 in a Hilbert space. The Hilbert space is complete in the sense that every possible physical state can be represented by a state vector in the space. 2 Physical properties Every measurable property or observable of a physical system is represented by a Hermitian operator on the Hilbert space associated with the system. A physical system has a determinate value for an observable if and only if it is in an eigenstate of the observable (this is often called the eigenvalue–eigenstate link).
The quantum state contains all accessible information about a system. However, for various reasons, the information is less complete than we might expect given the classical states we are used to. This suggests the possibility that the state itself contains a measure of entropy (lack of information), which could potentially arise in two ways. Firstly, quantum states contain an inherent uncertainty with respect to the measurement outcomes of conjugate variables. Secondly, due to limitations in the preparation process, we may not be sure which state a system is occupying. The best we can do in this situation is to assign each potential state a classical probability and construct a specialist mathematical representation to deal with the situation. This is known as the density matrix. It turns out to be a rather handy tool in general and crucial for the exploration of quantum entropy. So, in this chapter, we are going to extend our technical understanding of quantum theory and the representation of states. This will arm us sufficiently to venture into the physics of the measurement problem and the entropy of quantum systems, which we will tackle in Chapter 12 .
Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device. In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments , will be investigated in this chapter. There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process . A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements . Some other authors use the term indirect measurements . Three levels of description of measurements It is convenient to think of three levels of description when considering measurements.
We examine the entanglement and state disturbance arising in a protective measurement and argue that these inescapable effects doom the claim that protective measurement establishes the reality of the wave function. An additional challenge to this claim results from the exponential number of protective measurements required to reconstruct multi-qubit states. We suggest that the failure of protective measurement to settle the question of the meaning of the wave function is entirely expected, for protective measurement is but an application of the standard quantum formalism, and none of the hard foundational questions can ever be settled in this way. Introduction From the start, the technical result of protective measurement has been suggested to have implications for the interpretation of quantum mechanics. Consider how Aharonov and Vaidman [2] chose to begin their original paper introducing the idea of protective measurement: We show that it is possible to measure the Schrödinger wave of a single quantum system. This provides a strong argument for associating physical reality with the quantum state of a single system.… Since then, the pioneers of protective measurement seem to have taken a more moderate stance. Vaidman [42], in a recent synopsis of protective measurement, concedes that the protective measurement procedure is not a proof that we should adopt one interpretation instead of the other, but it is a good testbed which shows advantages and disadvantages of various interpretations. Notwithstanding this more subtle perspective and a number of critical studies of the technical and foundational aspects of protective measurement, Gao [21] has maintained, if not amplified, the force of Aharonov and Vaidman's original argument: An immediate implication is that the result of a protective measurement, namely the expectation value of the measured observable in the measured state, reflects the actual physical property of the measured system, as the system is not disturbed after this result has been obtained. […]