Classical ergodicity retains its meaning in the quantum realm when the employed measurement is protective. This unique measuring technique is re-examined in the case of post-selection, giving rise to novel insights studied in the Heisenberg rep-resentation. Quantum statistical mechanics is then briefly described in terms of two-state density operators.

Introduction

In classical statistical mechanics, the ergodic hypothesis allows us to measure position probabilities in two equivalent ways: we can either measure the appropriate particle density in the region of interest or track a single particle over a long time and calculate the proportion of time it spent there. As will be shown below, certain quantum systems also obey the ergodic hypothesis when protectively measured. Yet, since Schrödinger's wave function seems static in this case [1, 2, 3], and Bohmian trajectories were proven inappropriate for calculating time averages of the particle's position [4, 5], we will perform our analysis in the Heisenberg representation.

Indeed, quantum theory has developed along two parallel routes, namely the Schrödinger and Heisenberg representations, later shown to be equivalent. The Schrödinger representation, due to its mathematical simplicity, has become more common. Yet, the Heisenberg representation offers some important insights which emerge in a more natural way, especially when employing modular variables [6]. For example, in the context of the two-slit experiment it sheds a new light on the question of momentum exchange [7, 8, 9]. Recently studied within the Heisenberg representation are also the double Mach–Zehnder interferometer [10] and the N-slit problem [11]. As can be concluded from [11], the Heisenberg representation prevails in emphasizing the non-locality in quantum mechanics, thus providing us with insights about this aspect of quantum mechanics as well.

Equipped with the backward evolving state-vector within the framework of two-state-vector formalism (TSVF) [12], the Heisenberg representation becomes even more powerful since the time evolution of the operators includes now information from the two boundary conditions.

We generalize protective measurement for protective joint measurement of several observables. The merit of joint protective measurement is the determination of the eigenstates of an unknown Hamiltonian rather than the determination of features of an unknown quantum state. As an example, we precisely determine the two eigenstates of an unknown Hamiltonian by a single joint protective measurement of the three Pauli matrices on a qubit state.

Introduction

Protective measurement is one of the unexpected consequences of the strange structure of quantum mechanics. According to general wisdom, we cannot gain information on the unknown state ρ of a single quantum system unless we distort the state itself. In particular, we cannot learn the unknown state of a single system whatever test we apply to it. It came as a surprise that in weak measurements [1] the expectation value 〈Â〉 of an observable  can be tested on a large ensemble of identically prepared unknown states in such a way that the distortions per single systems stay arbitrarily small (see [2], too). An indirectly related surprise came with the so-called protective measurements [3, 4, 5] capable of testing 〈Â〉 at least in an unknown eigenstate of the Hamiltonian Ĥ at arbitrarily small distortion of the state itself. Interesting debates followed the proposal as to the merit of protective measurement in the interpretation of the wave function of a single system instead of a statistical ensemble (see, e.g., [6] and references therein).

My work investigates an alternative merit of protective measurement. First I construct joint protective measurements of several observables Â1, Â2, … and re-state the original equations for them in a general form. Then I show that the straightforward task that a single joint protective measurement solves on a single system is the determination of the eigenstates of an otherwise unknown Hamiltonian.

In this chapter we show how the weak values, are related to the T0µ(x, t) component of the energy–momentum tensor. This enables the local energy and momentum to be measured using weak measurement techniques. We also show how the Bohm energy and momentum are related to T0µ(x, t) and therefore it follows that these quantities can also be measured using the same methods. Thus the Bohm “trajectories” can be empirically determined, as was shown by Kocsis et al. (2011a) in the case of photons. Because of the difficulties with the notion of a photon trajectory, we argue the case for determining experimentally similar trajectories for atoms where a trajectory does not cause these particular difficulties.

Introduction

The notion of weak measurement introduced by Aharonov, Albert and Vaidman (1988) and Aharonov and Vaidman (1990) has opened up a radically new way of exploring quantum phenomena. In contrast to the strong measurement (von Neumann, 1955), which involves the collapse of the wave function, a weak measurement induces a more subtle phase change which does not involve any collapse. This phase change can then be amplified and revealed in a subsequent strong measurement of a complementary operator that does not commute with the operator being measured. This amplification explains why it is possible for the result of a weak spin measurement of a spin-1/2 atom to be magnified by a factor of 100 (Aharonov et al., 1988; Duck, Stevenson and Sudarshan, 1989). A weak measurement, then, provides a means of amplifying small signals as well as allowing us to gain new, more subtle information about quantum systems.

One of the new features that we will concentrate on in this chapter is the possible measurement of the T0µ (x, t) components of the energy–momentum tensor.

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”).

My view on the meaning of the quantum wave function and its connection to protective measurements is described. The wave function and only the wave function is the ontology of the quantum theory. Protective measurements support this view although they do not provide a decisive proof. A brief review of the discovery and the criticism of protective measurement is presented. Protective measurements with postselection are discussed.

Introduction

In the first graduate course of quantum mechanics I remember asking the question: “Can we consider the wave function as a description of a single quantum system?” I got no answer. Twelve years later, in South Carolina, after I completed my Ph.D. studies at Tel Aviv University under the supervision of Yakir Aharonov in which we developed the theory of weak measurements [1], I asked Aharonov: Can we use weak measurement to observe the wave function of a single particle?

At that time I had already become a strong believer in the many-worlds interpretation (MWI) of quantum mechanics [2] and had no doubt that a single system is described by the wave function. Yakir Aharonov never shared with me the belief in the MWI. When we realized that using what is called now protective measurement, we can, under certain conditions, observe the wave function of a single quantum system, he was really excited by the result. At 1992 I was invited to a conference on the Foundations of Quantum Mechanics in Japan where I presented this result: “The Schrödinger wave is observable after all!”[3]. Then I went home to Tel Aviv where I finished writing a letter which received mixed reviews in Phys. Rev. Lett., while Jeeva Anandan, working on the topic with Aharonov in South Carolina, wrote a paper accepted in Phys. Rev. A [4]. After acceptance of the PRA paper it was hard to fight the referees in PRL, but PLA accepted it immediately [5].

Quantum theory has traditionally – and not altogether unreasonably – been taken as a challenge to “realism” about physical theories. At the very least, the ways in which it is often formulated, presented and used suggest a non-realist understanding of the theory because the significance of the “state” of a system is described in terms of a catalogue of predictions. While protective measurement does not force one to give up on this standard view, it does support an alternative contention, namely, that the state of a physical system (a wave function, for example) has a somewhat more direct physical significance. I conclude with what I take to be the upshot of these observations for approaches to interpreting quantum theory and evaluating those interpretations.

Introduction

In 1993 I wrote an article (Dickson, 1995) about the scheme for protective measurement first described (to my knowledge) by Aharonov, Anandan and Vaidman (1993). There I claimed that protective measurement makes “realism” about quantum theory more attractive than it might otherwise have been. I still believe some form of that claim to be true, and I am grateful to the editor and to Cambridge University Press for the opportunity to say so in the manner that I'm inclined to now, 20 years later.

Of course, the debates about the interpretation of quantum theory have moved on quite a bit in the intervening time. We've seen a rise to prominence of “subjectivist” interpretations, largely based on developments in quantum information theory (but with its own historical precedents, as Timpson (2010) has pointed out – see Fuchs (2002) for a landmark paper in the modern development, and Timpson (2013) for a thorough overview and evaluation). The various “many-somethings” (worlds, minds, perspectives, whatever) interpretations have, arguably, seen a rise in fortune as well.

Introduction

The physical meaning of the wave function is an important interpretative problem of quantum mechanics. Notwithstanding nearly ninety years of development of the theory, it is still an unsolved issue. During recent years, more and more research has been done on the ontological status and meaning of the wave function (see, e.g. Monton, 2002; Lewis, 2004; Gao, 2011a, 2011b; Pusey, Barrett and Rudolph, 2012; Ney and Albert, 2013). In particular, Pusey, Barrett and Rudolph (2012) demonstrated that under certain non-trivial assumptions such as the preparation independence assumption, the wave function of a quantum system is a representation of the physical state of the system. This poses a further question, namely whether the reality of the wave function can be argued without resorting to non-trivial assumptions. Moreover, a harder problem is to determine the ontological meaning of the wave function, which is still a hot topic of debate in the realistic alternatives to quantum mechanics such as the de Broglie–Bohm theory or Bohmian mechanics (Belot, 2012).

In this chapter, we will first give a clearer argument for the reality of the wave function in terms of protective measurements, which does not depend on non-trivial assumptions and can also overcome existing objections. Next, based on an analysis of the mass and charge properties of a quantum system, we will propose a new ontological interpretation of the wave function. According to this interpretation, the wave function of an N-body system represents the state of ergodic motion of N particles.

In 1993, Yakir Aharonov, Lev Vaidman and Jeeva Anandan discovered an important new method of measurement in quantum mechanics, the so-called protective measurement. Distinct from conventional 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. In this way, theoretical analysis of protective measurement may lead to a new and deeper understanding of quantum mechanics. Moreover, its experimental realization may also be useful for quantum information technology.

This book is an anthology celebrating the 20th anniversary of the discovery of protective measurement. It begins with a clear and concise introduction to standard quantum mechanics, conventional measurement and protective measurement, and contains fourteen original essays written by physicists and philosophers of physics, including Yakir Aharonov and Lev Vaidman, the two discoverers. The topics include the fundamentals of protective measurement, its meaning and applications, and current views on the importance and implications of protective measurement. The book is accessible to graduate students in physics and chemistry. It will be of value to students and researchers with an interest in the meaning of quantum theory and especially to physicists and philosophers working on the foundations of quantum mechanics.

When I contacted potential contributors to this anthology, one of them replied, “Protective measurements are something I know nothing about.” Indeed, as one referee of this book also admitted, although protective measurement has attracted attention over the last 20 years and has raised many interesting questions, it is still an under-studied aspect of quantum mechanics. In recent years the associated field of weak measurement has seen significant increased activity, and the latest Pusey–Barrett–Rudolph theorem has also caused many people to revisit the question of the reality of the wave function.

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.

Introduction

It is not exaggeration to claim that one of the major divides in the foundations of non-relativistic quantum mechanics derives from the way physicists and philosophers understand the status of the wave function. On the instrumentalist side of the camp, the wave function is regarded as a mere instrument to calculate probabilities that have been established by previous measurement outcomes. On the other “realistic” camp, the wave function is regarded as a new physical entity or a physical field of some sort. While both sides agree about the existence of quantum “particles” (the so-called theoretical entities), and therefore reject the radical agnosticism about them preached by van Fraassen (1980), the various “realistic” (and consequently, instrumentalist) philosophies of quantum mechanics are typically formulated in different, logically independent ways, so their implications need to be further investigated.

For instance, on the one hand it seems plausible to claim that a realistic stance about the wave function is not the only way to defend “realism” about quantum theory. One can support a “flash” or a “density-of-stuff ” ontology (two variants of GRW), or an ontology of particles with well-defined positions (as in Bohmian mechanics), as primitive ontologies for observer-independent formulations of quantum mechanics (Allori, Goldstein, Tumulka and Zanghí, 2008). “Primitive ontologies”, as here are understood, are not only a fundamental ground for other ontological posits, but also entail a commitment to something concretely existing in spacetime (see also Allori, 2013). On the other hand, however, it is still debated whether such primitive ontologies can be autonomous from some form of realism about the wave function (Albert, 1996).

In order to discuss this problem, we begin with a preliminary clarification of the meaning of “realism” and “instrumentalism” in physics, which are often subject to ideological and abstract discussions that often have little to do with the practice of physics (Section 9.2).

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 explanatory role of empty waves in quantum theory

In this chapter we are concerned with two classes of interpretations of quantum mechanics: the epistemological (the historically dominant view) and the ontological. The first views the wave function as just a repository of (statistical) information on a physical system. The other treats the wave function primarily as an element of physical reality, whilst generally retaining the statistical interpretation as a secondary property. There is as yet only theoretical justification for the program of modelling quantum matter in terms of an objective spacetime process; that some way of imagining how the quantum world works between measurements is surely better than none. Indeed, a benefit of such an approach can be that “measurements” lose their talismanic aspect and become just typical processes described by the theory.

In the quest to model quantum systems one notes that, whilst the formalism makes reference to “particle” properties such as mass, the linearly evolving wave function ψ(x) does not generally exhibit any feature that could be put into correspondence with a localized particle structure. To turn quantum mechanics into a theory of matter and motion, with real atoms and molecules consisting of particles structured by potentials and forces, it is necessary to bring in independent physical elements not represented in the basic formalism. The notion of an “empty wave” is peculiar to those representatives of this class of extended theories which postulate that the additional physical element is a corpuscle-like entity or point particle. For clarity, we shall develop the discussion in terms of a definite model of this kind whose properties are well understood and which it is established reproduces the empirical content of quantum mechanics: the de Broglie–Bohm theory, a prominent representative of the class of ontological interpretations (Holland, 1993).