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:

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

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:

Introduction

It is a prima facie reasonable assumption that if a physical quantity is measurable, then it corresponds to a genuine physical property of the measured system. You can measure a person's mass because human beings have such a property. You can measure the average mass of a group of people because groups of people have such a collective property. And so on.

Now it would be truly surprising – miraculous, perhaps – if you could determine the average mass of a group of people by making measurements on just one of them. To ascribe such a statistical property to an individual looks like a category mistake. At first glance, protective measurements seem to pull off just such a miracle, determining, for example, the expectation value of position for an ensemble of particles via a measurement performed on one of them. The lesson we are supposed to draw, of course, is that expectation values are not statistical properties at all, despite their name. Rather than being an average over an ensemble of systems, the expectation value of position for a particle is a physical property of the individual system, and the wave function, as the bearer of these properties, is a physical entity (Aharonov, Anandan and Vaidman, 1993).

The protective measurement procedure has been challenged (Uffink, 1999; Gao, 2013; Uffink, 2013), but for present purposes I will assume that protective measurements exist, at least in principle, that are capable of revealing “statistical” properties like expectation values in a single measurement. My aim here is not to challenge the existence of such a physical procedure, but rather to explore the arguments that connect the existence of protective measurements with conclusions concerning the nature of physical reality. What protective measurements are sup- posed to show is that “epistemological” interpretations of the quantum state are untenable – that the wave function of a system must instead be interpreted “onto- logically” (Aharonov, Anandan and Vaidman, 1993: 4617).

Protective measurements illustrate how Yakir Aharonov's fundamental insights into quantum theory yield new experimental paradigms that allow us to test quantum mechanics in ways that were not possible before. As for quantum theory itself, protective measurements demonstrate that a quantum state describes a single system, not only an ensemble of systems, and reveal a rich ontology in the quantum state of a single system. We discuss in what sense protective measurements anticipate the theorem of Pusey, Barrett, and Rudolph (PBR), stating that, if quantum predictions are correct, then two distinct quantum states cannot represent the same physical reality.

Introduction

Although protective measurements [1, 2] are a new tool for quantum theory and experiment, they have yet to find their way into the laboratory; also theorists have not put them to best use, beyond a 1993 paper by Anandan on “Protective measurement and quantum reality” [3]. In Section 10.2, we point out that protective measurements offer new experimental tests of quantum mechanics, and we review recent experiments attempting to measure quantum wave functions. In Section 10.3, we present the Pusey–Barrett–Rudolph (PBR) theorem and discuss their conclusion that the quantum state represents physical reality, and in Section 10.4, we discuss in what sense protective measurements anticipate this conclusion.

Protective measurement: implications for experiment and theory

In 1926, Schrödinger postulated his equation for “material waves” in analogy with light waves: paths of material particles – which obey the principle of least action – are an approximation to material waves, just as rays of light – which obey the principle of least time – are an approximation to light waves [4]. But Born soon discarded “the physical pictures of Schr00F6;dinger” [5] and gave the “material wave” Ψ(x, t) a new interpretation: |Ψ(x, t)|2 is the probability density to find a particle at x at time t. Even Schrödinger was obliged to accept Born's interpretation.

In this chapter, the physical content of previous chapters is summarized, and the reader is suided towards some key areas of modern physics: the Feynman formulation, the unificaton of quantum theory with special relativity, and dualities in quantum mechanics. This very short chapter is intended to be a bridge aimed at motivating readers to attend first course in quantum field theory.

Towards an overall view

At this stage, the student or the general reader might want to know what picture of the physical world emerges from the material covered so far and from the areas related to it but not covered in our book, so as to be introduced to what is in sight not only for the beginner but also for the scientific community. For this purpose, we summarize very briefly the previous pictures and we arrive at three key concepts: the Lagrangian in quantum mechanics via the Feynman sum over histories; the unification of quantum mechanics and special relativity; and new duality symmetries in quantum mechanics.

From Schrödinger to Feynman

Quantum mechanics provides a probabilistic description of the world on atomic or sub-atomic scale. It tells us that, on such scales, the world can be described by a Hilbert space structure, or suitable generalizations. Even in the relatively simple case of the hydrogen atom, the appropriate Hilbert space is infinite-dimensional, but finite-dimensional Hilbert spaces play a role as well.

Approximation of eigen values and eigenvectors

Many realistic problems that arise in theoretical physics lead to equations which cannot be given exact general solutions. Therefore, many attempts have been made to produce satisfactory methods to provide approximate solutions. The most common ones search for solutions in terms of power series and in terms of asymptotic expansions. The main idea behind approximation methods is to try to find approximate solutions of a given system by means of exact solutions of an approximate system (a ‘nearby’ system). We start with the eigenvalue equation associated with the Hamiltonian H and consider an approximating one, say H0, whose equations of motion can be explicitly solved. We introduce an interpolating family of Hamiltonians Hε = H0 + εH1 such that, for a given value of ε, say ε, Hε ≡ H. Both eigenvalues and eigenvectors for Hε may be expanded as a series in ε and solved order by order. The simplest example to illustrate what we have in mind is provided by a second-order algebraic equation, e.g.

x2 − 3.03x + 2.02 = 0.

This equation may be written by means of an interpolating family

x2 − 3(1 + ε)x + 2(1 + ε) = 0,

which yields the equation we want to solve when ε = 10−2. One can consider the approximate equation to be given by ε = 0, i.e.

x2 − 3x + 2 = 0.

Aims and problems of quantum scattering theory

Scattering theory is the branch of physics concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world. In quantum mechanics, a typical scattering process involves a beam of particles prepared with a given energy and with a more or less defined linear momentum. One then studies either the collision with a similar wave packet, or the collision of the given wave packet against a fixed target. In particular, for a two-body elastic scattering problem with conservative forces, we work in the centre of mass frame, hence reducing the original problem to the analysis of a particle in an external field of forces. Indeed, in Section 7.2 we already used such a technique in the investigation of the hydrogen atom. The crucial difference with respect to Section 7.2 is that, in scattering problems, the continuous spectrum and its perturbations are studied, whereas the Balmer formula has to do with bound states. In the following section we outline the basic aspects of what is called time-dependent scattering.

Time-dependent scattering

The physical situation we would like to study is as follows: a beam of particles is prepared in a state that is approximately free and allowed to evolve towards the scatterer. After the collision we detect the scattered beam far away from the interaction region.

Since in both classical and quantum mechanics we are rarely dealing with explicitly solvable equations, this chapter is devoted to an elementary introduction to perturbation theory, Jeffreys–Wentzel–Kramers–Brillouin (hereafter JWKB) method and scattering theory, which are approximation methods that play a key role in all applications of quantum mechanics to atomic and nuclear physics.

In the course of teaching quantum mechanics at undergraduate and post-graduate level, we have come to the conclusion that there is another original book to be written on the subject. The abstract setting foreseen by Dirac and the geometric view pioneered by von Neumann are finding new realizations, leading to further progress both in physics and mathematics, while the applications to quantum computation are opening a new era in modern science. Our emphasis is mainly on structural issues and geometric ideas, moving the reader gradually from the concepts of classical mechanics to those of quantum mechanics, but we have also inserted many problems for students throughout the text, since the book is written, in the first place, for advanced undergraduate and graduate students, as well as for research workers.

The overall picture presented here is original, and also the parts in common with a previous monograph by some of us have been rewritten in most cases. The analysis of waves and particles (Chapter 3), the treatment of symmetries in quantum mechanics (in particular, the first half of Chapter 10), the assessment of modern pictures of quantum mechanics (Chapter 12) have never appeared before in any monograph, to the best of our knowledge. The material on experimental foundations is rather rich and it cannot easily be found to the same extent elsewhere. Our presentation of classical mechanics is original and the choice of topics is motivated by the subsequent development of quantum mechanics, expecially wave equations, Poisson brackets and harmonic oscillators.