Once upon a time everybody knew why measurements in quantum mechanics don't reveal pre-existing properties. It was because the act of acquiring knowledge unavoidably messes up the object being studied. What you learn is not intrinsic to the object, but a joint manifestation of the object and how you probe it to get your knowledge. In 1935 this state of happy innocence was forever dispelled by Einstein, who with Boris Podolsky and Nathan Rosen discovered how to learn about an object by messing up only some stuff it left behind in a faraway place. They concluded that knowledge acquired in this way was indeed about pre-existing properties of the object, revealed—not created—by the act of probing the stuff left behind. Bohr, however, insisted their conclusion was unjustified, and 30 years later John Bell proved that no assignment of such pre-existing properties could agree with the quantitative predictions of quantum mechanics. A couple of years ago Lucien Hardy [1] gave this tale an unexpected twist, by finding a charming variation of the Bell–EPR argument. Hardy's theorem is even simpler than the argument of Daniel Greenberger, Michael Home, and Anton Zeilinger that I enthused about in this column four years ago. The reason he was able to pull the trick off, and the reason, I suspect, nobody had noticed so neat an argument for so long, is that Hardy's analysis applies to data that are not correlated strongly enough to support the argument of EPR. But they do give rise to an argument every bit as seductive, which Hardy is then able to demolish with surprising ease. Parts of the formulation I give here of Hardy's gedankenexperiment are similar to those of Henry Stapp [2] and Sheldon Goldstein [3]. We consider two particles that originate from a common source and fly apart to stations at the left and right ends of a long laboratory. At the left station we can experimentally determine the answer to one of two yes–no questions, A or B . There is a choice of two other yes–no questions, M or N , to be answered by experiment on the right. Hardy provides questions A , B , M , and N , and a two-particle state |Ψ〉 for which the answers to the questions have the following features: […]
Once upon a time everybody knew why measurements in quantum mechanics don't reveal pre-existing properties. It was because the act of acquiring knowledge unavoidably messes up the object being studied. What you learn is not intrinsic to the object, but a joint manifestation of the object and how you probe it to get your knowledge. In 1935 this state of happy innocence was forever dispelled by Einstein, who with Boris Podolsky and Nathan Rosen discovered how to learn about an object by messing up only some stuff it left behind in a faraway place. They concluded that knowledge acquired in this way was indeed about pre-existing properties of the object, revealed—not created—by the act of probing the stuff left behind. Bohr, however, insisted their conclusion was unjustified, and 30 years later John Bell proved that no assignment of such pre-existing properties could agree with the quantitative predictions of quantum mechanics. A couple of years ago Lucien Hardy [1] gave this tale an unexpected twist, by finding a charming variation of the Bell–EPR argument. Hardy's theorem is even simpler than the argument of Daniel Greenberger, Michael Home, and Anton Zeilinger that I enthused about in this column four years ago. The reason he was able to pull the trick off, and the reason, I suspect, nobody had noticed so neat an argument for so long, is that Hardy's analysis applies to data that are not correlated strongly enough to support the argument of EPR. But they do give rise to an argument every bit as seductive, which Hardy is then able to demolish with surprising ease. Parts of the formulation I give here of Hardy's gedankenexperiment are similar to those of Henry Stapp [2] and Sheldon Goldstein [3]. We consider two particles that originate from a common source and fly apart to stations at the left and right ends of a long laboratory. At the left station we can experimentally determine the answer to one of two yes–no questions, A or B . There is a choice of two other yes–no questions, M or N , to be answered by experiment on the right. Hardy provides questions A , B , M , and N , and a two-particle state |Ψ〉 for which the answers to the questions have the following features: […]
We always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me … It has not yet become obvious to me that there is no real problem. I cannot define the real problem, therefore I suspect there is no real problem, but I'm not sure there's no real problem. Richard Feynman Max Born and quantum probabilities In this chapter, we will take a short break from our survey of successful applications of quantum mechanics to take a closer look at the foundations of this great edifice of modern physics. In a sense, we are now disobeying Feynman's warning about the results of the double slit experiment of chapter 1. We shall now ask Feynman's ‘forbidden’ question ‘But how can it be like that?’ Nobody disputes that quantum mechanics has been magnificently successful, enabling us to make correct quantitative calculations of atomic and nuclear effects. But there is a great divergence of opinion about the implications of quantum mechanics for the nature of matter, and indeed, of reality itself. To avoid becoming embroiled in a purely philosophical mire, we will focus on two famous paradoxes. The first of these is the ‘Einstein–Podolsky-Rosen’ paradox, named after its originators Albert Einstein, Boris Podolsky and Nathan Rosen, and usually abbreviated by the initials ‘EPR’. The second paradox is named after ‘Schrödinger's cat’.
Over fifty years ago Einstein, Podolsky, and Rosen published a striking argument that quantum theory provides only an incomplete description of physical reality – that is, that some elements of physical reality fail to have a counterpart in quantum-theoretical description. What made this a challenge to the quantum-theoretical Weltanschauung was the very mild character of the sufficient condition for the reality of a physical quantity on which their argument hinged: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” The argument consists in pointing out that it is possible to construct a quantum-mechanical state ψ with the following properties: The state ψ describes two noninteracting systems (I and II). In almost all discussions the two systems are taken to be two particles, and the absence of relevant interaction is built in by taking ψ to assign negligible probability to finding the particles closer together than some macroscopically large distance. By measuring an observable A of system I, one can predict with certainty the result of a subsequent measurement of a corresponding observable P of system II. If, instead, one chooses to measure a different observable B of system I, one can predict with certainty the result of a subsequent measurement of a corresponding observable Q of system II. The observables P and Q are represented in quantum theory by noncommuting operators, which means that they cannot both have definite values.
The subject of Einstein–Podolsky–Rosen correlations—those strong quantum correlations that seem to imply “spooky actions at a distance”—has just been given a new and beautiful twist. Daniel Greenberger, Michael Horne, and Anton Zeilinger have found a clever and powerful extension of the two-particle EPR experiment to gedanken decays that produce more than two particles [1]. In the GHZ experiment the spookiness assumes an even more vivid form than it acquired in John Bell's celebrated analysis of the EPR experiment, given over 25 years ago [2]. The argument that follows is my attempt to simplify a refinement of the GHZ argument given by the philosophers Robert Clifton, Michael Redhead, and Jeremy Butterfield [3]. Consider three spin-½ particles, named 1, 2, and 3. They have originated in a spin-conserving gedanken decay and are now gedanken flying apart along three different straight lines in the horizontal plane. (It's not essential for the gedanken trajectories to be coplanar, but it makes it easier to describe the rest of the geometry.) I specify the spin state |Ψ〉 of the three particles in a time-honored manner, giving you a complete set of commuting Hermitian spin-space operators of which |Ψ〉 is an eigenstate. Those operators are assembled out of the following pieces (measuring all spins in units of ½ ħ ): σ z i the operator for the spin of particle i along its direction of motion; σ x i the spin along the vertical direction; and σ y i , the spin along the horizontal direction orthogonal to the trajectory. (Any three orthogonal directions independently chosen for each particle would do. But we're going to be gedanken -measuring x and y components of each particle's spin, so it's nice to think of the x and y directions as orthogonal to the direction of motion, since the components can then be straightforwardly measured by passage through a conventional Stern–Gerlach magnet.) The complete set of commuting Hermitian operators consists of σ x 1 σ y 2 σ y 3 , σ y 1 σ x 2 σ y 3 , σ y 1 σ y 2 σ x 3 . Even though the x and y components of a given particle's spin anticommute—a fact of paramount importance in what follows—all three of the operators in (1) do indeed commute with one another, because the product of any two of them differs from the product in the reverse order by an even number of such anticommutations. Because they all commute, the three operators can be provided with simultaneous eigenstates.
This symposium commemorating the fiftieth anniversary of the paper of Einstein, Podolsky, and Rosen is a fitting place to review what that work and its sequels have taught us. Prima facie, the EPR paper appears to have been exceedingly counter-productive for the following reasons: (1) The work was quickly rebutted by Bohr, and this rebuttal was apparently accepted by most workers in the field. (2) Scientists who adopted the position advocated by Bohr have produced, in the intervening fifty years, a marvelous body of useful theory, whereas those following the course suggested by EPR have produced nothing of any certified practical value. (3) It has been shown by Bell that the conclusion reached by EPR is incompatible with their assumptions. Chemists and physicists have recently begun to examine the behavior of quantum mechanical systems that are very small, yet large enough to influence their environment in ways that appreciably modify their own behavior, vis-a-vis the behavior they would have if isolated. Because these systems are neither small enough to be treated as isolated (or as residing in a classically described environment) between preparation and detection, nor large enough to be treated classically, they do not conform to the format demanded by the Copenhagen interpretation. Indeed, the behavior of these systems depends on ontological considerations that were irrelevant in the situations covered by the Copenhagen interpretation, and that were systematically ignored in that interpretation. Scientists now face the task of enlarging the scope of quantum theory to cover these new situations, and comparing the empirical consequences of various ontological assumptions. 17 refs.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
In loving memory of Asher Peres, we discuss a most important and influential paper written in 1935 by his thesis supervisor and mentor Nathan Rosen, together with Albert Einstein and Boris Podolsky. In that paper, the trio known as EPR questioned the completeness of quantum mechanics. The authors argued that the then-new theory should not be considered final because they believed it incapable of describing physical reality. The epic battle between Einstein and Bohr intensified following the latter's response later the same year. Three decades elapsed before John S. Bell gave a devastating proof that the EPR argument was fatally flawed. The modest purpose of our paper is to give a critical analysis of the original EPR paper and point out its logical shortcomings in a way that could have been done 70 years ago, with no need to wait for Bell's theorem. We also present an overview of Bohr's response in the interest of showing how it failed to address the gist of the EPR argument.