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Cambridge University Press eBooks

What's wrong with these elements of reality, June 1990

N. David Mermin

Published Jan 5, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9781139162579.008

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Physical Review Letters•Mar 2, 1992•200 citations
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.
Abstract The Einstein‐Podolsky‐Rosen “gedanken‐experiment” does not imply non‐local interactions or an “action‐at‐a‐distance”. Contrary, the EPR proves the measurements at one particle does not have influences at canonical variables of the other particles if the quantum‐mechanical commutation relations are true. But, the EPR implices correlations between the particles which come in by “subjective knowledge”. These correlations are a priori informations about the relative motion or, complementarily, about the motion of the center of mass. The impression of an action‐at‐a‐distance is produced by the use of usual particle coordinates in the EPR‐arrangements. The discussion of the Einstein‐Podolsky‐Rosen “gedanken‐experiment” (EPR) has been going on over fifty years. EINSTEIN, PODOLSKY, and ROSEN formulated their famous paradox in 1935, and in the discussion between N. BOHR (1935, 1949) and A. EINSTEIN (1936, 1948); A. EINSTEIN (1948) made his point that the EPR implied an “action‐at‐a‐distance” for quantum‐mechanical particles (without obvious classical interactions). His argument is the starting point for the recent discussion about EPR and causality (see A. Aspect, 1981).
The long-standing puzzle of the nonlocal Einstein-Podolsky-Rosen correlations is resolved. The correct quantum mechanical correlations arise for the case of entangled particles when strict locality is assumed for the probability amplitudes instead of locality for probabilities. Locality of amplitudes implies that measurement on one particle does not collapse the companion particle to a definite state. Sixty five years ago, the most significant paper questioning a fundamental aspect of quantum phenomena was written by Einstein, Podolsky and Rosen (EPR) [1]. They addressed the question whether the wave-function represented a complete description of reality in quantum mechanics, and argued that it didn’t. Bohr’s reply to this paper [2] was not sufficient to resolve the fundamental issues raised by EPR. Decades later, Bohm rephrased the EPR problem [3] in terms of particles correlated in their spin and this helped enormously in analyzing the problem with clarity. John Bell analyzed the EPR problem in the early sixties and established the Bell’s inequalities obeyed by
In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen, confronted quantum physics with a profound objection concerning the quantessential property of entanglement. This led to a fierce debate between Bohr and Einstein closely followed by Schrödinger. In those days the problem was presented as a gedanken experiment involving a pair of spins or qubits which are entangled but widely separated in space. One may think of a spin-less particle at rest (a π 0 particle for example) decaying into two photons, because of momentum conservation both particles will fly off back to back and because of spin conservation the polarizations of the two photons have to be opposite. This means that without interactions the particles could separate and travel a long way, and we could imagine that one might arrive in New York and the other in Tokyo where Alice and Bob will make polarization measurements. The polarization state of the entangled pair is given by: II.4.1 https://www.w3.org/1998/Math/MathML" display="block"> | ψ N T 〉 = 1 2 ( | 1 , − 1 〉 − | − 1 , 1 〉 ) , where the first entry refers to the NY particle and the second to its Tokyo counterpart, and we for convenience have assumed the particles to be polarized in z-direction. Now Alice in New York decides to make a polarization measurement. Let us suppose that she chooses to do this along the x-axis, and let us also suppose that she finds a value +1. Then we know that the first spin is projected on the |+〉 state. But as the spins are opposite it follows that instantaneously the spin of the particle in Tokyo must have changed to the |−〉 state. That this indeed has to be the case follows from the fact that we could have written the initial state also in the form Figure II.4.1 The Myth of Depth, a 1984 painting by Mark Tensey. It makes you think of unusual, if not magical, ways information may propagate. It is the ‘Spooky action at a distance,' Einstein was so worried about. A painting depicts a group of people on a boat in turbulent water, with one person standing on the waves, symbolizing unusual information propagation. https://www.w3.org/1999/xlink" content-type="colour" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003701910/fd855ea7-f0f9-410f-a832-a3a1a2a2ce55/content/figII_4_1_C.jpg "/> (Source: ANP / Mark Garlick / Science Photo Library) https://www.w3.org/1998/Math/MathML" display="block"> | ψ N T 〉 = 1 2 ( | − , + 〉 − | + . − 〉 ) , and Alice’s projects on the first term as we discussed in the previous section, so after Alice’s measurement we have ψ NT ⇒ | +.−〉. If Bob also decides to measure along the x-axis, then he will obtain the value −1. It is clear that the probalities for measurement outcomes can be precisely calculated for all possible independent choices that Alice and Bob could make.
Although considerable discussion about the completeness of quantum mechanics followed the publication in 1935 of the Gedanken experiment of Einstein, Podolsky, and Rosen (EPR), it was not until Bell's work, published in 3 965, that it was realized that the issue could be rigorously formulated and put to experimental test. An even more provocative demonstration of the incompatibility of quantum mechanics with local realism was recently discovered by Greenberger, Horne, and Zeilinger (GHZ) in a Gedanken experiment employing more than two particles.1 Typical of EPR experiments that have been proposed to date, one has a metastable system that decays into n particles.
Introduction The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no ‘hidden variable’ interpretation of quantum mechanics is possible. These attempts have been examined elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed. That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions. Formulation With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern–Gerlach magnets, on selected components of the spins σ 1 and σ 2 .