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

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.