Gedankenexperiment for Three Entangled Electrons: Analog of GHZ Photon Experiments
The classic experiments of Greenberger, Horne, and Zeilinger (GHZ) [1, 2, 3] show that entangled systems of three or four photons conform exactly to the laws of quantum mechanics, thus removing any lingering doubts about any deterministic local reality or hidden-value alternatives to quantum mechanics. These experiments confirm the statistical results implying violations of Bell's inequalities, with the advantage that definite outcomes are obtained, rather than statistical distributions.[more]
We propose here an analog of the GHZ experiment, based on entangled electrons rather than photons. The ground state of atoms of the nitrogen group: N, P, As, Sb, and Bi, all have electronic ground states with an outer shell configuration containing three -electrons with parallel spins. There are four possible states for these spin systems, with . Bismuth appears to be a likely choice, since unbound Bi atoms can readily be produced from the solid element in a furnace. Using the highly developed technology of atomic-beam physics, a columnated beam of Bi atoms, chilled to their electronic ground states, in one of the four selected spin orientations, can be produced by the "black box" in the lower right of the figure. The beam then goes through an ionization chamber, where it is subjected to synchronized pulses slightly in excess of a 49.51-volt electric field, which is the sum of the first three ionization potentials of Bi. The geometry of the electrodes directs the ionized electrons (shown as green points) toward three Stern–Gerlach (SG) detectors, which can register electron-spin orientations of either ±1/2 (for or electrons, respectively). The axis of quantization ( axis) is taken to be the vertical direction.
A coincidence circuit will register only events in which one electron impinges on each of the three SG detectors. Everything might work out optimally for perhaps one out of every 100 atoms in the beam. (But beam physicists are known to be infinitely patient; the recently discovered Higgs particle was produced about once in every several trillion collisions at CERN's LHC.) The whole apparatus should be enclosed in a high-vacuum chamber to avoid spurious readings from extraneous electron collisions.
The Demonstration shows just a single successful event. In practice, the apparatus can be run continuously, discarding the majority of failures while recording the results of successful events.
The results for , producing either three or three readings, should not overly disturb Einstein. However, the states represent three parallel electron spins pointed at an angle oblique to the axis. Still, readings from any two of the SG detectors should unambiguously determine the reading of the third detector, which can only be accounted for by entanglement of the three electrons.[less]
Three parallel entangled electron spins not constrained by the Pauli principle can be represented by one of four possible spin wavefunctions: , , with the obvious analogs for and . In each of the states, two spins are entangled with parallel spins, while the third spin is antiparallel. This can happen in three different ways, which also reflects the indeterminacy of quantum mechanics.
 D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, "Bell's Theorem without Inequalities," American Journal of Physics, 58(12), 1990 pp. 1131–1143. doi:10.1119/1.16243.
 D. Mermin, "Quantum Mysteries Revisited," American Journal of Physics, 58(8), 1990 pp. 731–734. doi:10.1119/1.16503.
 S. M. Blinder, Introduction to Quantum Mechanics, Amsterdam: Elsevier, 2004 pp. 277–280.