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Nonseparability
The idea of something measured in one place "influencing" measurements far away challenged what Einstein thought of as "local reality." It came to be known as "nonlocality," but it always contained something else called "nonseparability." Einstein called it "spukhaft Fernwirkung" or "spooky action at a distance." Erwin Schrödinger called two particles "verschrankt" or "entangled" when their quantum properties had become correlated by an interaction. Entangled particles cannot be separated without an external interaction.
The question for Einstein and Schrödinger was how long the particles could retain their correlation as they traveled a great distance apart. Once de-correlated or "decohered," their two-particle wave function can be described as the product of two single-particle wave functions and there will no longer be any quantum interference between them. But entangled particles, it turns out, cannot be decohered without an external interaction of some kind (like a measurement). Einstein had objected to nonlocal phenomena as early as the Solvay Conference of 1927, when he criticized the collapse of the wave function as involving "instantaneous-action-at-a-distance" that prevents the spherical outgoing wave from acting at more than one place on the screen. He probably had seen nonlocality as early as his light-quantum hypothesis paper of 1905. Single-particle nonlocality can be defined in terms of the volume in phase space where the wave function has non-zero values. There are possibilities of finding the particle anywhere in this volume (with a calculable probability for each possibility).
A particle appears when one of those possibilities becomes actual and the particle is
We can now understand the nonseparability of two entangled particles in terms of this nonlocality. Two entangled particles are described by a two-particle wave function that can not be factored into the product of two single-particle wave functions. The entangled particles This means that either particle has the same possibility (with calculable probability) of appearing at any particular location. Just as with the single-particle nonlocality, we cannot say where the particles "are." Either one may be anywhere inside the nonlocality volume up to the moment of "collapse" of the two-particle wave function. So far this is what Richard Feynman called the "only mystery" in quantum mechanics. He mistakenly advised you not to try to understand it or visualize it, but information physics will help you to do both, for single particles, such as the two-slit experiment, and for the two-particle Einstein-Podolsky-Rosen thought experiment. When the entangled particles experience a random environmental interaction (described as "decoherence"), or an experimental measurement by an observer, the two-particle wave function "collapses." All the possibilities/probabilities that are not actualized go to zero, just as with the single particle wave function. But now, two particles appear, simultaneously in a special frame in which their center of mass is not moving. In other frames, one may appear to appear before the other. Just as with the single particle, the localization of the two particles can be anywhere there was a possibility. But now fundamental conservation principles constrain their local appearances.The two particles appear simultaneously, usually in a spacelike separation, now disentangled, and symmetrically located about the point of the interaction which entangled them.
Einstein's idea of "local reality" was that events at one point in spacetime could depend only on the values of continuous functions at that point. In a "complete" physical theory all physical variables should be locally determined by his four-dimensional continuous field of space-time. His 1905 light-quantum hypothesis, his 1909 study of wave-particle duality, and above all his 1927 illustration of a spherical wave hitting a screen as a single particle, showed Einstein that things appeared to happen simultaneously over a large distance in space, actions-at-a-distance, he called them. That appeared to violate his special theory of relativity. But these early concerns about nonlocality all involved just one photon or electron. In 1935, he raised another difficulty
In 1935 Einstein and his Princeton colleagues Boris Podolsky and Nathan Rosen proposed their "EPR" thought experiment that implied
Schrödinger wrote to Einstein immediately and explained that the two-particle wave function could not be "separated" (this came to be called "nonseparability," closely related to nonlocality). Schrödinger said they remain entangled until some interaction "disentangles" them. A measurement would be such an interaction. Einstein stubbornly insisted on what he called a "separation principle" (
But Schrödinger understood wave mechanics much better than Einstein. The wave function describes only the possibilities for particle locations (with calculable probabilities). In a two-particle wave function, the possibilities mean either particle can be found anywhere the two-particle wave function is non-zero. We cannot know where either one will be found until we make a measurement. At that moment, the other particle will instantly be located where the principles of conservation of energy, momentum, angular momentum, and spin require it to be. It is only And this means that any measurement that collapses the two-particle wave function measures both particles! It is not possible to measure "one" (now, here) and then the "other one" (far away, later). Because the particles are indistinguishable, either one could be anywhere just before the measurement, exactly as the single particle in Einstein's 1927 presentation (or in the two-slit experiment) can be anywhere just before the measurement. We cannot say that the two particles are separated beyond the possibility of speed-of-light contact before the measurement.
Einstein's Introduction of Asymmetry
Almost every presentation of the EPR paradox begins with something like "Alice observes one particle..." and concludes with the question "How does the second particle get the information needed so that Bob's later measurements correlate perfectly with Alice?" There is a fundamental asymmetry in this framing of the EPR experiment. It is a surprise that Einstein, who was so good at seeing deep symmetries, did not consider how to remove the asymmetry. Consider this reframing: Alice's measurement collapses the two-particle wave function. The two indistinguishable particles simultaneously appear at locations in a space-like separation. The frame of reference in which the source of the two entangled particles and the two experimenters are at rest is a special frame in the following sense. As Einstein knew very well, there are frames of reference moving with respect to the laboratory frame of the two observers in which the time order of the events can be reversed. In some moving frames Alice measures first, but in others Bob measures first. If there is a special frame of reference (not a preferred frame in the relativistic sense), surely it is the one in which the origin of the two entangled particles is at rest. Assuming that Alice and Bob are also at rest in this special frame and equidistant from the origin, we arrive at the simple picture in which any measurement that causes the two-particle wave function to collapse makes both particles appear simultaneously at determinate places with fully correlated properties (just those that are needed to conserve energy, momentum, angular momentum, and spin).
In the two-particle case (instead of just one particle making an appearance), when either particle is measured, we know instantly those properties of the other particle that satisfy the conservation laws, including its location equidistant from, but on the opposite side of, the source, and its other properties such as spin. We can also ask what happens if Bob is not at the same distance from the origin as Alice. This introduces a positional asymmetry. But there is still no time asymmetry from the point of view of the two-particle wave function collapse. When Alice detects the particle (with say spin up), at that instant the other particle also becomes determinate (with spin down) at the same distance on the other side of the origin. It now continues, in that determinate state, to Bob's measuring apparatus.
Einstein asked whether a particle has a determinate position just before it is measured. It does not, but we can say that before Bob's measurement the electron spin he measures was determined from the moment the two-particle wave function collapsed. The two-particle wave function describing the indistinguishable particles cannot be separated into a product of two single-particle wave functions. When either particle is measured, they both become determinate. Normal | Teacher | Scholar |