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Common Cause
We explore the idea that simultaneous measurement outcomes at two entangled particles widely separated in space can be
This idea opposes the standard view, developed over five decades by Albert Einstein, that some kind of faster-than-light
In his later years, Einstein famously called two-particle entanglement "spooky action at a distance" (
We shall critically examine the details of entanglement state preparations and the experimental data from particle measurements in
The first three (from the 1930's to 1960's) were thought ("
The first was Einstein's theoretical work starting in 1905 and culminating in the 1935 EPR paper, his most cited work and the touchstone for all subsequent research on entanglement. David Bohm's proposal in the 1950's that
The fourth and the sixth kind of experiments were Our fifth kind of entanglement study was the very popular thought experiment of David Mermin in the 1980's.
Although our focus is on experiments and their data correlations, we will also describe the mathematical quantum theory, the wave functions, superpositions or We will also indicate the initial symmetry, physical condition, or conservation laws in the entangled state preparations that may be a "common cause" for the subsequent measurements of correlated events. We cannot provide any physical mechanism or interactions between two particles that maintains the total angular momentum from moment to moment as they travel from state preparation to distant measurements, just as quantum mechanics cannot observe and describe the motions and interactions of electrons in and between different shells or orbitals in atoms and molecules. But we might say that the property of total electron spin zero is "local" in the sense that it is traveling along with each particle just as David Bohm's "hidden variables" or David Mermin's "instruction sets" were thought to do. And we can criticize the claim that the three components of spin angular momentum must exist and be defined in all spatial directions, x, y, and z (which is impossible), in order for experiments to find the spins opposite in all directions when measured. Finally, we will trace the history in quantum mechanics of the most powerful common cause of all, the idea that every physical event can be traced back to a chain of causes and effects that goes back to the creation of the universe.
1) Einstein and the EPR paradox.
Einstein first described In 1933, shortly before he left Germany to emigrate to America, Einstein attended a lecture on quantum electrodynamics by Léon Rosenfeld. Keep in mind that Rosenfeld was perhaps the most dogged defender of the Copenhagen Interpretation. After the talk, Einstein asked Rosenfeld, “What do you think of this situation?” Suppose two particles are set in motion towards each other with the same, very large, momentum, and they interact with each other for a very short time when they pass at known positions. Consider now an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum: then, from the conditions of the experiment, he will obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the first particle, he will be able tell where the other particle is.
It is most unfortunate that Einstein did not explain that measuring the momentum of the first particle allows us to deduce the momentum of the second particle because of the The same conservation principle explains, as Einstein says, "If, however, he chooses to measure the position of the first particle, he will be able to tell where the other particle is."
If Einstein had called this ability "
Einstein and colleagues Boris Podolsky and Nathan Rosen, proposed in 1935 a paradox (known by their initials as EPR or as the Einstein-Podolsy-Rosen paradox) to exhibit internal contradictions in the new quantum physics. They hoped to show that quantum theory could not describe certain intuitive "elements of reality" and thus was incomplete. They said that, as far as it goes, quantum mechanics is
Einstein was correct that quantum theory is " The most that can be said is that the particle can be found anywhere the probability amplitude is non-zero. This was the core idea of Einstein's claim of "incompleteness." For Bohr to deny this and call quantum mechanics "complete" was just to play word games, which infuriated Einstein.
Einstein was also correct that indeterminacy makes quantum theory an irreducibly
Einstein and his colleagues Erwin Schrödinger, Max Planck, (later David Bohm), and others hoped for a return to deterministic physics, and the elimination of mysterious quantum phenomena like the
What happens according to the information interpretation of quantum mechanics is an instantaneous change in the information about probabilities (actually complex probability amplitudes). Nothing physical (matter or energy) is moving anywhere.
As we've seen, Einstein had been bothered by "nonlocal" phenomena between a light quantum and its light wave (function) since his 1905 photoelectric paper. But this phenomenon was even more clearly exhibited in EPR experiments as the apparent transfer of something physical, an "action," from one particle to another particle faster than the speed of light.
The 1935 paper was based on Einstein's 1933 question to Leon Rosenfeld about two material particles fired in opposite directions from a central source with equal velocities. He imagined them starting at t to _{1}t the particles are in contact with one another.
_{1} + Δt
After the particles are measured and become entangled at _{12} that is not the simple product of two one-particle wave functions Ψ_{1} and Ψ_{2}.
Einstein said that at a later time without measuring it explicitly. And this is correct, just as after the collision of two billiard balls, measurement of one ball tells us exactly where the other one is due to conservation of momentum. But this is not "action at a distance." It is more nearly "knowledge at a distance."
Note that Einstein is
This idea of something measured in one place "influencing" measurements far away challenged what Einstein thought of as "local reality." Einstein thought that when the particles moved far enough apart they could be treated as
But Erwin Schrödinger quickly replied to the EPR paper, telling Einstein that his "separation principle" (
Ψ
_{12} ≠ Ψ_{1A} Ψ_{2B}
Instead, particles A and B are each
Ψ
_{12} = 1/√2 (Ψ_{1A} Ψ_{2B}) + 1/√2 (Ψ_{2A} Ψ_{1B})
Quantum mechanics says that the particles are not in "pure" quantum states, but a "mixture" of two states, which maintain the coherent phase relations that allow them to interfere with one another. Schrödinger used Paul Dirac's 1926 principle of superposition and John von Neumann's 1932 motion of "mixed states" in the "density matrix" to create two of the most popular and controversial ideas in quantum mechanics.
First, Schrödinger described the two particles as "entangled" at their first encounter. He called it
Second, Schrödinger introduced his famous cat, claiming it is in a
It was at this point in quantum history that the most controversial two-particle equation above appeared that combines the ontological chance that Einstein discovered in 1916 with the idea that one quantum state can be described as in a
In Dirac's theory, the squared coefficients of the two states give us the
The equation combines quantum
Note that the equation does not describe two These quantum mechanical wave functions, solutions to Schrödinger’s equations of motion (which replace Newton's equations of motion in classical mechanics), were thought by Schrödinger to be describing matter or energy, photons for light waves, mass and perhaps electric charge for electrons.
But in the quantum mechanics of Heisenberg, Jordan, Born, and Dirac the wave functions became “probability amplitudes,” whose absolute squares predict the Let’s look at the equation in its simplest form that describes the superposition state of Schrödinger’s cat.
| Cat > = ( 1/√2) | Live > + ( 1/√2) | Dead >
Although this equation predicts interference between the cat states, such interference is never seen in cats, though it has been measured in surprisingly large macroscopic objects. Nevertheless, squaring the coefficients 1/√2 tells us that there is 50% chance of finding such a cat in either the live or dead state, i.e., which is confirmed in principle.
Cats = (1/2) Live + (1/2) Dead.
Let's see how this simple equation also describes the two-slit experiment.
Ψ = ( 1/√2) | Left > + ( 1/√2) | Right >
The wave function past the two slits is a linear combination or Note that whichever slit the particle passes through (and it must go through just one, because a quantum particle cannot become two, violating conservation of mass and/or energy), the probabilities of finding it on the screen are determined by the two-slit superposition. If a particle was detected passing through the left slit, or if the right slit were closed, the interference pattern would depend only on that slit's wave function | Left >.
Given that the double-slit interference appears even if
This is the deepest metaphysical mystery in quantum mechanics. How can an abstract
Why interference patterns show up when both slits are open, even when particles go through just one slit, though we cannot know which slit or we lose the interference
Einstein criticized the collapse of the wave function as "instantaneous-action-at-a-distance." This criticism resembles the criticisms of Newton's theory of gravitation. Newton's opponents charged that his theory was "action at a distance" and instantaneous. Einstein's own field theory of general relativity shows that gravitational influences travel at the speed of light and are mediated by a gravitational field that shows up as curved space-time.
For Einstein, fields like gravitation and electromagnetism are " When a probability function collapses to unity in one place and zero elsewhere, nothing physical, neither matter nor energy, is moving from one place to the other. Only information changes. For a detailed history of Einstein's concerns about single-particle nonlocality over the thirty years before EPR, see this page.
2) David Bohm and hidden variables.
In our second kind of "thought experiment," David Bohm replaced Einstein's separating particles with a hydrogen molecule disassociating into two hydrogen atoms, each with ℏ/2 of spin angular momentum.
Instead of measuring linear momentum, Bohm proposed using two hydrogen atoms that are prepared in an initial state of known total spin angular momentum zero (the H
We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore
Note that when Bohm says "
Note also that our superposition equation for the two particles predicts a 50% chance that the first particle will be spin up ( ψ (2)) and a 50% chance of the reverse, that the first particle will be spin down (_{-}ψ (1)) and the second will be up (_{-}ψ (2)). In either case the total spin is always _{+}certain to be conserved as zero.
Next note that while the total spin is
Finally note that these amazing predictions of outcomes We can ask ourselves whether our first thought experiment (EPR) really needed some mechanism, some interaction, as Einstein feared, to keep the particles moving symmetrically away from their center? In the absence of an external asymmetric force, their motions are mirror images, preserving their original symmetry and their conservation of total linear momentum. If linear momentum can be conserved (by symmetry) without instantaneous interactions, isn't conservation of spin angular momentum a much more plausible explanation than impossible faster-than-light interactions? In 1964, John Bell made a study of EPR and David Bohm's suggestion of local hidden variables that could provide a mechanism to explain entanglement. Bell proposed an experiment using photons and polarizers that measures the angular dependence of the falloff in perfect correlations when experimenters at A and B (usually called Alice and Bob) don't set their polarizers at the same angle (which we argue is needed to preserve the symmetry of the initial entanglement and the conservation of critical properties like spin angular momentum). Correlations are perfect when they measure at the same (pre-agreed-upon) angle. When their polarizer angles differ by ninety degrees, all correlations are lost.
At intermediate angle differences θ, correlations diminish proportional to the square of the cosine of their angle difference - cos |