How a "Hidden" Constant of the Motion Can Explain Einstein's "Spooky Action at a Distance"
Preface
Albert Einstein had concerns for decades about apparently instantaneous interactions or "influences" between quantum particles or fields that he thought had been long "separated." He called it "spooky action at a distance."
Today hundreds of experiments with two entangled particles confirm that measurements made at arbitrarily large separations show the two particles' properties are perfectly correlated, even though the individual particle properties are found to be random.
Such widely separated events, happening simultaneously in a frame of reference that includes the original entanglement preparation, give the appearance of one event influencing the other at speeds much greater than the velocity of light. Such events are called "nonlocal."
Entanglement experiments are usually described with two widely separated experimenters, Alice and Bob. Between them is an apparatus that prepares entangled particles and ejects them in opposite directions heading towards the measurement devices, which can be arranged at different angles. As Alice and Bob look in the direction of the initial entanglement, they can agree on angles called "up" and "down," as well as "left" and "right" which are perpendicular to the "up-down" directions.
Each of the measuring devices can be freely rotated from the "up" direction, which we can call zero degrees, turning it clockwise around 90 degrees to the "right" direction," on to 180 degrees in the "down" direction, then to 270 degrees in the "left" direction, and finally ninety more degrees points the device back to the "up " direction.
John Bell's famous theorem made predictions about the results Alice and Bob should measure at various angle settings. But what do experimental results show? We can describe the results as "up" or "down," or plus or minus, or with digital bit sequences 1 or 0.
Alice's measurement sequences appear to her to be completely random, like this, with approximately equal numbers of 1's and 0's, approaching equality for longer bit sequences.
00010011011110101100011011000001
And Bob's sequence looks to him to be equally random, with 1's and 0's approaching 50/50.
11101100100001010011100100111110
But amazingly, should Alice send her bit sequence to Bob for comparison, he discovers that when he lines the bit strings up with one another, they are perfectly anti-correlated. Where Alice measured a 1, Bob measures 0, and vice versa. How can this be?
00010011011110101100011011000001
11101100100001010011100100111110
[Note that these random but perfectly correlated bit sequences are perfect for use as one-time pads for encrypting coded messages. And the sequences have not been "communicated" or "distributed" over an ordinary communication channel. They have been created independently and locally at Alice and Bob in a secure way that is invulnerable to eavesdroppers, solving the problem of quantum key distribution (QKD).]
Einstein knew that quantum mechanics predicts a random, uncertain outcome for the measurement of an individual particle, perhaps because of Werner Heisenberg's "uncertainty principle," or because of Paul Dirac's "projection postulate," which says the particle is statistically found in a state proportional to the probability for each possible state.
But if the two particle states are individually random, how can they be found jointly correlated?
Many physicists and philosophers of science who believe the laws of physics are deterministic, so past events completely determine the one possible future, think that a "common cause" from the past creates the correlations. But this is not an explanation.
Other scientists, starting with David Bohm in 1952, think there are "local hidden variables," traveling along with each particle, carrying "instructions" that tell each particle how to correlate with the other.
John Bell's test experiments were said to distinguish between "local" and "nonlocal." Bell concluded that if hidden variables do exist, they must be "nonlocal." But no reasonable explanation for how such variables work has been given.
Our proposed explanation for events that are independently random but are perfectly correlated jointly is a "condition" or "constraint" that travels along with the particles from their initial entanglement, so it is a local phenomenon.
The "condition" is a consequence of a deep principle that is fundamental to both quantum physics and classical physics. It's the law of conservation of certain quantities, like mass, charge, linear momentum, and angular momentum. As Emmy Noether taught us, conservation laws are the consequence of geometrical symmetries.
In our case, it is the spherical symmetry of the wave function for two particles in a so-called "singlet" state. The two 1s electrons in the filled shell of the helium atom are in such a spherically symmetric state. The atomic orbital is a sphere with the electrons equally likely to appear at any point, though if both were found they would most likely be located opposite one another because of their mutual repulsion. The total spin angular momentum is zero. Electron spins are exactly opposite at all times.
The two hydrogen atoms in the hydrogen molecule ground state 1Σg+ are rotationally symmetric about the molecule axis. David Bohm in 1952 started with a hydrogen molecule that was dissociating into two hydrogen atoms.
As the hydrogen atoms "separate," the quasi-molecular wave function remains rotationally symmetric around the molecular axis. The total spin angular momentum remains zero at all times, unless the atoms are disturbed by the environment or a measurement is made.
Our "condition" is this conservation of spin angular momentum equal to zero for all times, up to including the measurements made by Alice and Bob, but if, and only if, their two measurements are made at exactly the same measurement angles, preserving the overall symmetry. This is the case if they agree before their experiments on their free choice of angle in which to measure.
They must also measure at the same time (assuming the initial entanglement is centered between their measurement devices), to ensure they are measuring the correct pair of particles.
Should Alice and Bob measure at different angles, say angles separated by angle θ, they will lose the perfect correlations. Correlations will decline proportional to the square of the cosine of that angle difference, cos2θ. If they measure at right angles to one another, there will be no correlations, since the cosine of 90 degrees is zero.
John Bell predicted a different angular dependence of this falloff in correlations for quantum physics and for local hidden variables. We will discuss his results in chapter X.
Perhaps "hidden constant" is too cute, although it travels along with the particles just as
David Bohm's hidden variables or
David Mermin's "instruction sets" do.
A more modest name is
common cause.
Albert Einstein had concerns for decades about apparently instantaneous interactions or "influences" between quantum particles or fields that he thought had been long "separated." He called it "spooky action at a distance."
Today hundreds of experiments with two entangled particles confirm that measurements made at arbitrarily large separations show the two particles' properties are perfectly correlated, even though the individual particle properties are found to be random.
Such widely separated events, happening simultaneously in a frame of reference that includes the original entanglement preparation, give the appearance of one event influencing the other at speeds much greater than the velocity of light. Such events are called "nonlocal."
Entanglement experiments are usually described with two widely separated experimenters, Alice and Bob. Between them is an apparatus that prepares entangled particles and ejects them in opposite directions heading towards the measurement devices, which can be arranged at different angles. As Alice and Bob look in the direction of the initial entanglement, they can agree on angles called "up" and "down," as well as "left" and "right" which are perpendicular to the "up-down" directions.
Each of the measuring devices can be freely rotated from the "up" direction, which we can call zero degrees, turning it clockwise around 90 degrees to the "right" direction," on to 180 degrees in the "down" direction, then to 270 degrees in the "left" direction, and finally ninety more degrees points the device back to the "up " direction.
John Bell's famous theorem made predictions about the results Alice and Bob should measure at various angle settings. But what do experimental results show? We can describe the results as "up" or "down," or plus or minus, or with digital bit sequences 1 or 0.
Alice's measurement sequences appear to her to be completely random, like this, with approximately equal numbers of 1's and 0's, approaching equality for longer bit sequences.
00010011011110101100011011000001
And Bob's sequence looks to him to be equally random, with 1's and 0's approaching 50/50.
11101100100001010011100100111110
But amazingly, should Alice send her bit sequence to Bob for comparison, he discovers that when he lines the bit strings up with one another, they are perfectly anti-correlated. Where Alice measured a 1, Bob measures 0, and vice versa. How can this be?
00010011011110101100011011000001
11101100100001010011100100111110
[Note that these random but perfectly correlated bit sequences are perfect for use as one-time pads for encrypting coded messages. And the sequences have not been "communicated" or "distributed" over an ordinary communication channel. They have been created independently and locally at Alice and Bob in a secure way that is invulnerable to eavesdroppers, solving the problem of quantum key distribution (QKD).]
Einstein knew that quantum mechanics predicts a random, uncertain outcome for the measurement of an individual particle, perhaps because of Werner Heisenberg's "uncertainty principle," or because of Paul Dirac's "projection postulate," which says the particle is statistically found in a state proportional to the probability for each possible state.
But if the two particle states are individually random, how can they be found jointly correlated?
Many physicists and philosophers of science who believe the laws of physics are deterministic, so past events completely determine the one possible future, think that a "common cause" from the past creates the correlations. But this is not an explanation.
Other scientists, starting with David Bohm in 1952, think there are "local hidden variables," traveling along with each particle, carrying "instructions" that tell each particle how to correlate with the other.
John Bell's test experiments were said to distinguish between "local" and "nonlocal." Bell concluded that if hidden variables do exist, they must be "nonlocal." But no reasonable explanation for how such variables work has been given.
Our proposed explanation for events that are independently random but are perfectly correlated jointly is a "condition" or "constraint" that travels along with the particles from their initial entanglement, so it is a local phenomenon.
The "condition" is a consequence of a deep principle that is fundamental to both quantum physics and classical physics. It's the law of conservation of certain quantities, like mass, charge, linear momentum, and angular momentum. As Emmy Noether taught us, conservation laws are the consequence of geometrical symmetries.
In our case, it is the spherical symmetry of the wave function for two particles in a so-called "singlet" state. The two 1s electrons in the filled shell of the helium atom are in such a spherically symmetric state. The atomic orbital is a sphere with the electrons equally likely to appear at any point, though if both were found they would most likely be located opposite one another because of their mutual repulsion. The total spin angular momentum is zero. Electron spins are exactly opposite at all times.
The two hydrogen atoms in the hydrogen molecule ground state 1Σg+ are rotationally symmetric about the molecule axis. David Bohm in 1952 started with a hydrogen molecule that was dissociating into two hydrogen atoms.
As the hydrogen atoms "separate," the quasi-molecular wave function remains rotationally symmetric around the molecular axis. The total spin angular momentum remains zero at all times, unless the atoms are disturbed by the environment or a measurement is made.
Our "condition" is this conservation of spin angular momentum equal to zero for all times, up to including the measurements made by Alice and Bob, but if, and only if, their two measurements are made at exactly the same measurement angles, preserving the overall symmetry. This is the case if they agree before their experiments on their free choice of angle in which to measure.
They must also measure at the same time (assuming the initial entanglement is centered between their measurement devices), to ensure they are measuring the correct pair of particles.
Should Alice and Bob measure at different angles, say angles separated by angle θ, they will lose the perfect correlations. Correlations will decline proportional to the square of the cosine of that angle difference, cos2θ. If they measure at right angles to one another, there will be no correlations, since the cosine of 90 degrees is zero.
John Bell predicted a different angular dependence of this falloff in correlations for quantum physics and for local hidden variables. We will discuss his results in chapter X.
A conserved physical quantity that does not change as particles move around is called a "constant of the motion." Since we believe it can fulfill the function that David Bohm and John Bell thought their hidden variables could do, we have chosen to call the conserved total spin angular momentum zero a "hidden constant of the motion." One might say it is "hidden in plain sight."
Now there has not been any proposal as to how hidden variables might act on the two particles to produce that "action at a distance" that produces the particles with opposite spins.
Our "hidden constant" does not involve some action to create the opposite spins. As long as there is no asymmetric actions on the particles that would change their continuing conserved states of opposite spin, the "hidden constant" gives us .
.
Our "hidden constant" does not involve any actions to create the opposite spins. It only requires there are no asymmetric actions on the particles that would change their continuing conserved states of opposite spin.
Between 1905 and 1927 Einstein puzzled over "nonlocal," apparently faster-than light, effects between a light quantum (a photon) and its light wave, between a material particle (an electron) and its deBroglie matter wave, then between the abstract Schrödinger "wave function" ψ and a single particle. This is the modern problem of wave-particle duality and the so-called "collapse" of the wave function..
It was not until 1933 that Einstein first considered what he thought were "nonlocal" effects between two particles that had collided and separated with equal and opposite velocities. Measuring the position of one would tell us the position of the other, he said. This is true because the two particles have travelled equal distances from the collision, but not because they are still interacting, as Einstein thought. It is the conservation of total linear momentum they puts them at the same distance from the collision. We have "knowledge at a distance."
In 1935, Einstein's colleagues Boris Podolsky and Nathan Rosen suggested in their EPR paradox paper1 that quantum mechanics is not "complete" and that additional parameters might be needed to complete it.
What Einstein many years later called "spooky actions at a distance" conflicted with his view of physical reality as limiting interactions to "local" forces or fields, with any interactions proceeding at the speed of light or less.
But Erwin Schrödinger promptly told Einstein that the "entangled" two-particle wave function ψ12 cannot be "separated" into the product of independent single-particle wave functions ψ1 and ψ2 without a "disturbance" or a "measurement."
In 1952 David Bohm reformulated the two material particles of EPR as two atoms in a hydrogen molecule separating as the molecule dissociates. Bohm replace the continuous properties of position and momentum of EPR with the discrete ℏ/2π spins of the two atoms.
Until the two-particle wave function ψ12 has decohered and the particles have been disentangled by a disturbance or a measurement, Schrödinger said the particles are in a linear combination or "superposition" of states, according to the modern transformational theory of quantum mechanics formulated by Paul Dirac, which combined the 1925 matrix mechanics of Werner Heisenberg and the 1926 wave mechanics of Erwin Schrödinger.
ψ12 = 1/√2(ψ+1 ψ-2 - ψ-1 ψ+2) (1).
Measurements will find particles randomly in either the ψ+1 ψ-2 state or the ψ-1 ψ+2 state.
Quantum mechanics says that the resulting states of the two particles "did not exist" before the measurement. Dirac says the particles are "projected" into their states probabilistically, statistically, or indeterministically. Heisenberg said the states are "created" by the measurement.
We will demonstrate below that the final measured spin states of the individual particles could not have been determined at their original entanglement, but that the spherical symmetry of the two-particle wave function was determined at that time. And as long as the wave function is not decohered by the environment as it expands and the particles "separate," and as long as the measurements are made at A and B at the same measurement angle to preserve the spherical symmetry, the spin measurements will be opposite and perfectly correlated.
Each particle will be found 50/50 randomly in a + state (a spin-up state) or in a - state (a spin-down state).
But counterintuitively, the fact that two spins are always found opposite one another (either + - or - +, either up-down or down-up) is not random, indeterministic, or uncertain. It is determined because the total spin is conserved to be zero from the time of the particles initial entanglement. The total spin angular momentum of the two particles is zero from the initial entanglement up to and after the measurement, a "constant of the motion," unless the measurements are made asymmetrically, destroying the spherical symmetry of the original two-particle state.
Even in this case the fundamental conservation law for total spin angular momentum is not violated. The asymmetric measurements have acted on the particles differently, and their equal and opposite reactions have been different. Those differences between a particle and its measuring apparatus exactly balance and cancel one another. The total spin angular momentum of particles and measurement apparatuses is still conserved, as it must be.
So total spin angular momentum zero will be found conserved if, and only if, the spin measurements are made in exactly the same direction (detector angle or polarization angle), preserving the symmetry and the lack of any preferred direction at the initial entanglement.
Measurements made at different detector or polarization angles would introduce an asymmetry that reduces the correlations between distant measurements. This falloff in correlations as the angle of measurements between A and B increases is at the heart of John Bell's famous "inequality" theorem, which Bell said could distinguish between local and nonlocal phenomena.
If measurements are made at angles different by θ the total spin of the particles will no longer be zero and the correlations will no longer be perfect. Quantum mechanics predicts that the correlations will be reduced by cos2θ, the square of the cosine of the difference in measurement direction or in polarization angles . This has long been known as the "law of Malus" for crossed polarizers.
In 1952 David Bohm,2 inspired by Einstein's hope for "completeness," proposed "hidden variables" that could travel along with the particles (thus be considered "local") from their initial entanglement to ensure their perfect correlation of properties in any later measurements.
In 1964 John Bell4 proposed a theorem and experimental tests that could confirm (or deny) these "local hidden variables." Critical experimental tests made between 1972 (Stuart Freedman and John Clauser5) and 1982 (Alain Aspect6) denied the existence of local hidden variables by violating the Bell "inequalities."
The Bell-test experiments confirmed the existence of the mysterious "nonlocal" behavior and the failure of what Einstein called his "separability principle (Trennungsprinzip)."
Entanglement is thus a physically "real," if "weird," phenomenon which became the basis for the so-called "second quantum revolution," leading to quantum cryptography, quantum teleportation, and quantum computing.
The 2022 Nobel Prize in Physics was awarded to Clauser, Aspect, and Anton Zeilinger for their experiments confirming entanglement. The Nobel Prize committee mistakenly said that these experiments mean that "quantum mechanics cannot be replaced by a theory that uses hidden variables." But the work of John Bell and the dozens of experimenters who confirmed his predictions only denied the existence of "local" hidden variables. We will show that their work confirms the appearance of nonlocality, but not the actual reality of "faster-than light" nonlocal behavior, which is simply impossible according to Einstein's principle of special relativity.
We propose the conservation of spin angular momentum zero in the original spherically symmetric singlet state is a "constant of the motion," necessarily true at all times as long as the particles are not disturbed by the environment. This "constant of the motion" determines that the particle spins will jointly be found opposite one another at the distant points A and B, despite the randomness for each of the individual particles' spin direction when measured. Particle measurements are independently random but joint measurements are determined.
This is not an "instantaneous" interaction or "influence" between the distant particles at the time of measurement, but a pair of interdependent actions that were "simultaneous" in the past in the frame of reference and at the time of the original entangling state preparation. These initial entangling actions can be considered "local," happening inside the atomic or molecular wave functions of the particles.
The "constant of the motion" can be "understood" as a "common cause" coming from the past light cone which explains the perfect correlations found in Bell-test experiments, because of the "condition" on total spin imposed on the particles at their creation. It is not a mechanical model of instantaneous particle "interactions" pushing and pulling, which would violate relativity and is no more useful than Niels Bohr's semi-classical model of electron "orbits" in atoms.
The perfectly correlated opposite spins are also not the specific spin directions created at initial entanglement. Such specific spin directions would only accidentally align with the experimenters' freely chosen directions at A and B. The constant of the motion is the total spin zero state, symmetric in all directions, which means as long as the same measurement direction is freely chosen at A and B (by prior agreement), the particles will be found in perfectly correlated opposite states!
Exactly how the bit strings of data at A and at B are indeterministically random, even as the combined A and B results appear to be deterministically correlated, is only explained by the "collapse" of the "superposition" of states above. Whether spin is up or down for each particle did not exist before the measurements. The particles spins are individually random but jointly determined. That is the core of the EPR paradox.
Let's look more closely at the extraordinary mathematical predictions of quantum mechanics. Equation 1 can be written more simply with Dirac's <bra| and |ket> notation as
ψ12 = 1/√2 (| + > | - > +/- | - > | + >) (2).
[The plus sign in the middle applies to photons and other bosons which are symmetric under interchange of identical particles and the minus sign applies to electrons and other fermions that are anti-symmetric (they change sign) when interchanged.]
We square the coefficient 1/√2 to predict that 1/2 of the measurement outcomes will be | + > | - > and the other half will be | - > | + >.
The individual particles are randomly | + > or | - >, but the total spin is determined to be always zero, always opposite spins, whether | + > | - > or | - > | + >.
Quantum mechanics does not predict outcomes like | + > | + > or | - > | - >. They are never seen in "perfect" experiments, although they may be found in practice because of system noise or other experimental errors. For example, the measurement directions might not be exactly parallel.
Should outcomes like | + > | + > or | - > | - > be observed, they would violate a conservation law, and conservation laws are the most fundamental laws of both quantum physics and classical physics.
The History of Single-particle Nonlocality and Two-particle Entanglement
We argue that the problems of nonlocality and entanglement began with Einstein's photoelectric effect in 1905 and cannot be fully understood by starting thirty years later with the famous Einstein-Podolsky-Rosen paper, as is done by most commentators on entanglement.
Introduction
As Louisa Gilder has written,11 we live in "The Age of Entanglement." Entanglement is the basis of powerful new technologies like quantum cryptography, quantum teleportation, and quantum computing, which Lars Jaeger20 and others call "The Second Quantum Revolution."
Although these technologies are new, the "weirdness" of entanglement is not something new.
We will show that it is partly based on what Richard Feynman called the "central mystery of quantum mechanics," the "general mystery," and the "only mystery" in quantum mechanics, which led Feynman to famously say "nobody understands quantum mechanics."
Feynman's central mystery is the two-slit experiment, the problem of a particle (or its "wave") appearing to go through two slits and "interfering" with itself to produce characteristic diffraction patterns on a distant screen.
Feynman told us we cannot hope to "understand" what is going on in quantum mechanics in terms of processes we have learned from our everyday experiences of classical mechanics. But he said he could describe what is happening for us in the two-slit experiment.
In a similar vein, Paul Dirac argued that the non-intuitive concepts of quantum mechanics, though impossible to understand in terms of continuous classical concepts, could be mastered through long familiarity with them. He wrote...
The new theories, if one looks apart from their mathematical setting, are built up from physical concepts which cannot be explained in terms of things previously known to the student, which cannot even be explained adequately in words at all. Like the fundamental concepts (e.g. proximity, identity) which every one must learn on his arrival into the world, the newer concepts of physics can be mastered only by long familiarity with their properties and uses.
By contrast with Feynman and Dirac, Niels Bohr and Werner Heisenberg, in their Copenhagen interpretation of quantum mechanics, said that the results of quantum measurements must be expressible in classical concepts because it is the language that humans can understand. Sadly, the Copenhagen Interpretation has led to a great deal of confusion.
Following Feynman's and Dirac's suggestion of simply making you familiar with quantum phenomena, we try to describe, with words, diagrams, and a minimum of equations, several essential quantum phenomena...
1) Feynman's "only mystery," the two-slit experiment.
2) Wave-particle duality.
3) Erwin Schrödinger's "wave function" ψ that describes the wave as well as
4) | ψ |2, the absolute square of the wave function,
which first Einstein and later Max Born interpreted as giving us the probabilities of finding the particle at one of its possible locations (which is now the "standard" interpretation of quantum mechanics).
Einstein said this about the wave function...
|ψ|2 expresses the probability that there exists at the point considered a particular particle of the cloud, for example at a given point on the screen.
(Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, G. Bacciagaluppi and A. Valentini, 2009. pp.440-442)
5) the so-called "collapse" of the wave function
which describes the random appearance of a particle at a certain location and with random properties when measured. This indeterminism and probabilistic or statistical aspect of quantum mechanics greatly bothered Einstein, who hoped such properties existed before their measurement.
But we are not there yet. It takes two particles to be entangled.
The two-slit experiment involves a single particle and its wave (function) going through two slits.
Entanglement involves two particles measured when their two-particle wave function ψ12 has spread out in all directions at the speed of light, so that when measured they are most likely to be found a great distance apart, but not yet physically "separated!"
Schrödinger said that a wave function describing two entangled particles must be in what Paul Dirac called a
6) Superposition of two-particle states.
Consider two particles we can number 1 and 2 which can each be in one of two states, + or -.
| ψ
12 > = 1/√2) | 1
+ 2
- > - 1/√2) | 1
- 2
+ >
| ψ
12 > = (1/√2) | + - > - (1/√2) | - + >
∆p∆x = ℏ
We may think about Heisenberg's uncertain position as the dimension in position space where Schrödinger's wave function is non-zero, and the uncertain momentum as the dimension of momentum space where Schrödinger's wave function is non-zero, so the dimension of so-called phase space (position x momentum) as ℏ, and the minimum phase-space volume that can be occupied by two spin 1/2 particles is ℏ3, the volume of the first electron shell in atoms.
We should note that Heisenberg's uncertainty in 1927 was already implicit in Schrödinger's wave function, which Einstein had long interpreted as the probability of finding particles, though he gave full credit to Max Born for the idea, which became known as Born's Rule.
And Einstein's interpretation of quantum theory as probabilistic and quantum experiments as statistical goes back to his 1916 discovery of ontological chance in the emission and absorption of light quanta by atoms. But Einstein called this fundamental randomness a "weakness in the theory" and spent the rest of his life exploring the possibility of a deterministic theory underlying indeterministic quantum mechanics, making him the foremost critic of quantum mechanics, though all his life he was more concerned about nonlocal effects between particles and between particles and their waves.
Finally, we need to describe principles that are fundamental to both quantum mechanics and classical mechanics.
8) Conservation Laws
The principle is the conservation of some fundamental properties of physics, like mass, energy, and momentum. According to Emmy Noether these conserved properties emerge from certain symmetries in space.
Our proposed explanation of entanglement as resulting from the spherical symmetry of the two-particle wave function and the conservation of spin angular momentum as a "constant of the motion" will therefore not be simple, but we hope to make it as clear as possible.
Chapters (of book) Parts (of this web page)
In part 1 we show that Einstein's concerns about nonlocality can be glimpsed as early as his "miracle year" of 1905 in his photoelectric paper. See Photoelectric Effect
In 1909 Einstein described how a high-energy electron could eject a high energy light wave from a metal plate and radiate in all directions. A small part of the wave could hit a second plate and eject an electron with energy almost equal to the original electron, but in a random direction. See Wave-Particle
Einstein drew an illustration of the nonlocality problem on a blackboard at the Solvay conference in 1927. See Einstein at Solvay
He challenged Leon Rosenfeld to explain two-particle nonlocality in 1933. See Einstein at Solvay
This was just two years before Einstein's colleagues Boris Podolsky and Nathan Rosen published the famous "EPR Paradox," the most cited scientific paper in history. In a letter to Max Born in 1947 Einstein described it as "spooky action at a distance." And in his 1949 "Autobiography" he said the measurement of one particle "telepathically changes the real situation" of the second particle.
It is unfortunate that Einstein
In part 2, we discuss the famous EPR paradox of 1935 and Einstein's fears about possible violations of special relativity.
In part 3, we see that Erwin Schrödinger replied to the EPR paper by saying that his two-particle wave function ψ12 would keep the particles "entangled" until a measurement projects them (i.e., collapses the two-particle wave function) into a product of two single-particle wave functions ψ1 and ψ2, only after which the particles would be "disentangled" (decohered) and could then actually "separate," as Einstein insisted.
In part 4, we see how David Bohm in 1952 replaced Einstein's continuous variables of position and momentum with the discrete variables of electron spins, making future experimental measurements practically possible. Bohm also revived Louis de Broglie's "pilot-wave" theory which offered a way of imagining hidden variables that are "guiding" the particles and suggested a new (Bohmian) mechanics that could restore determinism, eliminating quantum indeterminism.
And we can now show that Dirac's principle of superposition of states <1/√2) | 1+2- > + 1/√2) | 1-2+ >, which for Schrödinger led to his famous "cat both dead and alive," can explain both the 50/50 randomness of outcomes and the perfect correlations (or anti-correlations) of spin measurements.
In part 5, we explain why the perfectly correlated electron spins could not have been determined at the time of the initial entanglement preparation, with one spin up and the other down in a particular direction. Some physicists said that the electron spins would have to have been defined in all three dimensions - x-spin, y-spin, and z-spin. But this is impossible. In 1957 Bohm and Aharonov showed that that electron spins could be defined in at most a single direction, and this would not be the random freely chosen direction of the experimenters...
In quantum theory, a difficulty arises, in the interpretation of the above experiment, because only one component of the spin of each particle can have a
definite value at a given time. Thus, if the x component
is definite, then the y and z components are indeterminate and we may regard them more or less as in a
kind of random fluctuation.
"Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky," Physical Review 108, 4, p.1070 (1957)
But having opposite spins in even a single preferred direction would not explain the experimental data!
Had there been a single preferred direction (or angle) for electron spins in the initial entanglement preparation, there is essentially zero chance that Alice and Bob could observe in precisely that (unknown) direction or angle. And observations at angle θ to that initial preferred direction will produce decorrelated results + + and - - proportional to sin2θ.
In part 6, we will show that no preferred direction exists in the initial entanglement preparation. The original entangled state with total spin angular momentum zero is rotationally and spherically symmetric, the same in all directions. The preferred direction of measurement or angle for Alice and Bob is introduced by them, agreed to before they measure! Total spin angular momentum is conserved as zero at all times from the initial entanglement to their measurements (assuming that no environmental interaction disturbs either particle). As long as Alice and Bob now measure in the same plane (same direction, same angle), a new planar symmetry replaces the spherical symmetry and defines the one component of electron spin that Bohm and Aharonov say is allowed. The conservation of total angular momentum zero continues. As long as Alice and Bob measure in the chosen plane (preserving the symmetry), they maintain that conservation of total spin angular momentum. The conserved total spin zero is a candidate for a "hidden constant of the motion."
In part 7, we examine John Bell's theorem predicting how correlations between measurements made at differing angles could distinguish between local hidden variables and quantum mechanics. We show how the correlations would be reduced if Alice and Bob's measurement angles differ by θ degrees. And we will examine Bell's hypothesis about how the falloff of correlations with increasing angle θ is different for local hidden variables (straight line function of θ) and for quantum mechanics (curved cosθ function).
In part 8, we comment on the origin of the flawed idea that something must be traveling from Alice to Bob. Assuming, for the moment, that the entangled particles do not have perfectly correlated opposing spins, what kind of interaction could Alice's measurement produce to bring Bob's spin angular momentum into agreement. Would it not first have to detect the amount of angular deviation? And then what kind of torque could it exert from far away to rotate the remote particle precisely into agreement?
In part 9, we criticize the argument that perfectly opposed spins at the time of the Alice and Bob measurements (and in the arbitrary direction chosen by them) would require that the spins be defined in all three axial directions, which is impossible. When spins are defined in one direction, they are necessarily undefined in the other two directions.
In part 10, we review the work of some physicists, starting with Einstein, who used conservation principles to explain some aspects of entanglement, but without them seeing exactly how conservation of momentum could produce the appearance of Einstein's "spooky action at a distance."
In part 11, we discuss the relation between conservation principles and geometric symmetries. Einstein was normally quite aware of symmetries and he greatly favored theories based on "principles" over those that are "constructed" to fit experimental data.
In part 12, we comment on the idea of "instruction sets" accompanying the particles as "hidden variables." We trace the origin of this idea to Schrödinger's suggestion that the particles seemed to know the answers to all possible questions that would be asked of them when measured. What process could prepare such information and what mechanism could read the information in the "instructions" and then change the electron spin of the distant particle as required?
In part 13, we conclude with a summary of the "hidden constant" hypothesis.
Part 1 Einstein and Nonlocality, from 1905 to 1927
We show that Einstein's concerns about nonlocality can be glimpsed as early as his "miracle year" of 1905 in his light quantum/photoelectric effect paper. Einstein wrote...
How can energy spread continuously over a large volume and later be absorbed in its entirety, without contradicting
his principle of relativity?
Einstein sees this here, but does not say so explicitly until 1927.
The energy
of a ponderable body cannot be subdivided into
arbitrarily many or arbitrarily small parts, while
the energy of a beam of light from a point source
(according to the Maxwellian theory of light or,
more generally, according to any wave theory) is
continuously spread over an ever increasing
volume.
The wave theory of light, which operates with
continuous spatial functions, has worked well in
the representation of purely optical phenomena
and will probably never be replaced by another
theory. It should be kept in mind, however, that
the optical observations refer to time averages
rather than instantaneous values. In spite of the
complete experimental confirmation of the theory
as applied to diffraction, reflection, refraction,
dispersion, etc., it is still conceivable that the
theory of light which operates with continuous
spatial functions may lead to contradictions
with experience when it is applied to the phenomena
of emission and transformation of light.
It seems to me that the observations associated
with blackbody radiation, fluorescence, the
production of cathode rays by ultraviolet light,
and other related phenomena connected with the
emission or transformation of light are more
readily understood if one assumes that the energy
of light is discontinuously distributed in space.
In particular, the photoelectric effect showed discontinuous and discrete light quanta absorbed by single electrons, though it was doubted until the discovery of the Compton Effect in 1923.
In accordance with the assumption to be considered here, the energy of a light ray spreading
out from a point source is not continuously
distributed over an increasing space but consists
of a finite number of energy quanta which are
localized at points in space, which move without
dividing, and which can only be produced and
absorbed as complete units.
In 1909 Einstein described how a high-energy electron could eject a high energy light wave radiating in all directions, a small part of which could hit a plate and eject an electron with energy almost equal to the original electron. In this work, Einstein gave us the first clear picture of wave-particle duality.
In a presentation at the Salzburg conference in September, 1909, Einstein argued that the interaction of radiation and matter involves elementary processes that are not "invertible," a deep insight into the irreversibility of natural processes. While incoming spherical waves of radiation are mathematically possible, they are not practically achievable. Nature appears to be asymmetric in time. He speculated that the continuous electromagnetic field might be made up of large numbers of light quanta - singular points in a field that superimpose collectively to create the wavelike behavior.
Although he could not formulate a mathematical theory that does justice to both the continuous oscillatory waves and the discrete particle pictures, Einstein argued that they could be "fused" and made compatible. This was over a decade before Erwin Schrödinger's wave mechanics and Werner Heisenberg's quantum mechanics. And because gases behave statistically, he knows that the connection between the wave and particles may involve probabilistic behavior.
When light was shown to exhibit interference and diffraction, it seemed almost certain that light should be considered a wave.
The greatest advance in theoretical optics since the introduction of the oscillation theory was Maxwell's brilliant discovery that light can be understood as an electromagnetic process...One became used to treating electric and magnetic fields as fundamental concepts that did not require a mechanical interpretation.
This path leads to the so-called relativity theory. I only wish to bring in one of its consequences, for it brings with it certain modifications of the fundamental ideas of physics. It turns out that the inertial mass of an object decreases by L / c2 when that object emits radiation of energy L...the inertial mass of an object is diminished by the emission of light.
Now Einstein looks for symmetry and equivalent treatment for interchangeable matter and energy.
The energy given up was part of the mass of the object. One can further conclude that every absorption or release of energy brings with it an increase or decrease in the mass of the object under consideration. Energy and mass seem to be just as equivalent as heat and mechanical energy.
Relativity theory has changed our views on light. Light is conceived not as a manifestation of the state of some hypothetical medium, but rather as an independent entity like matter. Moreover, this theory shares with the corpuscular theory of light the unusual property that light carries inertial mass from the emitting to the absorbing object. Relativity theory does not alter our conception of radiation's structure; in particular, it does not affect the distribution of energy in radiation-filled space.
Einstein is about to tell us that the distribution of energy in radiation-filled space may be similar in some respects to the distribution of particles in matter-filled space!
Nevertheless, with respect to this question, I believe that we stand at the beginning of a development of the greatest importance that cannot yet be surveyed. The statements that follow are largely my personal opinion, or the results of considerations that have not yet been checked enough by others. If I present them here in spite of their uncertainty, the reason is not an excessive faith in my own views, but rather the hope to induce one or another of you to deal with the questions considered.
In the kinetic theory of molecules, for every process in which only a few elementary particles participate (e.g., molecular collisions), the inverse process also exists. But that is not the case for the elementary processes of radiation.
Incoming spherical waves (the advanced potential considered by Wheeler and Feynman in 1945) are never observed in nature. Radiation is
irreversible, one of the
arrows of time
In his remarks after the talk, Johannes Stark confirmed that he had observed a single X-ray that traveled as far as ten meters and ejected a similar energy electron from P2.
If the above is really true then, because of the independence of the secondary cathode rays' speed on the primary cathode rays' intensity, we must assume that an electron impinging on P1 will either cause no electrons to be produced at P2, or else a secondary emission of an electron whose speed is of the same order as that of the initial electron impinging on P1. In other words, the elementary process of radiation seems to occur in such a way that it does not scatter the energy of the primary electron in a spherical wave propagating in every direction, as the oscillation theory demands.
That energy is possibly available "somewhere else" is the key idea of nonlocality that Einstein will present in 1927 at the Solvay conference
Rather, at least a large part of this energy seems to be available at some place on P2, or somewhere else. The elementary process of the emission of radiation appears to be directional. Moreover, one has the impression that the production of X-rays at P1 and the production of secondary cathode rays at P2 are essentially inverse processes.
Therefore, the constitution of radiation seems to be different from what our oscillation theory predicts. The theory of thermal radiation has given important clues about this, mostly by the theory on which Planck based his radiation formula...
Planck's theory leads to the following conjecture. If it is really true that a radiative resonator can only assume energy values that are multiples of hν, the obvious assumption is that the emission and absorption of light occurs only in these energy quantities. On the basis of this hypothesis, the light-quanta hypothesis, the questions raised above about the emission and absorption of light can be answered. As far as we know, the quantitative consequences of this light-quanta hypothesis are confirmed. This provokes the following question. Is it not thinkable that Planck's radiation formula is correct, but that another derivation could be found that does not rest on such a seemingly monstrous assumption as Planck's theory? Is it not possible to replace the light-quanta hypothesis with another assumption, with which one could do justice to known phenomena? If it is necessary to modify the theory's elements, couldn't one keep the propagation laws intact, and only change the conceptions of the elementary processes of emission and absorption?
As far as I know, no mathematical theory has been advanced that does justice to both its oscillatory structure and its quantum structure...
Anyway, this conception seems to me the most natural: that the manifestation of light's electromagnetic waves is constrained at singularity points, like the manifestation of electrostatic fields in the theory of the electron. It cannot be ruled out that, in such a theory, the entire energy of the electromagnetic field could be viewed as localized in these singularities, just like the old theory of action-at-a-distance. I imagine to myself, each such singular point surrounded by a field that has essentially the same character as a plane wave, and whose amplitude decreases with the distance between the singular points. If many such singularities are separated by a distance small with respect to the dimensions of the field of one singular point, their fields will be superimposed, and will form in their totality an oscillating field that is only slightly different from the oscillating field in our present electromagnetic theory of light. Of course, it need not be emphasized that such a picture is worthless unless it leads to an exact theory. I only wished to illustrate that the two structural properties of radiation according to Planck's formula (oscillation structure and quantum structure) should not be considered incompatible with one another.
In 1927, Einstein drew an illustration of the nonlocality problem on a blackboard at the fifth Solvay Conference in Brussels, Belgium. Sadly, despite Einstein's two decades of pioneering work on the interaction of photons and electrons, his ideas and concerns were given little attention at this Solvay, even though the conference was dedicated to electrons and photons!
The conference was dominated by papers on the new quantum theory delivered by Louis de Broglie, Niels Bohr, Max Born, and Werner Heisenberg. It is best known for Einstein's after-hours suggestions to Bohr and Heisenberg probing for faults in the uncertainty principle. Accounts of these events have been told largely by the victors (there are no holes in uncertainty) but Einstein has said they often missed or ignored his important point. That point was the nonlocal behavior of a spherical light wave as it collapses to get absorbed by a single electron. This was Einstein's only contribution mentioned in the published proceedings.
Here are the notes on Einstein's original remarks at the conference and Bohr's brief and confused response. They contain much of Einstein's future 1935 EPR paper, except in 1927 only one particle is involved. Entanglement in EPR requires two identical particles.
Notice how Einstein's diagram clearly shows his concerns of over two decades about reconciling a spherical wave (his example is now an electron) and its collapse to being measured at just one point where it appears as a particle. At this point in the history of quantum mechanics, wave-particle duality is seen as the debate between Schrödinger's wave mechanics and Heisenberg's particle mechanics.
MR ElNSTEIN. - Despite being conscious of the fact that I have not entered
deeply enough into the essence of quantum mechanics, nevertheless I want to
present here some general remarks.
One can take two positions towards the theory with respect to its postulated
domain of validity, which I wish to characterise with the aid of a simple example.
Let S be a screen provided with a small opening O, and P a hemispherical
photographic film of large radius. Electrons impinge on S in the direction of the arrows. Some of these go through O, and because of the smallness of O and the speed of the particles, are dispersed uniformly over the directions of the hemisphere, and act on the film.
Both ways of conceiving the theory now have the following in common. There are de Broglie waves, which impinge approximately normally on S and are diffracted at O. Behind S there are spherical waves, which reach the screen P and
whose intensity at P is responsible [massgebend] for what happens at P.
We can now characterise the two points of view as follows.
The waves give the probability or possibilities for a single electron being found at different locations in an ensemble of identical experiments. A wave does not describe a cloud of electrons as Schrödinger had hoped. Einstein's ensemble theory is the correct interpretation.
1. Conception I. - The de Broglie-Schrödinger waves do not correspond to a single electron, but to a cloud of electrons extended in space. The theory gives no information about individual processes, but only about the ensemble of an infinity of elementary processes.
Quantum theory is not complete in this sense. Representing each particle as a narrow wave packet aiming at the point P is mistaken.
2. Conception II. - The theory claims to be a complete theory of individual processes. Each particle directed towards the screen, as far as can be determined by its position and speed, is described by a packet of de Broglie-Schrödinger waves of short wavelength and small angular width. This wave packet is diffracted and, after diffraction, partly reaches the film P in a state of resolution [un etat de resolution].
According to the first, purely statistical, point of view | ψ |2 expresses the
probability that there exists at the point considered a particular particle of the cloud, for example at a given point on the screen.
If by the same particle, Einstein means that the one individual particle has a possibility of being found at more than one (indeed many) locations on the screen. This is so, but this seems to be conception I?
According to the second, | ψ |2 expresses the probability that at a given instant
the same particle is present at a given point (for example on the screen). Here,
the theory refers to an individual process and claims to describe everything that is governed by laws.
The second conception goes further than the first, in the sense that all the
information resulting from I results also from the theory by virtue of II, but
the converse is not true. It is only by virtue of II that the theory contains the
consequence that the conservation laws are valid for the elementary process; it is only from II that the theory can derive the result of the experiment of Geiger and Bothe, and can explain the fact that in the Wilson [cloud] chamber the droplets stemming from an α-particle are situated very nearly on continuous lines.
Einstein is right that the one elementary process has a
possibility of action elsewhere, but that could not mean producing an
actual second particle. That would contradict conservation laws.
The "mechanism" of action-at-a-distance is simply the disappearance of possibilities elsewhere when a particle is actualized (localized) somewhere.
Note that Einstein is using
Max Born's new rule that the square of the wave function | ψ |
2 gives us the
probabilities of finding a particle somewhere. Born later said he based his rule on an older idea of Einstein's that Einstein called a "ghost field" (
Gespensterfeld) or a "guiding field" (
Führungfeld)
When a particle appears - just one of the multiple nonlocal possibilities becomes actual or localized - at a specific point P , what becomes of the wave that was going off in all other directions? Its "collapse" - the instantaneous going to zero of probabilities - mistakenly appears to Einstein to violate his relativity principle.
In my opinion, one can remove this objection only in the following way, that one
does not describe the process solely by the Schrödinger wave, but that at the same
time one localises the particle during the propagation. I think that Mr de Broglie is
right to search in this direction. If one works solely with the Schrödinger waves,
interpretation II of | ψ |2 implies to my mind a contradiction with the postulate of relativity.
The permutation of two identical particles does not produce two different points in multidimensional (configuration space).
Einstein and Bose discovered the new quantum statistics and indistinguishability. Dirac and Fermi extended it to electrons.
For example, interchange of the two electrons in the filled first electron shell, 1s2, just produces a change of sign for the two-particle wave function.
I should also like to point out briefly two arguments which seem to me to speak against the point of view II. This [view] is essentially tied to a multi-dimensional representation (configuration space), since only this mode of representation makes
possible the interpretation of | ψ |2 peculiar to conception II. Now, it seems to me
that objections of principle are opposed to this multi-dimensional representation.
In this representation, indeed, two configurations of a system that are distinguished only by the permutation of two particles of the same species are represented by two different points (in configuration space), which is not in accord with the new results in statistics. Furthermore, the feature of forces of acting only at small spatial distances finds a less natural expression in configuration space than in the space of three or four dimensions.
(Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, G. Bacciagaluppi and A. Valentini, 2009. p.442)
Bohr's reaction to Einstein's presentation has been preserved. He didn't understand a word! He disingenuously claims he does not know what quantum mechanics is. His response is vague and ends with his ideas on complementarity and the inability to describe a causal spacetime reality.
Does Bohr really not understand? As we have seen, Einstein has been making this general point for many years. And only recently had Bohr started taking Einstein's concept of light quanta seriously.
MR BOHR. I feel myself in a very difficult position because I don't understand what precisely is the point which Einstein wants to [make]. No doubt it is my fault.
As regards general problem I feel its difficulties. I would put [the] problem in [an]other way. I do not know what quantum mechanics is. I think we are dealing with some mathematical methods which are adequate for description of our experiments. Using a rigorous wave theory we are claiming something which the theory cannot possibly give. [We must realise] that we are away from that state where we could hope of describing things on classical theories. [I] Understand [the] same view is held by Born and Heisenberg. I think that we actually just try to meet, as in all other theories, some requirements of nature, but [the} difficulty is that we must use words which remind [us] of older theories. The whole foundation for causal spacetime description is taken away by quantum theory, for it is based on [the] assumption of observations without interference. ... excluding interference means exclusion of experiment and the whole meaning of space and time observation ... because we [have] interaction [between object and measuring instrument] and thereby we put us on a quite different standpoint than we thought we could take in classical theories. If we speak of observations we play with a statistical problem There are certain features complementary to the wave pictures (existence of individuals). ...
The saying that spacetime is an abstraction might seem a philosophical triviality but nature reminds us that we are dealing with something of practical interest. Depends on how I consider theory. I may not have understood, but I think the whole thing lies [therein that the] theory is nothing else [but] a tool for meeting our requirements and I think it does.
(Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, G. Bacciagaluppi and A. Valentini, 2009. pp.440-442)
Twenty-two years later, in his contribution to the Schilpp memorial volume on Einstein, Bohr had no better response to Einstein's 1927 concerns. But he does remember vividly and provides a picture of what Einstein drew on the blackboard.
Here is Bohr's 1949 recollection of Einstein's 1927 presentation:
At the general discussion in Como, we all missed the presence of Einstein,
but soon after, in October 1927, I had the opportunity to meet him in Brussels
at the Fifth Physical Conference of the Solvay Institute, which was devoted
to the theme "Electrons and Photons."
Note that they wanted Einstein's reaction to their work, but actually took little interest in Einstein's concern about the nonlocal implications of quantum mechanics, nor did they look at his work on electrons and photons, despite the conference title.
At the Solvay meetings,
Einstein had from their beginning been a most prominent figure, and several
of us came to the conference with great anticipations to learn his reaction
to the latest stage of the development which, to our view, went far in
clarifying the problems which he had himself from the outset elicited so
ingeniously. During the discussions, where the whole subject was reviewed
by contributions from many sides and where also the arguments mentioned
in the preceding pages were again presented, Einstein expressed, however,
a deep concern over the extent to which a causal account in space and time
was abandoned in quantum mechanics.
To illustrate his attitude, Einstein referred at one of the sessions
to the simple example, illustrated by Fig. 1, of a particle (electron or
photon) penetrating through a hole or a narrow slit in a diaphragm placed
at some distance before a photographic plate.
On account of the diffraction of the wave connected with the motion
of the particle and indicated in the figure by the thin lines, it is under
such conditions not possible to predict with certainty at what point the
electron will arrive at the photographic plate, but only to calculate the
probability that, in an experiment, the electron will be found within any
given region of the plate.
Bohr's labels for points A and B are helpful. The "
nonlocal" effects at point B are just that the
probability of an electron being found at point B goes to zero instantly (not an "action at a distance") when an electron is localized at point A
("Discussion with Einstein," by Niels Bohr, Albert Einstein, Philosopher-Scientist, P. A. Schilpp, 1949. pp.211-213)
Although Bohr seems to have missed Einstein's point completely, Werner Heisenberg at least came to explain the 1927 presentation well. In his 1930 lectures at the University of Chicago, Heisenberg presented a critique of both particle and wave pictures, including a new example of nonlocality that Einstein had apparently developed since 1927. It includes Einstein's concern about "action-at-a-distance" that might violate his principle of relativity, and Heisenberg anticipates the Einstein-Podolsky-Rosen paradox. Heisenberg wrote:
In relation to these considerations, one other idealized experiment (due to Einstein) may be considered. We imagine a photon which is represented by a wave packet built up out of Maxwell waves. It will thus have a certain spatial extension and also a certain range of frequency. By reflection at a semi-transparent mirror, it is possible to decompose it into two parts, a reflected and a transmitted packet. There is then a definite probability for finding the photon either in one part or in the other part of the divided wave packet. After a sufficient time the two parts will be separated by any distance desired; now if an experiment yields the result that the photon is, say, in the reflected part of the packet, then the probability of finding the photon in the other part of the packet immediately becomes zero. The experiment at the position of the reflected packet thus exerts a kind of action (reduction of the wave packet) at the distant point occupied by the transmitted packet, and one sees that this action is propagated with a velocity greater than that of light. However, it is also obvious that this kind of action can never be utilized for the transmission of signals so that it is not in conflict with the postulates of the theory of relativity.
(The Physical Principles of the Quantum Theory, W. Heisenberg, 1930. p.39)
Part 2 EPR, Entanglement, and Suspected Violations of Special Relativity
Just two years before his colleagues Boris Podolsky and Nathan Rosen published the famous "EPR Paradox," the most cited scientific paper in history, in 1933 Einstein challenged Leon Rosenfeld to explain two-particle nonlocality. You will see that EPR is a version of this challenge to Rosenfeld.
Shortly before Einstein left Germany to emigrate to America,
Einstein attended a lecture on quantum electrodynamics by Leon
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.
(Niels Bohr, His Life and Work as seen by His Friends and Colleagues, 1967, S. Rozental, pp.128-129)
We can diagram a simple case of Einstein’s question as follows after the particles have interacted and separate from the center. We use electrons instead of Einstein's generic particles, an anachronism introduced by
David Bohm in 1952.
Recall that it was Einstein who discovered in 1924 the
identical nature,
indistinguishability, and
interchangeability of some quantum particles. He found that identical particles are not
independent, altering their quantum statistics.
Note the anachronism of electrons as Einstein's generic particles. It was
David Bohm in 1952 who proposed that Einstein's EPR problem use electrons. Today many if not most accounts of the EPR paradox describe it with electrons. Many mistakenly say that Einstein was describing electrons.
After the particles interact at
t1, quantum mechanics describes them with a single two-particle wave function that is not the product of
independent single-particle wave functions. In the case of electrons, which are indistinguishable and interchangeable particles, it is not proper to say electron 1 goes this way and electron 2 that way. (Nevertheless, it is convenient to label the particles, as we do in the illustration.)
Einstein then asked Rosenfeld, “How can the final state of the second
particle be influenced by a measurement performed on the first
after all interaction has ceased between them?” This was the germ
of the
EPR paradox, and ultimately the paradoxical problem of two-particle
entanglement.
Why does Einstein question Rosenfeld and describe this as an “influence,” suggesting an “action-at-a-distance?”
One Einstein concern could see a paradox in the context of Rosenfeld’s
Copenhagen
Interpretation, since the second particle is not itself measured and
yet we know something about its properties, whereas the Copenhagen Interpretation
says we
cannot know any properties without an explicit measurement...
Einstein was clearly correct to tell Rosenfeld that at a later time
t2, a measurement of one particle's position would instantly establish the position of the other particle -
without measuring it. Einstein was implicitly using the
conservation of linear momentum to calculate (and to know) the position of the second particle.
Another of Einstein's concerns may have been Werner Heisenberg's
uncertainty principle. It suggests that knowing the exact position of the second particle is not possible. In any case, if the second particle were to be measured, it should be found in a range of positions Δx ≥ h/2πΔp. And this might contradict the conservation of linear momentum, as Nathan Rosen appears to have considered in the later EPR paper.
Two years later, reacting to EPR, Schrödinger described two such particles as becoming "entangled" (
verschränkt) at their first interaction, so "nonlocal" phenomena are also known as "quantum entanglement."
Although conservation laws are rarely cited as the explanation, we argue that they are the physical reason that entangled particles
always produce correlated results for all properties, linear momentum, angular momentum, and energy. If the results were not always correlated, the implied violation of a fundamental conservation law would cause a much bigger controversy than entanglement itself, as puzzling as that phenomenon is.
The hypothesis that the conservation principle gives us a "hidden constant" of the motion like the above linear momentum (perhaps three hidden constants to include angular momentum and energy?), can explain the above results.
In his 1935
EPR paper, Einstein cleverly introduced
two particles instead of one, and in the same year Erwin Schrödinger would create a two-particle wave function that describes both particles. EPR described the particles as widely separated, one "here" and measurable "now" and the other distant and to be measured "later."
Here we must explain the
asymmetry that Einstein mistakenly introduced into a perfectly symmetric situation, making entanglement such a mystery.
The two entangled particles must normally be measured at the same time (synchronously) to ensure that just one entangled pair is involved. This means the measuring devices are placed equidistant (symmetrically) from the central location of initial entanglement. Most accounts of entanglement say that Alice measures "first" and then must interact instantaneously with Bob's particle to align its spin with hers. This is a false asymmetry and an impossible interaction, as it is a violation of special relativity.
At creation, the wave function of the entangled pair is rotationally and spherically symmetric.
The measurement devices must be arranged symmetrically. in the same plane with the origin. If the measurements differ by an angle θ, the perfect correlations would be reduced by cos
2θ.
The normal interactions between the separating particles (electromagnetic forces) do not exceed the speed of light and have the translational, rotational, and temporal invariance symmetries needed to conserve the total spin zero (as well as total linear momentum zero and total energy) created at initial entanglement.
All these symmetries are essential to creating perfectly correlated outcomes.
What could have led Einstein to see an asymmetry as an
instantaneous action a distance. In his 1905 photoelectric study it was likely the view that the light wave was radiation energy spread out in space that had to collect itself into the photoelectron. In his 1909 wave-particle duality electron article, it was likely the view shared by Schrödinger that the wave was spread out electric charge. It lies behind the current concern with the
collapse of the wave function.
The EPR paradox has been reframed since David Bohm in 1952 as two entangled electrons instead of generic particles conserving their total linear momentum zero. In this case, the particles are identical, indistinguishable, and interchangeable, with indeterminate positions according to the uncertainty principle.
We can diagram a simple case of indistinguishable electrons separating from their point of entanglement to be measured at a later time when they are far apart.
As we saw in part 1, this 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." In 1947, Einstein called it "
spukhaft Fernwirkung" or "spooky action at a distance." And in his 1949 "Autobiography" he said the measurement of one particle "telepathically changes the real situation" of the second particle. From our standpoint of information philosophy, it is better to think of it as "knowledge-at-a-distance." And according to our hypothesis of a "hidden constant of the motion," the measurement of one particle sends no signal, telepathic or otherwise, to the second particle. The electrons produce perfectly correlated outcomes because they conserve their total spin angular momentum as zero.
As we also saw, Einstein had considered nonlocal phenomena as early as 1905 and objected publicly to nonlocal phenomena at the Solvay Conference of 1927, when he criticized the
collapse of the wave function as "instantaneous-action-at-a-distance." He said that the probability wave | ψ |
2 could not act simultaneously at different places on the screen, which are in a "space-like" separation, without violating his theory of relativity.
Einstein's criticism resembles the criticisms made of Newton's theory of gravitation. Newton's opponents charged that his theory was instantaneous "action at a distance." Einstein's new 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. An allowable "action-at-a-distance" is one that is caused by "local" events, those in its past light-cone.
Both gravitation and electromagnetism are
field theories, in which physical variables are functions of the four "local" space-time coordinates of Einstein's theory of relativity. Any disturbance at one point in the field can only "influence" another distant point by propagating to that point at the speed of light or less. But the quantum-mechanical wave-function is different. It is neither matter nor energy, nothing physical, only information and a "probability amplitude" whose square gives us probabilities and intensities of quantum phenomena.
Note that when a probability function collapses to unity in one place and zero elsewhere, nothing physical, no matter or energy, is moving from one place to another. When the nose of one horse crosses the finish line, its probability of winning goes to certainty, and the finite probabilities of the other horses, including the one in the rear, instantaneously drop to zero. This happens faster than the speed of light, since the last horse is in a "space-like" separation.
Quantum theory, especially
Paul Dirac's "transformation theory," describes the two electrons as in a
superposition of states, either state equally likely to be randomly found by an experiment. The first state |
+ - > is spin up ( + ) and spin down ( - ), the second |
- + > is spin down ( - ) and spin up ( + ),
| ψ12 > = (1/√2) | + - > - (1/√2) | - + >
The coefficients 1/√2, when squared, tell us the probabilities of either state being found is 1/2.
Dirac's
superposition) can provide only the
probabilities of finding the entangled system in either the |
+ - > state or the |
- + > state. Quantum mechanics does not and cannot describe the paths or the spins of the individual particles as the wave function expands at the speed of light. Note that should measurements result in |
+ + > or |
- - > state, that would violate the conservation of angular momentum. We call the conserved total spin zero a
"hidden constant of the motion". It is the implicit reason why David Bohm, John Bell, and many others say that when one particle is measured spin-up, we instantly know the other must be spin-down.
Part 3 Erwin Schrödinger, Entanglement, and his Two-Particle Wave Function
Erwin Schrödinger replied to the EPR paper by telling Einstein that his two-particle wave function ψ
12 would keep the particles coherently "entangled" until a measurement projects them into a product of single-particle wave functions ψ
1 and ψ
2. We show that the Schrödinger/Dirac
superposition of states <1/√2) |
1+2- > - 1/√2) |
1-2+ > can explain both the 50/50 randomness of outcomes and the perfect correlations (or anti-correlations) of measurements, provided the spin angular momentum is conserved.
Schrödinger said that the entangled particles have jointly shared properties that cannot be changed independently, until after they have become disentangled, decoherent, and independent of one another. Though Schrödinger did not mention it, one of those properties is total linear momentum, key to the EPR paper.
But like Einstein's other reactions to quantum mechanics, he could not accept that particles he regarded as
separated could change their properties simultaneously. Einstein did not
explicitly recognize that the joint properties of the two particles would need to obey conservation laws, though he had
implicitly used the conservation of linear momentum in 1933 and 1935, when he said the second particle position must be symmetrically placed.
Conservation laws are the consequence of extremely deep properties of nature that arise from simple considerations of symmetry. We regard these laws as "cosmological principles." Physical laws do not depend on the absolute place and time of experiments, nor their particular direction in space. Conservation of linear momentum depends on the translation invariance of physical systems, conservation of energy the independence of time, and conservation of angular momentum the invariance under rotations. Conservation laws are the consequence of symmetries, as explained by
Emmy Noether.
Recall that the EPR experiment (Bohm version) starts with two electrons (or photons) prepared in an entangled state that is a mixture of pure two-particle states, each of which conserves the total angular momentum and, of course, conserves the linear momentum as in Einstein's original EPR example. This information about the linear and angular momenta is established by the initial state preparation (a measurement).
Quantum mechanics describes the probability amplitude wave function
Ψ12 of the two-particle system as in a
superposition of two-particle states. It is
not a product of single-particle states, and there is no information about the identical
indistinguishable electrons traveling along
distinguishable paths. With slightly different notation, we can write equation (1) as
Ψ12 = 1/√2) | 1+2- > - 1/√2) | 1-2+ > (2)
The probability amplitude wave function
Ψ12 travels away from the source (at the speed of light or less). Let's assume that at
t0 observer A finds an electron (e
1) with spin up.
At the time of this "first" measurement, by observer A or B, new information comes into existence telling us that the wave function
Ψ12 has "collapsed" into the state |
1+2- >
(or into |
1-2+ >). Just as in the
two-slit experiment, probabilities have now become certainties, one possibility is now an actuality. If the first measurement finds a particular component of electron 1 spin is up, so the same spin component of entangled electron 2 must be down to conserve angular momentum.
And conservation of linear momentum tells us that at
t0 the second electron is equidistant from the source in the opposite direction.
As with any
wave-function "collapse", the probability amplitude information changes (it does not "travel" anywhere). Nothing really "collapses." Nothing is moving. Only information is changing.
Unlike the
two-slit experiment, where the "collapse" goes to a specific point in 3-dimensional configuration space, the "collapse" here is a "jump" (Bohr) or a "projection" (Dirac) or a "reduction" (Heisenberg) into one of the two possible 6-dimensional two-particle quantum states |
+ - > or |
- + >. This makes "visualization" (Schrödinger's
Anschaulichkeit) difficult or impossible, but the parallel with the collapse in the two-slit case provides an intuitive insight of sorts. In both cases, the local values of the probability wave function change instantly over a wide range of space.
This is what Einstein saw in 1905 with the light wave spread out everywhere collapsing into a photon, in 1927 with the probability of a particle being found everywhere on a spherical surface collapses to the spot where it is found, and again in 1933 where observing a particle in one place tells us exactly where the other one is far away.
If the measurement finds an electron (call it electron 1) as spin-up, then at that moment of new information creation, the two-particle wave function collapses to the state |
+ - > and electron 2 "jumps" into a spin-down state with probability unity (certainty). The results of observer B's measurement at a later time
t1 is therefore
determined to be spin down.
Notice that
Einstein's intuition that the result seems already "determined" or "fixed" before the second measurement is in fact correct. The result is determined by the law of conservation of momentum.
But as with the distinction between
determinism and
pre-determinism in the
free-will debates, the measurement by observer B was not pre-determined
before observer A's measurement.
It was simply
determined by her measurement.
Part 4 David Bohm, Atoms, Electron Spins, and Hidden Variables
As we saw in part 2, the first practical and workable experiments to test the EPR paradox were suggested in 1952 by
David Bohm (though experiments were not realized for almost two decades). Instead of using
continuous linear momenta in opposite directions, Bohm proposed using two atoms (or electrons) in
discrete electronic spin states. The atoms are prepared in an initial
molecular state of known total spin zero.
Bohm also revived
Louis de Broglie's "pilot-wave" theory which offered a way of imagining hidden variables "guiding" the particles and suggested a new (Bohmian) mechanics that could restore
determinism and eliminate quantum
indeterminism.
The Bohm version of EPR starts with a hydrogen molecule in a spherically symmetric "singlet" state, with total spin angular momentum zero, which disassociates into two hydrogen atoms. Bohm-EPR is describable by a two-particle wave function we can call
ψ12 which is the
linear combination of two terms, each the product of two single-particle wave functions,
ψ1 and
ψ2.
ψ12 = 1/√2(ψ1ψ2) - 1/√2(ψ2ψ1)
The coefficients 1/√2, when squared, tell us there is a 50-50 chance of finding the separated atoms in
ψ1ψ2 or in
ψ2ψ1. In either case the atomic spins are always found in opposite directions, when measurements are made in the same direction, preserving the symmetry, and conserving the total spin angular momentum as zero, the same as the original molecule.
It is this physical fact that let Bohm in 1951 write
Suppose that we have a molecule containing two atoms in a state in which the total spin is zero and that the spin of each atom is ℏ/2. Roughly speaking, this means that the spin of each atom points in a direction exactly opposite to that of the other, insofar as the spin may be said to have any definite direction at all. [Indeed, we can only say that the total spin zero of the two atoms means that their two-particle wave function ψ12 is spherically symmetric with no preferred direction - the singlet state.] Now suppose that the molecule is disintegrated by some process that does not change the total angular momentum. The two atoms will begin to separate and will soon cease to interact appreciably. [But Erwin Schrödinger tells us that they will not become independent and "cease to interact" until some measurement disentangles them, decohering their phases, and allowing their two-particle wave function to be replaced with the product of two single-particle wave functions ψ1ψ2 or ψ2ψ1.] Their combined spin angular momentum, however, remains equal to zero, because no torques have acted on the system...
Suppose now that one measures the spin angular momentum ofany one of the particles, say No.1. Because of the existence of correlations we can immediately conclude that the angular momentum vector of the other particle (No.2) is equal and opposite to that on No.1. In this way, we can indirectly measure the angular momentum of particle No.2 by measuring the corresponding vector of particle No.1.
Quantum Theory, p.614
Bohm and Yakir Aharonov said the equivalent in 1957
Then, because the total spin is still zero, it can immediately
be concluded that the same component of the
spin of the other particle (B) is opposite to that of A.
"Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky,” Physical Review, vol.108, no.4. p.1070, 1957
Finally, in
John Bell's first paper, he wrote in 1964
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 σ1 and σ2. If measurement of the component σ1 • a, where a is some unit vector, yields the value + 1 then, according to quantum mechanics, measurement of σ2 • a must yield the value — 1 and vice versa.
Now why, we may ask, did not Bohm, or Bell, or modern commentators on Bell's theorem, consider the conservation of total spin angular momentum zero as a "common cause" explanation for the perfect measurement correlations? The conservation is true at all times for the "separating" particles (now known as a "quasi-molecule" because it is described with a molecular wave function and not the product of two atomic wave functions). Conservation will remain true as long as the forces involved in Stern-Gerlach spin measurements are symmetric, the two SG devices oriented in the same plane to preserve the symmetry. When the devices measure in planes at an angle to one another, the correlations fall off as the cosine squared of that angle (Malus's law).
This "common cause" is just a shared/joint property of the two particles, true at all times, not just for measurements. It is not one particle's measurement acting (instantaneously at a distance) on the other particle.
The constant total spin zero is thus a "constant of the motion." Since it achieves the goal of hypothetical "local hidden variables" traveling with the particles, we might call it a "
hidden constant of the motion."
Part 5 Were the Spin Variables Created at the Initial Entanglement?
We comment on the flawed idea that something (hidden variables?) must be traveling between Alice and Bob. Assuming, for the moment, that the entangled electrons do not already have perfectly correlated opposing spins (which they will have if the electrons are conserving total spin angular momentum zero), 1) what kind of sensor would Alice need to measure the deviation between their electron spins and 2) what kind of interaction could Alice's measurement generate and send to Bob to bring his spin angular momentum into agreement? Alice must first detect the amount of Bob's angular deviation from her measurement angle. (Keep in mind that an electron spin measuring device, for example, a Stern-Gerlach device, would change the electron spin to be in the direction of the SG device. But Alice could not send an SG device to Bob, especially instantaneously!) Otherwise, what kind of torque could Alice exert on Bob's remote electron from far away to rotate it precisely into agreement? Any such two-part interaction, first signals about his orientation (pure information) from Bob to Alice and then mechanical forces back from Alice to Bob, applied to his electron in a distant space-like separation. Both violate the no-signaling theorem and the principle of relativity.
We explain why the perfectly correlated electron spins in the direction of measurements could not have been determined at the time of the initial entanglement preparation, with one spin up and the other down
in the particular direction of measurements. Some physicists say that the electron spins would have to have been defined in all three dimensions - x-spin, y-spin, and z-spin. But this is impossible.
Bohm in 1951 and Bohm and Aharonov in 1957 argued that electron spins could be defined in at most a single direction, and this direction would very likely not be the random freely chosen direction of the experimenters. This is a serious misunderstanding...
Here, the investigator can measure either the x, y, or z component of the spin of article No.1, but not more than one of these components, in any particular experiment. Nevertheless, it still turns out as we shall see that whichever component is measured, the results are correlated, so that if the same component of the spin of No.2 is measured, it will always turn out to have the opposite value. This means that a measurement of any component of the spin of atom No.1 provides, as in classical theory, an indirect measurement of atom No.2. Since, by hypothesis, the two particles no longer interact, we have obtained a way of measuring an arbitrary component of the spin of particle No.2 without in any way disturbing that particle. Since this can be accomplished without in any way disturbing the second atom, we conclude that...precisely defined elements of reality must exist in the second atom, corresponding to the simultaneous definition of all three components of its spin. [This is a gravely flawed conclusion. It is not that components of spin are defined in all (3) direction, but that total spin zero means the spins will be found with opposite values in whatever agreed-upon direction they are measured!] ...Because the wave function can specify, at most, only one of these components at a time with complete precision, we are then led to the conclusion that the wave function does not provide a complete description of all elements of reality existing in the second atom. [The spherically symmetric (singlet state) wave function actually has no preferred direction, so as long as measurements are made symmetrically by the Stern-Gerlach devices, the atoms will be found to have perfectly correlated opposite spin values in that direction (which the observers must freely choose by previous agreement to be in the same direction, otherwise correlations will be reduced by the square of the cosine of the angle between them.]
Quantum Theory, pp.614-615
In 1957, Bohm and Aharonov expanded this description...
In quantum theory, a difficulty arises, in the interpretation of the above experiment, because only one component of the spin of each particle can have a definite value at a given time. Thus, if the x component
is definite, then the y and z components are indeterminate and we may regard them more or less as in a
kind of random fluctuation.
"Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky," Physical Review 108, 4, p.1070 (1957)
But having opposite spins in even a single preferred direction would not explain the experimental data!
Had there been a single preferred direction (or angle) for electron spins in the initial entanglement preparation, there is essentially zero chance that Alice and Bob could observe in precisely that (unknown) direction or angle. And observations at angle θ to that initial preferred direction will produce decorrelated results + + and - - proportional to sin
2θ.
Part 6 No Preferred Direction in the Spherically Symmetric Wave Function
No preferred direction exists in the initial entanglement preparation. The original entangled state with total spin angular momentum zero is rotationally and spherically symmetric, the same in all directions. The arbitrary direction of measurement or angle for Alice and Bob is introduced by them, agreed to before they measure! Total spin angular momentum is conserved as zero at all times from the initial entanglement to their measurements (assuming that no environmental interaction disturbs either particle).
As long as Alice and Bob now measure in the same plane (same direction, same angle), a new planar symmetry replaces the spherical symmetry and defines the one component of electron spin that Bohm and Aharonov say is allowed. The conservation of total spin angular momentum zero continues. As long as Alice and Bob measure in their "freely chosen" plane (preserving the symmetry), conservation of total spin angular momentum continues. There are no "local hidden variables" traveling with the particles to maintain perfect correlations, but conserved total spin zero might be called a "hidden constant of the motion" that provides the needed "common cause" for the correlations.
Part 7 John Bell and his Inequalities Theorem
We examine John Bell's theorem predicting how correlations between measurements made at differing angles could distinguish between local hidden variables and quantum mechanics. Bell predicted how the correlations would be reduced if Alice and Bob did not measure in the same direction, but chose measurement angles different by θ degrees. And we will examine Bell's hypothesis about how the falloff of correlations with increasing angle θ is different for local hidden variables (a straight line function of θ) and for quantum mechanics (a curved cosθ function).
We can illustrate the straight-line predictions of Bell's inequalities for local hidden variables, the cosine curves predicted by quantum mechanics and conservation of angular momentum, and the odd "kinks" at angles 0°, 90°, 180°, and 270°, with what is called a "Popescu-Rorhlich box."
The "PR Box" shows Bell’s local hidden variables prediction as four straight lines of the inner square. The circular region of quantum mechanics correlations are found outside Bell's straight lines, "violating" his inequalities. Quantum mechanics and Bell's inequalities meet at the corners, where Bell's predictions show a distinctly non-physical right-angle that Bell called a "kink."
All experimental results have been found to lie along the curved quantum predictions called the "Tsirelson bound."
In 1976, Bell gave us this diagram of the "kinks" in his local hidden variables inequality. He says,
Unlike the quantum correlation, which is stationary in θ at θ = 0, at the hidden variable correlation must have a kink there
("Einstein-Podolsky-Rosen Experiments," republished in Speakable and Unspeakable in Quantum Mechanics," 1987, p. 85)
In his famous 1981 article on "Bertlmann's Socks," Bell explains that the predictions for his "
ad hoc" model are
linear in the angle difference |a - b|, and he notes the fact that his inequality only agrees with the quantum predictions at the corners of the square of linear predictions above, and not
at intermediate angles.
To account then for the Einstein-Podolsky-Rosen-Bohm correlations we have only to assume that the two particles emitted by the source have oppositely directed magnetic axes. Then if the magnetic axis of one particle is more nearly along (than against) one Stern-Gerlach field, the magnetic axes of the other particle will be more nearly against (than along) a parallel Stern- Gerlach field. So when one particle is deflected up, the other is deflected down, and vice versa. There is nothing whatever problematic or mind-boggling about these correlations, with parallel Stern-Gerlach analyzers, from the Einsteinian point of view.
So far so good. But now go a little further than before, and consider non-parallel Stern-Gerlach magnets. Let the first be rotated away from some standard position, about the particle line of flight, by an angle a. Let the second be rotated likewise by an angle b. Then if the magnetic axis of either particle separately is randomly oriented, but if the axes of the particles of a given pair are always oppositely oriented, a short calculation gives for the probabilities of the various possible results, in the ad hoc model,...
P(up, down) = P(down, up) = 1/2 - |a-b|/2π
where ‘up’ and ‘down’ are defined with respect to the magnetic fields of the two magnets. However, a quantum mechanical calculation gives
P(up, down) = P(down, up) = 1/2 - 1/2(sin(a - b)/2)2 [= 1/2(cos(a - b)/2)2]
Thus the ad hoc model does what is required of it (i.e., reproduces quantum mechanical results) only at (a — b) = 0, (a - b) = π/2 and (a — b) = π, but not at intermediate angles.
("Bertlmann's Socks and the Nature of Reality," republished in Speakable and Unspeakable in Quantum Mechanics," 1987, p. 146)
What was Bell's "short calculation" that gives "the probabilities of possible results" in his
ad hoc model as
linearly proportional to the angle |a-b|?
Bell does not give us any underlying physical reasons for the linear dependence on angle. He clearly knows that his linear "inequality" is a strong challenge to the curved cosine prediction of quantum mechanics.
And Bell's odd prediction of sharp corners or "kinks" where his straight lines turn ninety degrees (it is only at these corners where his linear inequality agrees with the curving quantum mechanics), surely should have prompted Bell to give us a deeper physical explanation of his theorem?
And what exactly was Bell's "quantum mechanical calculation" that gives us probabilities proportional to the cosine of the angle |a-b| squared? This more physical picture coincides with falloff of light polarization when polarizers are crossed at an angle (Malus's Law).
Conservation of total spin angular momentum zero (our "hidden constant of the motion") perfectly matches Bell's quantum mechanics predictions.
Could the measurement outcomes have been predetermined at the initial entanglement?
John Bell considered this a possibility. In his very first paper on EPR, he wrote...
Since we can predict in advance the result of measuring any chosen component of σ2, by previously measuring the same component of σ1, it follows that the result of any such measurement must actually be predetermined.
Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.
Over a decade later, Bell still considered predeterminism...
Some people find this situation paradoxical. They may, for example, have come to think of quantum mechanics as fundamentally indeterministic. In particular they may have come to think of the result of a spin measurement on an unpolarized particle (and each particle, considered separately, is unpolarized here) as utterly indefinite until it has happened. And yet here is a situation where the result of such a measurement is perfectly definitely known in advance.
It did become
determined (but it was not
predetermined beforehand) by the measurement which
collapses the entangled two-particle wave function
During a mid-1980's interview by BBC Radio 3 organized by P. C. W. Davies and J. R. Brown, Bell proposed the idea of a "
superdeterminism" that could explain the correlation of results in two-particle experiments without the need for faster-than-light signaling. The two experiments need only have been
pre-determined by causes reaching both experiments from an earlier time.
I was going to ask whether it is still possible to maintain, in the
light of experimental experience, the idea of a deterministic universe?
You know, one of the ways of understanding this business is to
say that the world is super-deterministic. That not only is inanimate
nature deterministic, but we, the experimenters who
imagine we can choose to do one experiment rather than another,
are also determined. If so, the difficulty which this experimental
result creates disappears.
Free will is an illusion - that gets us out of the crisis, does it?
That's correct. In the analysis it is assumed that free will is
genuine, and as a result of that one finds that the intervention of
the experimenter at one point has to have consequences at a
remote point, in a way that influences restricted by the finite
velocity of light would not permit. If the experimenter is not free
to make this intervention, if that also is determined in advance,
the difficulty disappears.
(The Ghost in the Atom, P.C.W. Davies and J. Brown, ch.3, p.47)
The freely chosen direction of measurement angle previously agreed upon by Alice and Bob can only be implemented at the time of their measurements and not before. As shown in parts 5 and 6, the only dependence on the original entanglement is that its spherically symmetric singlet state has total spin zero and
no preferred direction. The specific measurement direction/angle, freely chosen by Alice and Bob, and the specific correlations at that measurement angle (specifically either + - or - +), results from the "
collapse" of the Dirac/Schrödinger two-particle wave function (a "superposition" of + - and - + states) one of those states which is randomly realized.
These random results are "created" long after the particles were together at entanglement, when the particles have apparently become "separated" by arbitrarily large distances. The phenomenon therefore
appears distinctly "
nonlocal," but it does not involve any physical interaction or information transfer between the particles. It is "caused" by a condition (the conserved total spin angular momentum zero) that has been true from the moment of initial entanglement, as long as neither particle is disturbed by any asymmetric environmental interaction.
Part 9 Do Spins Found Correlated in One Direction Mean Correlated in All?
In his 1951 textbook Quantum Theory, Bohm wrote...
Suppose that we have a molecule containing two atoms in a state in which the total spin is zero and that the spin of each atom is ℏ/2. Roughly speaking, this means that the spin of each atom points in a direction exactly opposite to that of the other, insofar as the spin may be said to have any definite direction at all. [Indeed, we can only say that the total spin zero of the two atoms means that their two-particle wave function ψ12 is spherically symmetric with no preferred direction - the singlet state.] Now suppose that the molecule is disintegrated by some process that does not change the total angular momentum. The two atoms will begin to separate and will soon cease to interact appreciably. [But Erwin Schrödinger tells us that they will not become independent and "cease to interact" until some measurement disentangles them, decohering their phases, and allowing their two-particle wave function ψ12 to be replaced with the product of two single-particle wave functions ψ12ψ2 or ψ2ψ1.] Their combined spin angular momentum, however, remains equal to zero, because no torques have acted on the system...
Suppose now that one measures the spin angular momentum of any one of the particles, say No.1. Because of the existence of correlations we can immediately conclude that the angular momentum vector of the other particle (No.2) is equal and opposite to that on No.1. In this way, we can indirectly measure the angular momentum of particle No.2 by measuring the corresponding vector of particle No.1.
Quantum Theory, p.614
The Bohm and Aharonov argument that perfectly opposed spins at the time of the Alice and Bob measurements (and in the arbitrary direction chosen by them) would require that the spins would have to be defined in all three axial directions (which is impossible). This is a flawed argument. When spins are measured in one direction, they are necessarily undefined in the other two directions, as Bohm and Aharonov later said.
The original entangled state with total spin zero is rotationally symmetric, with no preferred direction, so that the preferred plane or direction of measurement for Alice and Bob is introduced by them. This symmetry ensures the conservation of total angular momentum zero until the time of measurement. As long as Alice and Bob measure in the same plane (preserving the symmetry), they maintain the conservation of angular momentum, our proposed "hidden constant of the motion."
Part 10 Physicists Who May Have Seen Conservation Principles
We review the work of some physicists to show they were clearly using conservation principles to explain some aspects of entanglement, but without their necessarily seeing how conservation of linear or angular momentum could produce the
appearance of Einstein's "spooky action at a distance" without any interaction or "influence" between the particles.
David Bohm
Bohm and his Israeli student Yakir Aharonov reformulated the original EPR argument in terms of electron spin. They said experimental tests with
continuous variables would be much more difficult than tests with
discrete quantities, such as the spin of electrons or polarization of photons.
Bohm and Aharonov described the preparation of two particles, such that a measurement of one at a later time determines a measurement
in the same direction of the other particle at any distance away.
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
ψ = (1/√2) [ ψ+ (1) ψ- (2) - ψ- (1) ψ+ (2) ] (1)
where ψ+ (1) refers to the wave function of the atomic
state in which one particle (A) has spin +ℏ/2, etc. The
two atoms are then separated by a method that does
not influence the total spin. After they have separated
enough so that they cease to interact, any desired component
of the spin of the first particle (A) is measured.
Then, because the total spin is still zero, it can immediately
be concluded that the same component of the
spin of the other particle (B) is opposite to that of A.
"Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky,” Physical Review, vol.108, no.4. p.1070, 1957
Their statement "because the total spin is still zero, it can immediately
be concluded that the same component of the
spin of the other particle (B) is opposite to that of A." surely is true because of the conservation of spin angular momentum.
Eugene Wigner
In an important article written after Bohm and before Bell's 1964
"theorem" paper,
Eugene Wigner in 1963 said very clearly that the symmetrically placed positions (for the
EPR paper) are caused by
conservation of linear momentum and the perfectly opposite electron spins (for
David Bohm's version of
nonlocality with electrons) are the result of conservation of
angular momentum. Wigner wrote
If a measurement of the momentum of one of the particles is carried out — the possibility of this is never questioned — and gives the result p, the state vector of the other particle suddenly becomes a (slightly damped) plane wave with the momentum -p. This statement is synonymous with the statement that a measurement of the momentum of the second particle would give the result -p, as follows from the conservation law for linear momentum. The same conclusion can be arrived at also by a formal calculation of the possible results of a joint measurement of the momenta of the two particles.
One can go even further: instead of measuring the linear momentum of one particle, one can measure its angular momentum about a fixed axis. If this measurement yields the value mℏ, the state vector of the other particle suddenly becomes a cylindrical wave for which the same component of the angular momentum is -mℏ. This statement is again synonymous with the statement that a measurement of the said component of the angular momentum of the second particle certainly would give the value -mℏ. This can be inferred again from the conservation law of the angular momentum (which is zero for the two particles together) or by means of a formal analysis.
8
John Bell
We can ask why John Bell, with the prior writings of Bohm and Wigner, never
explicitly considered the conservation of linear momentum in the EPR experiment, and conservation of spin angular momentum as explaining the perfectly opposite electron spins found in Bohm's version of EPR. But we can see in Bell's description that, like Bohm, he is
implicitly using conservation of angular momentum.
In his 1964 paper "
On the Einstein-Podolsky-Rosen Paradox," Bell made the case for
nonlocality.
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 σ1 and σ2. If measurement of the component σ1 • a, where a is some unit vector, yields the value + 1 then, according to quantum mechanics, measurement of σ2 • a must yield the value — 1 and vice versa.
Here Bell is implicitly conserving total spin angular momentum. His hypothesis that one measurement does not "influence" the other is correct.
Now we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other.
During a mid-1980's interview by BBC Radio 3 organized by P. C. W. Davies and J. R. Brown, Bell proposed the idea of a "
superdeterminism" that could explain the correlation of results in two-particle experiments without the need for faster-than-light signaling. The two experiments need only have been
pre-determined by causes reaching both experiments from an earlier time.
I was going to ask whether it is still possible to maintain, in the
light of experimental experience, the idea of a deterministic universe?
You know, one of the ways of understanding this business is to
say that the world is super-deterministic. That not only is inanimate
nature deterministic, but we, the experimenters who
imagine we can choose to do one experiment rather than another,
are also determined. If so, the difficulty which this experimental
result creates disappears.
(The Ghost in the Atom, P.C.W. Davies and J. Brown, ch.3, p.47)
But the obvious fact that the total spin angular momentum at zero is conserved for all times since the initial entanglement means that the two particles are indeed "
determined by causes reaching both experiments from an earlier time," though not "
superdetermined from the origin of the universe.
C.S.Unnikrishnan
In 2005,
C.S.Unnikrishnan of the Tata Institute of Fundamental Research in Mumbai, India was an exception. He proposed
explicitly that the conservation law of angular momentum can correlate measurements of entangled electrons, explaining the perfect correlations of entangled particles, without the faster-than-light interactions-at-a-distance or "hidden variables" often invoked to explain nonlocaiity.
Unnikrishnan wrote
Bell’s inequalities can be obeyed only by violating a conservation law.
"Correlation functions, Bell’s inequalities and the fundamental conservation laws,"
arxive.org, 2004
Unnikrishnan argues that conservation of angular momentum (electron spin) produces the same perfect correlations (or anti-correlations) found in all Bell test experiments when both experimenters measure at the same (pre-agreed upon) measurement angle.
However, Unnikrishnan is concerned that "for individual measurements of the two-point correlation, the conservation law cannot be invoked, since only the conditional probabilities are predicted by quantum mechanics." He uses instead averages of measurements.
Unnikrishnan's apparent concern is that individual measurements will have random outcomes of up-down, down-up, and possibly even some up-up and down-down. He mistakenly thinks quantum mechanics predicts separate probabilities for each electron. The latter two product states (up-up and down-down) would violate conservation of angular momentum.
Part 10 Conservation Laws and Constants of the Motion
Conservation laws are the consequence of extremely deep properties of nature that arise from simple considerations of symmetry. We regard these laws as "cosmological principles." Physical laws do not depend on the absolute place and time of experiments, nor their particular direction in space. Conservation of linear momentum depends on the translation invariance of physical systems, conservation of energy the independence of time, and conservation of angular momentum the invariance under rotations. Conservation laws are the consequence of these spatial symmetries, as explained by
Emmy Noether.
David Bohm's version of the EPR experiment starts with two electrons (or photons) prepared in an entangled state that is a
superposition of two-particle states, each of which conserves the total angular momentum and, of course, conserves the linear momentum as in Einstein's original
EPR example.
Quantum mechanics describes the probability amplitude wave function
Ψ12 of the two-particle system as in a
superposition of two-particle states. It is
not a product of single-particle states, as
Erwin Schrödinger told Einstein in his reaction to the EPR paper. We can write this as
Ψ12 = 1/√2) | 1+2- > + 1/√2) | 1-2+ > (1)
The probability amplitude wave function
Ψ12 travels away from the source (at the speed of light or less). The total spin zero wave function is rotationally symmetric and isotropic, the same in all directions.
Let's assume that at
t0 an observer finds a particle with spin up in the x direction. This measurement breaks the rotational symmetry. The new symmetry is planar, including the chosen x direction and the z direction back to the origin of the entangled particles.
Before the measurement, the spin has a
possibility of being found in any direction. Rotational symmetry says the probability is the same in all directions. This does not mean a particle has spins in all directions at all times, which is impossible.
At the time of this "first" measurement, say by observer A, new information comes into existence telling us that the wave function
Ψ12 has "collapsed" into the state |
1+2- >. Probabilities have now become certainties, one possibility is now an actuality. If the first measurement finds particle 1's x-component spin is up, so the same spin component of entangled particle 2 must be down to conserve total angular momentum.
And conservation of
linear momentum tells us that at
t0 the second electron is equidistant from the source in the opposite direction.
As with any
wave-function "collapse", the probability amplitude information changes. Nothing really "collapses." Nothing physical, no matter or energy, is moving. Only information is changing. The wave function is updated to reflect the new information that comes into existence as the result of the measurement.
When the "first" measurement finds particle 1 as spin-up at t
0, at that moment of new information creation, particle 2 will be found (if measured simultaneously or later) in a spin-down state with probability unity (certainty). And the results of observer B's measurement at t
0 or any later time
t1 is therefore
determined to be spin down (if and only if, B measures in a pre-agreed upon same direction).
Notice that
Einstein's intuition that observer B's result seems already "determined" or "fixed" before the second measurement is in fact correct. Observer B's outcome is determined by the law of conservation of momentum.
But the measurement by observer B was not
pre-determined before observer A's measurement. It was simply
determined by her measurement. The measured values of particle 1 spin-up and particle 2 spin-down
did not exist before the "free choice" of observer A brought them into existence, as
Werner Heisenberg insisted.
Which of the two-particle quantum states |
+ - > or |
- + > occurs is completely random. It is the result of "Nature's choice," as
Paul Dirac described it.
Note also that before the measurement the two-particle wave function was rotationally symmetric. No preferred angular direction existed. The preferred angle also comes into existence as a result of what Heisenberg called the "free choice" of the ("first") experimenter. Of course neither Alice nor Bob measures first, as their measurements of a single entangled pair reaching their experiments are normally made simultaneously.
This "free choice" of a measurement angle breaks the rotational symmetry of the original two-particle wave function. As
Erwin Schrödinger described it to Einstein in his 1935 response to the EPR paper, the measurement
disentangles the particles and projects the pure-state superposition into a mixed-state product of single-particle wave functions, either
|
+ - > or |
- + >.
Many critics have pointed out that the two particles cannot already have opposite spin values in a specific direction long before the measurements. That would require them to have spin values in all three x, y, and z directions, say some physicists (e.g., David Bohm) which is impossible, they say. They only need to
acquire opposite spins when measured along an agreed upon direction that breaks the rotational symmetry of the two-particle wave function.
Finally note that conservation of total spin zero requires no
instantaeous superluminal influence or interaction by one particle on the other. It only requires that interactions between the particles preserve the symmetry needed for the conservation law.
Many other critics have insisted that some mechanism will be found to explain the mysterious entanglement
That symmetry has become linear and planar, as the rotational symmetry disappears, leaving symmetry only along the plane between the electrons that includes the chosen measurement direction.
If the two particles did not conserve total spin zero (and every Bell test shows that they do conserve total spin), the violation of the conservation law would likely be met with more criticism than hypothetical superluminal interactions, which are of course impossible.
What about the uncertainty principle? In the case of EPR measuring the position x (or the momentum p) of the two particles, won't their values be "fuzzy" (ΔxΔp ~ħ) and therefore not conserve momentum
exactly, but only
statistically for large numbers of examples? No, the conservation laws require that if x
1 is found less than the expected value by an amount -δ, that x
2 would be greater by the opposite amount +δ, so that the two particles are equidistant from the origin. This ensures the conservation of linear momentum, just as Einstein in 1924 proved that the
Bohr-Kramers-Slater theory that energy is only conserved statistically was wrong. Momentum is conserved exactly in every measurement, although the uncertainty principle may prevent this from being shown experimentally.
Furthermore, in the Bohm version of nonlocality the quantities are
discrete spins, not
continuous positions or momenta. Electron spins are always measured to be either up or down, with nothing fuzzy about these values. And all the experimental results from all the Bell tests have always found the two spins opposite as long as both measurements are made in exactly the same direction, thus conserving total spin zero.
There is still
indeterminism (uncertainty) in the spin measurement results. We don't know which electron will be up and which down. It is this property that "does not exist" before the measurement.
Physics has not found any hidden variables, local or nonlocal, as the cause of the perfect opposite spins. Is the conserved total spin zero acceptable as what we call a "hidden constant of the motion" that completely accounts for the perfectly opposite spins?
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We can take the x-axis to be vertical along 0° to 180°, the y-axis horizontal along 270° to 90°, and the z-axis to be into the diagram along the line from observer A through the origin to observer B. Observer A (red line) breaks the rotational symmetry by choosing to measure along angle 0° to 180°, finding her spin in the plane x-z, shown as the line from 0°to 180°. If observer B measures along the same pre-agreed angle 0° to 180°, then both measurements are in the plane x-z, shown superposed (purple line) from 0° to 180°. The observers have perfectly correlated spins.
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If both observers measure along a different pre-agreed angle 45° to 225°, then rotational symmetry is again broken, replaced by a new planar symmetry at the 45° angle, but again with perfectly correlated measurements. We see that electron spins cannot exist at 0° and 45° at the same time, but the rotational symmetry of the total spin zero state can be broken and replaced by planar symmetry in any direction.
|
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If observer A measures at 0° to 180° (red line) and observer B measures at a different angle 45°, shown as the blue line, their perfect correlations are lost, reduced by the factor cos45° (wave amplitude), and by cos245° (intensities).
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John Bell's theorem maintains that his inequalities for local hidden variables should be a straight-line function of the angle between observer A and observer B, shown as the facets of the square LHV polytope inscribed in the circle of quantum physics predictions. At angle 45°, if Bell's predicted inequality is less than or equal to 1, the distance to the quantum prediction is √2 ≈ 1.414. All the Bell experiments to date have found the the Bell inequalities have been violated by approximately 41%, an amazing confirmation of the accuracy of quantum mechanics.
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Part 11 David Mermin's Instruction Sets as Hidden Variables
We trace the modern idea of "instruction sets" accompanying the particles as "hidden variables" to Erwin Schrödinger's suggestion that the particles seem to know the answers to all possible questions that would be asked of them when measured. What process could prepare such information for the particles and what mechanism could read the information in the "instructions" and then change the particle properties as required?
In his 1936 paper "Property Relations Between Separated Systems," Schrödinger wrote
Years ago I pointed out that when two systems separate far enough
to make it possible to experiment on one of them without interfering with the
other, they are bound to pass, during the process of separation, through stages
which were beyond the range of quantum mechanics as it stood then...
It seems worth noticing that the paradox could be avoided by a very simple
assumption, namely if the situation after separating were described by the
expansion [ψ (x,y) = Σ ak gk(x) fk(y), as assumed in EPR], but with the additional statement that the knowledge of the phase relations between the complex constants ak has been entirely lost in consequence of the process of separation.
When some interaction, like a measurement, causes a separation, the two-particle wave function Ψ12 collapses, the system decoheres into the product Ψ1Ψ2, losing their coherent phase relation so there is no longer "interference," and we have a mixed state rather than a pure state.
This would mean that not only the parts,
but the whole system, would be in the situation of a mixture, not of a pure state.
It would not preclude the possibility of determining the state of the first system
by suitable measurements in the second one or vice versa. But it would utterly
eliminate the experimenters influence on the state of that system which he does
not touch.
Schrödinger refers to the central claim of the EPR paradox, in terms of the information available to the particles...
the result of measuring p1 serves to predict the result for p2 and vice versa.
But of course every one of the four observations in question, when actually performed, disentangles the systems, furnishing each of them with an independent representative of its own. A second observation, whatever it is and on whichever system it is executed, produces no further change in the representative of the other system.
Yet since I can predict either x1 or p1 without interfering with system No. 1 and since system No. 1, like a scholar in examination, cannot possibly know which of the two questions I am going to ask it first: it so seems that our scholar is prepared to give the right answer to the first question he is asked, anyhow. Therefore he must know both answers; which is an amazing knowledge.
(ibid.)
Part 12. Summary of the "Hidden Constant" hypothesis
Standard quantum mechanics plus the
principle of angular momentum conservation (true for quantum and classical mechanics) predicts:
- A measurement of either particle will project both particles into a random product of spin states that did not pre-exist the measurement.
- Conservation is a joint or shared property of the two particles, not a pre-existing property of the individual particles.
- Opposite spin outcomes conserve the total spin zero, without superluminal influences of one particle on the other.
- This conservation of total spin zero has been true at all times from the initial entanglement to the measurements, as long as neither particle is disturbed.
- The usual assumption, made by David Bohm, John Bell, and nearly every philosopher of science and science writer who has commented on entanglement, that the two particles are spinning in random directions, so that one must act instantaneously at a distance to align the second particle, is unjustified by theory and the experimental data.
- The normal interactions between the particles (electromagnetic and gravitational forces) as they separate do not exceed the speed of light and have the translational, rotational, and temporal invariance symmetries needed to conserve the total spin zero created at initial entanglement.
- The particular angle or direction in which to measure the two spins is the "free choice" of the experimenters, as Werner Heisenberg called it. It did not exist before this choice and its implementation.
- Moreover, which spin is up, which down, did not exist before the measurements and is "Nature's choice," as Paul Dirac called it. Before the measurements, quantum physics describes the particles as in a quantum "superposition," a linear combination of up/down and down/up< states, with equal probabilities of being found in either state.
- These predicted outcomes (both random individually and perfectly correlated) have been experimentally confirmed by thousands of Bell test experiments, as long as the entangled particles are measured at the (pre-agreed upon) same angle.
- Should the measurement angles differ by angle θ, the correlations fall off as cosθ (wave amplitude), and by cos2θ (intensities), as known from measurements of light polarizers crossed at different angles (the "law of Malus").
- The measured falloff in correlations with angle is said to violate the Bell "inequalities" and confirm the predictions of quantum mechanics.
- One could not hope for better agreement between hypothesis and experiment.
- The hypothesis of a "hidden constant" is based on deep (conservation) "principles" and is not what Einstein called a "constructed" theory, fitted to experimental data.
- This hypothesis explains the perfectly correlated random bit strings needed for quantum cryptography.
- The quantum teleportation of qubits uses subluminal communication of information. It does not depend on superluminal interaction at a distance, so is consistent with the hypothesis of a "hidden constant of the motion."
- Quantum computing involves very short distances for qubit manipulations and also is unaffected by a "hidden constant" as the explanation for entanglement.
- None of these three parts of the "second quantum revolution" depend on "spooky action-at-a-distance."
- The hypothesis of a "hidden constant" changes nothing about quantum physics but it eliminates the "weirdness" popularly associated with entanglement.
- Do you see a flaw in hypothesis of a "hidden constant?" Please write bobdoyle@informationphilosopher.com.
- Of course the constant is "hidden in plain sight" of the conservation law!
The "hidden constant of the motion" was introduced in
chapter 45 of
My God, He Plays Dice, How Albert Einstein Invented Most of Quantum Mechanics, 2019, p.376.
If the conservation of angular momentum (spin) is not the "common cause" of the perfect correlations, the vast experimental evidence for those correlations, so critical to the twin random bit strings needed for quantum cryptography, tells us that conservation of spin is an experimentally proven fact of the matter!
We can also note that a subsequent measurement by either observer at a different angle will destroy the planar symmetry. The original linear combination or
superposition of states
Ψ12 = 1/√2) | 1+2- > + 1/√2) | 1-2+ > (1)
will become a
mixture of product states that have
decohered(their
pure-state coherent phase relations lost). They are
disentangled, as
Erwin Schrödinger argued in 1936. Measuring one can still tell us something about the other. he said. But they have
decohered and can no longer interfere with one another and cannot remain correlated after future measurements. They have then truly
separated and "independent," but they had not become independent before the disentanglement, as Einstein had hoped with his separability principle (
Trennungsprinzip).
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Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of "hidden" variables. II.
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