Local causal interactions with the entangling apparatus put entangled particles in a spherically symmetric two-particle quantum state Ψ₁₂ that Erwin Schrödinger said cannot be represented as a product of two independent single-particle states. As the particles travel away from the central apparatus C in opposite directions toward measurement devices at A and B the two-particle wave function is in a spherically symmetric singlet state with total spin angular momentum zero. This total spin zero is conserved as a constant of the motion until separate local causal interactions at A and B collapse the two-particle wave function, decohering it into single-particle states Ψ₁ and Ψ₂ that are now disentangled. As long as local measurements are made at the same pre-agreed upon angle their planar symmetry will maintain total spin zero, the particles will have correlated opposite spin states up-down or down-up, and individual spin states will be randomly up or down. Should measurements at A and B differ by angle θ, perfect correlations will be reduced by cos²θ, as quantum mechanics predicts. We note that no information is communicated (at any speed) between A and B. Single bits of information are created at A and at B, caused by the initial entanglement at C.

The initial local causes are symmetries imposed at their entanglement state preparation in the causal center C, the condition is a conservation law that maintains those symmetries as the particles travel to A and B, and the final local causes are the perfectly correlated measurements. There is no faster-than-light communication or interaction between the separated measurement events at A and B.

This model opposes the standard view, developed over five decades by Albert Einstein and others, that some kind of faster-than-light instantaneous and nonlocal interaction between a particle and its wave function, or between two particles, is needed to explain entanglement.

Einstein first saw an instantaneous (nonlocal) change across the front of a widespread light wave in his 1905 photoelectric effect work.

In his later years, Einstein famously called two-particle entanglement "spooky action at a distance" (spukhafte Fernwirkung) in a March 3, 1947 letter to Max Born.

Although our focus is on experiments and their data correlations, we will also describe the mathematical quantum theory, its wave functions, and their superpositions or linear combinations of product states that perfectly predict the puzzling correlations.

Our model of a "common cause" explaining entanglement is a causal chain of events that begins with the local causes in the entangling apparatus that establishes the initial symmetry of the entangled particles. The particles are put into a spherically symmetric two-particle quantum state Ψ₁₂ that Erwin Schrödinger said cannot be represented as a product of two independent single-particle states Ψ₁ and Ψ₂.

As the particles travel away from the central apparatus C in opposite directions toward measurement devices at A and B the two-particle wave function Ψ₁₂ is in a spherically symmetric singlet state with total spin angular momentum zero. This total spin zero is conserved as a constant of the motion.

Conservation is not a causal process in the sense of a causal interaction. Indeed, it demands the lack of any causal interaction with the environment which might decohere the two-particle wave function. But we might say that the conserved property of total electron spin zero is also "local" in the sense that it is traveling along with each particle just as David Bohm's "hidden variables" or David Mermin's "instruction sets" were thought to do.

Galileo's law of Inertia, which became Newton's first law of motion, is a conservation principle. If a particle experiences no causal interactions it will maintain its state of motion, including remaining at rest.

The final causal interactions in our "causal chain" of events are the two measurements at A and B, which create two bits of digital information. As long as local measurements are made at the same pre-agreed upon angle their planar symmetry will maintain total spin zero, the particles will have correlated opposite spin states up-down or down-up, and individual spin states will be randomly up or down. Should measurements at A and B differ by angle θ, perfect correlations will be reduced by cos²θ, as predicted by quantum mechanics.

We can criticize the claim by Bohm¹ and Mermin² that the three components of spin angular momentum must exist and be defined in all spatial directions, x, y, and z (which is impossible), in order for experiments to find the spins perfectly correlated and opposite in all directions when measured. The correct requirement is that there be no preferred spin direction (spherical symmetry) at initial entanglement. The arbitrary but agreed upon direction chosen for the two final measurements maintains the spin symmetry. Each final spin measurement creates a single bit of information. No information is communicated from A to B as Einstein feared.

We shall critically examine the details of entanglement state preparations and the experimental data from particle measurements in six different kinds of entanglement experiments.

The first three (from the 1930's to 1960's) were thought ("gedanken") experiments (really hypothetical theories) involving material particles, atoms and electrons.

The first was Einstein's theoretical work starting in 1905 and culminating in the 1935 EPR paradox paper, his most cited work and the touchstone for all subsequent research on entanglement. David Bohm's proposal in the 1950's that hidden variables could explain entanglement is the second kind of thought experiment. The third is the great work of John Stewart Bell in the 1960's, including his Bell Theorem and his claim that if hidden variables exist they must be non-local.

The fourth and the sixth kind of experiments were actual physical experiments done in laboratories with light particles (photons) entangled by atomic cascades in calcium atoms. In the 1970's John Clauser and Stuart Freedman did the first experiments that violated Bell's inequality. They confirmed the correctness of quantum physics and showed that any "hidden variable" must be non-local. In the 1980's Alain Aspect used fast switching to ensure the particles were already in flight when the choice of measurment angle is made. In the 1990's Anton Zeilinger and his colleagues in Vienna developed spontaneous parametric down conversion(SPDC) techniques to generate entangled photons that are now used in most entanglement experiments. Clauser, Aspect, and Zeilinger received the 2022 Nobel Prize in Physics for their work.

Our fifth kind of entanglement study was the very popular thought experiment of David Mermin in the 1980's.

Einstein first described two particles exhibiting nonlocal behavior in a conversation with Leon Rosenfeld at a meeting in Berlin in 1933. Before this, nonlocality was between a single light quantum and its light wave.

In 1933, shortly before he left Germany to emigrate to America, Einstein attended a lecture on quantum electrodynamics by Léon Rosenfeld. Keep in mind that Rosenfeld was perhaps the most dogged defender of the Copenhagen Interpretation. After the talk, Einstein asked Rosenfeld,

“What do you think of this situation?” Suppose two particles are set in motion towards each other with the same, very large, momentum, and they interact with each other for a very short time when they pass at known positions. Consider now an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum: then, from the conditions of the experiment, he will obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the first particle, he will be able tell where the other particle is.

(Niels Bohr, His Life and Work as seen by His Friends and Colleagues, 1967, S. Rozental, pp.128-129)

It is most unfortunate that Einstein did not explain that measuring the momentum of the first particle allows us to deduce the momentum of the second particle because of the conservation of linear momentum.

The same conservation principle explains, as Einstein says, "If, however, he chooses to measure the position of the first particle, he will be able to tell where the other particle is."

If Einstein had called this ability "knowledge (information) at a distance," instead of "spooky action at a distance," entanglement might never have been thought "spooky". It would be just a correlation of physical properties resulting from a common cause and a conservation law.

Einstein and colleagues Boris Podolsky and Nathan Rosen, proposed in 1935 a paradox (known by their initials as EPR or as the Einstein-Podolsy-Rosen paradox) to exhibit internal contradictions in the new quantum physics. They hoped to show that quantum theory could not describe certain intuitive "elements of reality" and thus was incomplete. They said that, as far as it goes, quantum mechanics is correct, just not "complete."

Einstein was correct that quantum theory is "incomplete" relative to classical physics, which has twice as many dynamical variables that can be known with arbitrary precision. But half of this information is missing in quantum physics, due to the indeterminacy principle which allows only one of each pair of non-commuting observables (for example momentum or position) to be known with arbitrary accuracy. Even more important, an individual particle, cannot be said to have a known position before a measurement, since evolution described by the unitary and deterministic Schrödinger equation provides us only probabilities.

The most that can be said is that the particle can be found anywhere the probability amplitude is non-zero. This was the core idea of Einstein's claim of "incompleteness." For Bohr to deny this and call quantum mechanics "complete" was just to play word games, which infuriated Einstein.

Einstein was also correct that indeterminacy makes quantum theory an irreducibly discontinuous and statistical theory. Its predictions and highly accurate experimental results are statistical in that they depend on an ensemble of identical experiments, not on any individual experiment. Einstein wanted physics to be a continuous field theory, in which all physical variables are completely and locally determined by the four-dimensional field of space-time in his theory of relativity.

Einstein and his colleagues Erwin Schrödinger, Max Planck, (later David Bohm), and others hoped for a return to deterministic physics, and the elimination of mysterious quantum phenomena like the superposition of states, the mysterious "collapse" of the wave function, and Schrödinger's famous cat. EPR continues to fascinate determinist philosophers of science who hope to prove that quantum indeterminacy does not exist.

The 1935 paper was based on Einstein's 1933 question to Leon Rosenfeld about two material particles fired in opposite directions from a central source with equal velocities. He imagined them starting at t₀ some distance apart and approaching one another with high velocities. Then for a short time interval from t₁ to t₁ + Δt the particles are in contact with one another.

After the particles are measured and become entangled at t₁, quantum mechanics describes them with a single two-particle wave function Ψ₁₂ that is not the simple product of two one-particle wave functions Ψ₁ and Ψ₂.

Einstein said 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 explicitly. And this is correct, just as after the collision of two billiard balls, measurement of one ball tells us exactly where the other one is due to conservation of momentum. But this is not "action at a distance." It is more nearly "knowledge at a distance."

Note that Einstein is implicitly using conservation of linear momentum to know the position of the second particle. Although conservation laws are rarely cited as the explanation, they may be the physical reason that entangled particles always produce correlated results. If the results were not always correlated, the implied violation of a fundamental conservation law would be a much bigger story than mysterious entanglement itself, as interesting and important as that is.

This idea of something measured in one place "influencing" measurements far away challenged what Einstein thought of as "local reality." Einstein thought that when the particles moved far enough apart they could be treated as separated, with independent wave functions Ψ₁ and Ψ₂.

But Erwin Schrödinger quickly replied to the EPR paper, telling Einstein that his "separation principle" (Trennungsprinzip) was not correct. Today this is known as nonseparability.

The particles can not be separated into a product of independent wave functions, for example, particle A definitely in state 1, particle B in state 2,

Instead, particles A and B are each randomly found in state 1 or 2. But Particle B is certain to be in state 2 if particle A is measured in state 1, and vice versa.

Quantum mechanics says that the particles are not in "pure" quantum states, but a "mixture" of two states, which maintain the coherent phase relations that allow them to interfere with one another.

Schrödinger used Paul Dirac's 1926 principle of superposition and John von Neumann's 1932 motion of "mixed states" in the "density matrix" to create two of the most popular and controversial ideas in quantum mechanics.

First, Schrödinger described the two particles as "entangled" at their first encounter. He called it verschränkt in German. Verschränkt means something like cross-linked. It describes someone standing with arms crossed, where each arm reaches out to touch the other. Today EPR is the classic example of entanglement.

Second, Schrödinger introduced his famous cat, claiming it is in a mixed state or a superposition of live and dead cats!

It was at this point in quantum history that the most controversial two-particle equation above appeared. It combines the ontological chance that Einstein discovered in 1916 with the idea that one quantum state can be described as in a linear combination or superposition of two other states that Dirac introduced in his 1927 "transformation theory" of quantum mechanics.

In Dirac's theory, the coefficients of the two states (1/√2) when squared (1/2) give us the 50% probabilities of finding the particles in the one or the other state.

The equation combines quantum randomness with quantum interference. Interference was made famous in the two-slit experiment, which Richard Feynman almost thirty years after EPR called the “one mystery,” the only mystery at the heart of all quantum mechanics.

Note that the equation does not describe two material particles interacting with one another but their abstract immaterial wave functions Ψ₁ and Ψ₂ interfering with one another.

These quantum mechanical wave functions, solutions to Schrödinger’s equations of motion (which replace Newton's equations of motion in classical mechanics), were thought by Schrödinger to be describing matter or energy, photons for light waves, mass and perhaps electric charge for electrons.

But in the quantum mechanics of Heisenberg, Jordan, Born, and Dirac the wave functions became “probability amplitudes,” whose absolute squares predict the probability of finding the values of observable quantum properties. And those predictions have been confirmed with extraordinary accuracy by countless experiments.

Let’s look at the equation in its simplest form that describes the superposition state of Schrödinger’s cat.

Although this equation predicts interference between the cat states, such interference is never seen in cats, though it has been measured in surprisingly large macroscopic objects.

Nevertheless, squaring the coefficients 1/√2 tells us that there is 50% chance of finding such a cat in either the live or dead state, i.e., which is confirmed in principle.

Let's see how this simple equation also describes the two-slit experiment.

The wave function beyond the two slits is a linear combination or superposition of the wave function passing the left slit | Left > with the wave function passing the right slit | Right >.

Note that whichever slit the particle passes through (and it must go through just one, because a quantum particle cannot become two (violating conservation of mass and/or energy), the probabilities of finding it on the screen are determined by the two-slit superposition. If a particle was detected passing through the left slit, or if the right slit were closed, the interference pattern would depend only on that slit's wave function | Left >.

Given that the double-slit interference appears even if only one particle at a time is incident on the two slits, we see why many say that the particle interferes with itself. But it is the wave function alone that is interfering with itself. Whichever slit the particle goes through, it is the probability amplitude ψ, whose squared modulus |ψ|² gives us the probability of finding a particle somewhere, the interference pattern. It is what it is because the two slits are open.

This is the deepest metaphysical mystery in quantum mechanics. How can an abstract immaterial probability wave influence the material particle paths to show interference when large numbers of independent particles are collected?

When there is only one slit open (here the left slit), the probabilities pattern has one large maximum (directly behind the slit) and small side fringes. If only the right slit were open, this pattern would move behind the right slit. If we add up the results of some experiments with the left slit open and others with the right open we don't see the multiple fringes that appear with two slits open.	When both slits are open, the maximum is now at the center between the two slits, there are more interference fringes, and these probabilities apply whichever slit the particle enters. The solution of the Schrödinger equation depends on the boundary conditions - different when two holes are open. The "one mystery" remains - how these "probabilities" can exercise causal control (statistically) over matter or energy particles. Feynman's path integral formulation of quantum mechanics suggests the answer. His "virtual particles" explore all space (the "sum over paths") as they determine the variational minimum for least action, thus the resulting probability amplitude wave function can be said to "know" which holes are open.