Entanglement is a mysterious quantum phenomenon that is widely, but mistakenly, described as capable of transmitting information over vast distances faster than the speed of light. It has proved very popular with science writers, philosophers of science, and many scientists who hope to use the mystery to deny some of the basic concepts underlying quantum physics.

Entanglement depends on two quantum properties that are simply impossible in "classical" physics. One is called nonlocality. We shall argue that Albert Einstein first caught a glimpse of nonlocality as early as 1905 in his photoelectric paper, when he questioned how a light wave spread out in space could instantly collapse all its energy to be localized in the quantum of energy needed to eject an electron.

Einstein made a clear public statement about nonlocality at the 1927 Solvay conference, but was misunderstood by Niels Bohr. Einstein's concerns about nonlocality were ignored by most physicists until 1935 and the appearance of the famous Einstein-Podolsky-Rosen paper.

The other "impossible" quantum property is nonseparability, which Einstein was first to see, even as he attacked the idea. This was just as Einstein had reacted to his discovery of indeterminism in 1916. He attacked the appearance of chance (Zufall) as a "weakness in the theory".

In the EPR paper Einstein extended nonlocality beyond the relation between a light-quantum particle and its wave function. Nonlocality was now extended from one particle and it wave, to correlations between one particle and another with which it had interacted in the past. Erwin Schrödinger called them "entangled."

A couple of these somewhat weird claims about measurements of entangled particles can be illustrated and explained, as we shall see. One deep philosophical claim is that the minds of physicists are manipulating "quantum reality," that there is nothing "really" there until we look at it. The half-truth is that our "free choice" as to which property to measure in an experiment can create a value of a property that did not exist before the experiment. But we cannot force the value to be specific, for example +1 or -1. That is determined by "Nature's choice," ontological randomness ("chance") which was discovered in the emission of photons by Einstein in 1916.

Einstein was the first to see nonlocal behavior in quantum phenomena. He may have seen it as early as 1905 in the photoelectric effect, the same year he published his special theory of relativity. But it was perfectly clear to him 22 years later (ten years after his general theory of relativity and his explanation of how quanta of light are randomly emitted and absorbed by atoms), when he described nonlocality with a diagram on the blackboard at an international conference of physicists in Belgium in 1927 at the fifth Solvay conference.

In his contribution to the 1949 Schilpp memorial volume on Einstein, Niels Bohr gave us a picture of what Einstein drew on that blackboard.

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.

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.

The "nonlocal" effect at point (B) is just the probability of an electron being found at point (B) going to zero instantly (as if an action at a distance) the moment an electron is found at point A

The apparent difficulty, in this description, which Einstein felt so acutely, is the fact that, if in the experiment the electron is recorded at one point A of the plate, then it is out of the question of ever observing an effect of this electron at another point (B), although the laws of ordinary wave propagation offer no room for a correlation between two such events.

("Discussion with Einstein," by Niels Bohr, Albert Einstein, Philosopher-Scientist, P. A. Schilpp, 1949. pp.211-213)

Bohr is telling us that in 1927 Einstein saw instantaneous "correlations" of events widely separated ("as if actions at a distance"), which exactly describes today's perfect "nonlocal" correlations of widely separated entangled particles.

Then in 1935, Einstein, Boris Podolsky, and Nathan Rosen proposed a thought experiment (known by their initials as EPR) 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 either incomplete or, as they might have hoped, demonstrably incorrect.

Einstein and his colleagues Erwin Schrödinger, Max Planck, and others hoped for a return to deterministic physics, and the elimination of mysterious quantum phenomena like superposition of states and "collapse" of the wave function. EPR continues to fascinate determinist philosophers of science hoping to prove that quantum indeterminacy
(ontological randomness) does not exist.

Beyond the problem of nonlocality, the EPR thought experiment introduced the problem of "nonseparability." This mysterious phenomenon appears to transfer something physical faster than the speed of light. What happens actually is merely instantaneous (simultaneous) knowledge of a distant particle's properties by measurement of a local particle that interacted with the distant particle sometime in the past.

The 1935 EPR paper was based on an earlier question of Einstein's about two 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 equal high velocities. Then for a short time interval from t₁ to t₁ + Δt the particles are in contact with one another.

Einstein described this situation to Léon Rosenfeld in 1933. Shortly before he 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, which maintains that a particle has no position until it is measured.

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 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 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 problem of two-particle entanglement.

Why does Einstein question Rosenfeld and describe this as an “influence,” suggesting an “action-at-a-distance?”

It is only paradoxical in the context of Rosenfeld’s Copenhagen Interpretation, since the second particle is not itself measured and yet we know something about its properties, which the Copenhagen Interpretation says we cannot know 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 obviously used conservation of linear momentum implicitly to calculate (and know) the position of the second particle.

Two years later, after 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, they are the physical reason that entangled particles always produce correlated results for all properties. 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 is.

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." Einstein called it a "spukhaft Fernwirkung" or "spooky action at a distance."

We prefer to describe this phenomenon as "knowledge at a distance." No action has been performed on the distant particle simply because we learn about its position. Note that this assumes the distant particle has not been disturbed by an intermediate interaction with the environment.

In the year following the Einstein-Podsky-Rosen paper, Erwin Schrödinger looked more carefully at Einstein's "separability" assumption (Trennungsprinzip) that an entangled system can be separated enough to be regarded as two systems with independent wave functions:

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. For it seems hard to imagine a complete separation, whilst the systems are still so close to each other, that, from the classical point of view, their interaction could still be described as an unretarded actio in distans. And ordinary quantum mechanics, on account of its thoroughly unrelativistic character, really only deals with the actio in distans case. The whole system (comprising in our case both systems) has to be small enough to be able to neglect the time that light takes to travel across the system, compared with such periods of the system as are essentially involved in the changes that take place...

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 a_k g_k(x) f_k(y),
as assumed in EPR], but with the additional statement that the knowledge of the phase relations between the complex constants a_k 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 "influence" on one another, but not the acquisition of information about the second system by (nonlocsl) measurements on the first.

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.

("Probability Relations between Separated Systems," Proceedings of the Cambridge Physical Society 1936, 32, issue 2, p.446-452)

Schrödinger says that the entangled system may become disentangled (Einstein's separation) and yet some perfect correlations between later measurements might remain. Note that the entangled system could simply decohere as a result of interactions with the environment, as proposed by decoherence theorists. The perfectly correlated results of Bell-inequality experiments might nevertheless be preserved, depending on the interaction.

Schrödinger tells us that the two-particle wave function Ψ₁₂ will be separated into the product of single-particle wave functions Ψ₁ and Ψ₂ by a measurement of either particle, for example, by either Alice's or Bob's measurements in the case of Bell's Theorem.

As we saw, Einstein had objected to nonlocal phenomena as early as the Solvay Conference of 1927, when he criticized the collapse of the wave function as "instantaneous-action-at-a-distance" that prevents the wave from "acting at more than one place on the screen."

The simultaneous events at points A and B in Einstein's 1927 Figure 1 above are the same kind of nonlocality as the two entangled particles acquiring perfectly correlated properties while in a spacelike separation that he suggested to Rosenfeld in 1933, and which Podolsky and Rosen developed into the EPR paradox in 1935.

Einstein's 1927 concern was based on the idea that the light wave might contain some kind of ponderable energy. At that time Schrödinger thought it might be distributed electricity. In these cases the instantaneous "collapse" of the wave function might violate Einstein's principle of relativity, a concern he first expressed in 1909. When we recognize that the wave function is only pure information about the probability of finding a particle (or particles) somewhere, we see that there is no matter or energy (or in particular no information or signal of any kind) traveling faster than the speed of light.

Einstein's criticism somewhat resembles the criticisms by Descartes and others about Newton's theory of gravitation. Newton's opponents charged that his theory was "action at a distance" and instantaneous. Einstein's own theory of general relativity shows that gravitational influences travel at the speed of light and are mediated by a gravitational field that can be described as curved space-time.

When a probability function collapses to unity in one place and zero elsewhere, nothing physical is moving from one place to the other. 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. But it does not violate relativity.

The first practical and workable experiments to test the 1935 "thought experiments" of Einstein, Podolsky, and Rosen (EPR) were suggested by David Bohm in 1952. Instead of measuring linear momentum, Bohm proposed using two electrons that are prepared in an initial state of known total spin. Momentum and position are continuous variables. Spin is discrete. Bohm argued that measurements of discrete variables would be more precise. Bohm also proposed local "hidden variables" might be needed to explain the correlations. Here is Bohm's description

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) ]

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

Note that when Bohm says "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," he is implicitly using the conservation of total spin.

In 1964, John Bell put limits on Bohm's "hidden variables" that might restore a deterministic physics in the form of what he called an inequality, the violation of which would confirm standard quantum mechanics. Here is Bell's description. As with Bohm, conservation is not mentioned explicitly, but it involves spin components measured in the same direction

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 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.

"pre-determination" is too strong a term. The first measurement just "determines" the later measurement. We shall see that the second measurement is synchronous with the "first" in a "special" frame

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.

("On the Einstein-Podolsky-Rosen Paradox," Physics, 1.3, p.195.)

Just like Bohm, Bell is implicitly using the conservation of total spin.

If one electron spin is 1/2 in the up direction and the other is spin down or -1/2, the total spin is zero. The underlying physical law of importance is not conservation of linear momentum (as Einstein used), Bohm and Bell use the conservation of angular momentum (or spin). If electron 1 is prepared with spin down and electron 2 with spin up, the total angular momentum is zero. This is called the singlet state.

Bohm and Bell agree that quantum theory describes the two electrons as in a superposition of spin up ( + ) and spin down ( - ) states,

The principles of quantum mechanics say that the prepared system is in a linear combination (or superposition) of these two states, and can provide only the probabilities of finding the entangled system in either the | + - > state or the | - + > state.

The 1/√2 coefficients of the probability amplitude for each term, when squared, give us the probabilities (1/2) that the system will be found in the state | + - > or in the state | - + >. The actual outcome is random (Paul Dirac called it "Nature's choice." But the individual electron spin outcomes are not individually and separately random, because the particles are not independent. One is always up and the other down, as the conservation law requires.

Should measurements ever show both spins in the same state, either | + + > or | - - >, that would violate the conservation of angular momentum. Quantum mechanics does not include such terms in the wave function. So they are not predicted and they are never observed.

EPR tests can be done more easily with polarized photons than with electrons, which require complex magnetic fields. The first of these was done in 1972 by Stuart Freedman and John Clauser at UC Berkeley. They used oppositely polarized photons (one with spin = +1, the other
spin = -1) coming from a central source. Again, the total photon spin of zero is conserved. Their data, in agreement with quantum mechanics, violated the Bell's inequalities to high statistical accuracy, thus providing strong evidence against local hidden-variable theories. If hidden variables exist, they must be non-local, said Bell.

For more on superposition of states and the physics of photons, see the Dirac 3-polarizers experiment.

John Clauser, Michael Horne, Abner Shimony, and Richard Holt (known collectively as CHSH) and later Alain Aspect did more sophisticated tests. The outputs of the polarization analyzers were fed to a coincidence detector that records the instantaneous measurements, described as + -, - +, + +, and - - . The first two ( + - and - + ) conserve the spin angular momentum and are the only types ever observed in these nonlocality/entanglement tests.

With the exception of some of Holt's early results that were found to be erroneous, no evidence has so far been found of any failure of standard quantum mechanics. And as experimental accuracy has improved by orders of magnitude, quantum physics has correspondingly been confirmed to one part in 10¹⁸, and the transfer speed of the probability information between particles has a lower limit of 10⁶ times the speed of light. There has been no evidence for local "hidden variables."

Nevertheless, experimenters continue to look for possible "loopholes" in the experimental results, such as detector inefficiencies that might be hiding results favorable to Einstein's picture of "local reality."

Nicolas Gisin and his colleagues have extended the polarized photon tests of EPR and the Bell inequalities to a separation of 18 kilometers near Geneva. They continue to find 100% correlation and no evidence of the "hidden variables" sought after by Einstein and David Bohm.

An interesting use of the special theory of relativity was proposed by Gisin's colleagues, Antoine Suarez and Valerio Scarani. They use the idea of hyperplanes of simultaneity. Back in the 1960's, C. W. Rietdijk and Hilary Putnam argued that physical determinism could be proved to be true by considering the experiments and observers A and B in the above diagram to be moving at high speed with respect to one another. Roger Penrose developed a similar argument in his book The Emperor's New Mind. He called it the Andromeda Paradox.

Suarez and Scarani showed that for some relative speeds between the two observers A and B, observer A could "see" the measurement of observer B to be in his future, and vice versa.

Because the two experiments have a "spacelike" separation (neither is inside the causal light cone of the other), each observer thinks he does his own measurement before the other. Gisin tested the limits on this effect by moving mirrors in the path to the birefringent crystals and showed that, like all other Bell experiments, the "before-before" suggestion of Suarez and Scarani did nothing to invalidate quantum mechanics.

These experiments were able to put a lower limit on the speed with which the information about probabilities collapses, estimating it as at least thousands - perhaps millions - of times the speed of light and showed empirically that probability collapses are essentially instantaneous.

Despite all his experimental tests verifying quantum physics, including the "reality" of nonlocality and entanglement, Gisin continues to explore the EPR paradox, considering the possibility that signals are coming to the entangled particles from "outside space-time."

Information physics starts with the fact that measurements bring new stable information into existence. In EPR the information in the prepared state of the two particles includes the fact that the total linear momentum and the total angular momentum are zero.

New information requires an irreversible process that also increases the entropy more than enough to compensate for the information increase, to satisfy the second law of thermodynamics. It is this moment of irreversibility and the creation of new observable information that is the "cut" or Schnitt" described by Werner Heisenberg and John von Neumann in the famous problem of measurement

Note that the new observable information does not require a "conscious observer" as Eugene Wigner and some other scientists thought. The information is ontological (really in the world) and not merely epistemic (in the mind). Without new information, there would be nothing for the observers to observe.

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.

Recall that the EPR experiment 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

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 (e₁) 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 instantly). Nothing really "collapses."

Unlike the two-slit experiment, where the "collapse" goes to a specific point in 3-dimensional configuration space, the "collapse" here is a "jump" or "projection" 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. It is what Einstein saw in 1927 and again in 1933.

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.

Although Einstein mentioned conservation in the original EPR paper, it is noticeably absent from later work. Bohm and Bell are obviously using it without an explicit mention. A prominent exception is Eugene Wigner, writing on the problem of measurement in 1963:

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.

Writing a few years after Bohm, and one year before Bell, Wigner explicitly describes Einstein's conservation of momentum example as well as the conservation of angular momentum (spin) that explains perfect correlations between angular momentum (spin) components measured in the same direction

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.

(The Problem of Measurement, Eugene Wigner, in Wheeler and Zurek, p,340)

Schrödinger said that his "Wave Mechanics" provided more "visualizability" (Anschaulichkeit) than the Copenhagen school and its "damned quantum jumps" as he called them. He was right.

But we must focus on the probability amplitude wave function of the prepared two-particle state, and not attempt to describe the paths or locations of independent particles - at least until after some measurement has been made. We must also keep in mind the conservation laws that Einstein used to discover nonlocal behavior in the first place. Then we can see that the "mystery" of nonlocality is primarily the same mystery as the single-particle collapse of the wave function.

As Richard Feynman said, there is only one mystery in quantum mechanics (the collapse of probability and the consequent statistical outcomes).

We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by "explaining" how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.

(The Feynman Lectures on Physics, vol III, 1-1)

In his 1935 paper, Schrödinger described the two particles in EPR as "entangled" in English, and verschränkt in German, which means something like cross-linked. It describes someone standing with arms crossed.

In the time evolution of an entangled two-particle state according to the Schrödinger equation, we can visualize it - as we visualize the single-particle wave function - as collapsing when a measurement is made. The discontinuous "jump" is also described as the "reduction of the wave packet." This is apt in the two-particle case, where the superposition of | + - > and | - + > states is "projected" or "reduced: to one of these states, and then further reduced to the product of independent one-particle states | + > and | - >.

In the two-particle case (instead of just one particle making an appearance), when either particle is measured we know instantly those properties of the other particle that satisfy the conservation laws, including its location equidistant from, but on the opposite side of, the source, and its other properties such as spin.

Here is an animation showing the two particles simultaneously acquiring their opposite spins when either is measured.

Some commentators say that nonlocality and entanglement are a "second revolution" in quantum mechanics, "the greatest mystery in physics," or "science's strangest phenomenon," and that quantum physics has been "reborn." They usually quote Erwin Schrödinger as saying

"I consider [entanglement] not as one, but as the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."

Schrödinger knew that his two-particle wave function Ψ₁₂ cannot have the same simple interpretation as the single particle, which can be visualized in ordinary 3-dimensional configuration space. And he is right that entanglement exhibits a richer form of the "action-at-a-distance" and nonlocality that Einstein had already identified in the "collapse" of the single particle wave function.

But the main difference is that two particles acquire new properties instead of one particle, and they do it instantaneously (at faster than light speeds), just as the single-particle wave function changes everywhere in the case of a single-particle measurement.

Nonlocality and entanglement are thus another manifestation of Richard Feynman's "only" mystery in the two-slit experiment.

Here we must explain the asymmetry that Einstein and Schrödinger have introduced into a perfectly symmetric situation, making entanglement such a mystery. Every follower of their early thinking introduces this false asymmetry.

The classic EPR idea is completely symmetric about the origin of the state preparation. Einstein introduced the mistaken idea of measuring one particle "first" and then asking how it influences subsequent measurements of the "second" particle. By contrast, Schrödinger's two-particle wave function "collapses" at all positions in an instant of time. Both particles then appear in a space-like separation.

The perfectly symmetric picture shows that neither Alice nor Bob can in any way influence the other's experiment, as can be seen best in what we can call a special frame.

There is a special frame in which the collapse of the two-particle wave function is best visualized. It is not a preferred frame in the special relativistic sense (e.g., an inertial frame). But observers in all other frames in relative motion along the experiment axis will see one of the measurements before the other. Relativity contributes confusion to what is going on.

Almost every presentation of the EPR paradox begins with something like "Alice observes one particle..." and concludes with the question "How does the second particle get the information needed so that Bob's measurements correlate perfectly with Alice?"

There is a fundamental asymmetry in this framing of the EPR experiment. It is a surprise that Einstein, who was so good at seeing deep symmetries, did not consider how to remove the asymmetry. Even more puzzling, why did he introduce it? Why do most all subsequent scientists accept it without question?

Consider this reframing: Alice's measurement collapses the two-particle wave function. The two indistinguishable particles simultaneously appear at locations in a space-like separation. The frame of reference in which the source of the two entangled particles and the two experimenters are at rest is a special frame in the following sense.

As Einstein knew very well, there are frames of reference moving with respect to the laboratory frame of the two observers in which the time order of the events can be reversed. In some moving frames Alice measures first, but in others Bob measures first.

If there is a special frame of reference (not a preferred frame in the relativistic sense), surely it is the one in which the origin of the two entangled particles is at rest. Assuming that Alice and Bob are also at rest in this special frame and equidistant from the origin, we arrive at the simple picture in which any measurement that causes the two-particle wave function to collapse makes both particles appear simultaneously at determinate places with fully correlated properties (just those that are needed to conserve energy, momentum, angular momentum, and spin).

In the two-particle case (instead of just one particle making an appearance), when either particle is measured, we know instantly those properties of the other particle that satisfy the conservation laws, including its location equidistant from, but on the opposite side of, the entangling interaction, and all other properties such as spin. It's just "knowledge-at-a-distance."

Although we find no need for "hidden variables," whether local or non-local, we might say that the conservation laws give us "hidden constants." Conservation of a particular property is often described as a "constant of the motion." These constants might be viewed as "local," in that they travel along with particles at all times, or as "global," in that they are a property of the two-particle probability amplitude wave function Ψ₁₂ as it spreads out in space.

This agrees with Bohm, and especially with Bell, who says that the spin of particle 2 is "predetermined" to be found up if particle 1 is measured to be down.

But recall that the Copenhagen Interpretation says we cannot know a spin property until it is measured. So some claim that the spins are in an unknown combination of spin down and spin up until the measurements. It is this that suggests the possibility that both spins might be found in the same direction, violating conservation.

Although electron spins in this situation are never found experimentally in the same direction, the Copenhagen view gave rise to the idea of a hidden variable as some sort of signal that could travel to particle 2 after the measurement of particle 1, causing it to change its spin to be opposite that of particle 1. What sort of signal might this be? And what mechanism exists in a bare electron that could cause it to change a property like its spin without an external force of some kind?

Clearly, Wigner's explicit view, and the implicit claims of Bohm and Bell that the electron spins were prepared (entangled) in opposite states, are the simplest and clearest explanations of the entanglement mystery.

Despite accepting that a particular value of some "observables" can only be known by a measurement (knowledge is an epistemological problem) Einstein asked whether the particle actually (really, ontologically) has a path and position, even other properties, before we measure it? His answer was yes.

So Einstein would likely agree with Wigner, Bohm, and with Bell to assume that the two particles have opposite spins from the time of their entangling interaction.

Here is an animation illustrating the assumption that the two electrons are prepared, one in a spin-up, the other in a spin-down state. They remain in those states no matter how far they separate, provided neither interacts with anything else until the measurements at A and B.

Two "hidden constants" of the motion, one spin up, one down, completely explain the fact of perfect correlations of opposing spins. That "Nature's" initial choice of up-down versus down-up is quantum random explains why the bit strings can be used in quantum encryption.

The simple and intuitive idea that the two particles acquired their specific opposite spins when they were prepared, at the moment of entanglement interaction above is unfortunately wrong.

This simple, almost classical, picture does not capture the deep and mysterious quantum phenomenon that Einstein called spooky action-at-a-distance in 1927. The preferred spin directions for Alice and Bob do not appear until one of them makes a measurement.

What Schrödinger saw in 1935 is that neither particle has a definite spin direction until a measurement is made of either particle. Then, because the two-particle wave function Ψ₁₂ has total spin zero, when they are disentangled and become the product of two single-particle wave functions Ψ₁ and Ψ₂, if Ψ₁ is measured with spin particular direction, then Ψ₂ must have spin down in the exact opposite direction, to conserve total spin (angular momentum).

We can know the Ψ₂ spin without measuring it. It is "knowledge-at-a-distance."

We can still describe this situation as a single "hidden constant" of the motion. It is the total spin zero of the two-particle wave function Ψ₁₂. Spin is a constant because of the principle of conservation of angular momentum based on the rotational symmetry of that wave function.

So we can establish the fact that there is no preferred spatial direction for the two-particle entangled wave function. The founders of quantum mechanics, especially Werner Heisenberg, insists that the experimenter has a "free choice" as to which direction (or component of spin) to measure. It is therefore Alice's "free choice" that introduces the preferred direction into the EPR problem. And it is only measurements by Bob in that same (or opposite) direction that will yield the perfectly correlated (or anti-correlated) values needed for quantum encryption.

This preferred direction did not exist before Alice's measurement. And note that although Alice can choose the direction, she cannot choose the outcome, spin up or spin down. As Paul Dirac showed, the outcome is indeterministic, a matter of chance that he called "Nature's Choice."

There is no interaction or action-at-a-distance from Alice to Bob. When the two-particle Ψ₁₂ collapses into disentangled Ψ₁ and Ψ₂, the new single-particle wave functions have opposite spins in the direction Alice chose to measure.

Here is a crude two-dimensional animation of this picture,

If you move the timeline playhead slowly you can see the two spins are oscillating back and forth, always keeping the total spin zero. Richard Feynman described them as arrows spinning randomly in all directions, but none of these visualizations do justice to the underlying fact that there is no preferred direction for the rotationally symmetric total spin zero state of the entangled Ψ₁₂ wave function.

In his 1933 essay, "On the Method of Theoretical Physics," Albert Einstein argued that the greatest physical theories would be built on "principles," not on constructions derived from physical experience. His theory of special relativity was based on the principle of relativity, that the laws of physics are the same in all inertial frames, along with the constant velocity of light in all frames.

Our explanation of entanglement as the result of "hidden constants" of the motion is based on conservation laws, which, as Emmy Noether showed, are based on still deeper principles of symmetry.

This explanation is, of course, also based solidly on the empirical fact that electron spins are always found in opposite directions.

As Einstein should have seen in his discussion with Leon Rosenfeld, the conservation of total zero momentum of identical particles separating with equal but opposite velocities does not depend on any interaction between the particles. Neither particle is "influencing" the motion of the other one to keep their momenta perfectly opposite. They each conserve their own momentum.

The case is similar with two quantum particles, whether electrons, photons, atoms, or "buckyballs. " The two-particle wave function Ψ₁₂ describes the probability of finding the two particles somewhere if a measurement is made. As with any quantum wave function, particles can be found anywhere the squared modulus |Ψ²| is not zero.

Just as we can say nothing about where a single particle is located before measurement, so we cannot know where the two particles will be found when observed. But we can know that their relative spin directions will always be found to be exactly opposite one another, just as with Einstein's two particles with opposite linear momentum.

The perfect rotational symmetry of the two-particle wave function Ψ₁₂ ensures that every direction angle is equally probable. As Werner Heisenberg and Pascual Jordan liked to describe it, the properties of particles are "created" by the measurement, when one of the possible locations and angle directions becomes actual. Alice's "free choice" of direction to measure ensures that the spin will be found along that angle direction, but randomly up or down in that direction, according to Paul Dirac's idea that this is "Nature's choice." As long as Bob measures at the same pre-agreed angle, he will always find his spin opposite to the direction that Alice found, establishing the perfect anti-correlation of bits needed for quantum encryption.

The proper quantum description is that the two-particle wave function is in a linear combination of up-down and down-up states.

Either of these possible states can become actual when measured.

We might attempt a rough visualization (Schrödinger's Anschaulichkeit) of the spins of the entangled particles. It is something like an ice-skating couple holding hands while they rotate around one another. But the analogy Is weak, because quantum spins are not in ordinary 3-dimensional space.

Critics say that we mistakenly assume that the entangled particles acquire their (perfectly correlated) properties (spin values, positions) before Alice's or Bob's measurement.

The Copenhagen Interpretation and many others agree that quantum properties (spin, position, momentum) normally do not exist before their measurement. (Exceptions are cases of a "state preparation" where a system is put into a definite state. A subsequent measurement finds it in the same state - Pauli measurements of the first kind.)

Properties acquired during a measurement are cases in which we say that we "create reality." And we agree that Alice's measurement of spin-up or spin-down in some direction did not exist beforehand. It was truly random (Dirac calls it Nature's choice"). But note that her choice of measurement direction (angle) was her own (Heisenberg) "free choice."

Whether a random result or one deliberately chosen, Alice is creating the reality of the spin in her chosen direction. Bob will only get a perfectly correlated (or anti-correlated) result if he measures at exactly the same angle (by pre-agreement with Alice). Only in this case do Alice and Bob general the sequences of random bits needed for quantum cryptography.

So we agree with our critics who say the specific properties created by Alice and Bob (spin and direction angle) do not pre-exist their measurements. However, the critics are also correct that our explanation does require something to exist before the measurements. But what pre-exists is not individual particle properties.

It is instead a shared or joint property that the entangled particles acquired when initially entangled, and which they carry with them as they travel. We see this shared property as a "hidden constant of the motion" that functions just like the "hidden variables" that David Bohm and John Bell hoped for.

We believe this "hidden constant" fully explains the non-local behavior of entangled particles.

The "hidden constant" for entangled electrons or photons is the total spin zero state the particles are in. Without any external interaction, the total spin remains zero at all times by the law of conservation of angular momentum.

Critics are correct that we do not know (epistemology) and the particles do not have (ontology) specific spin directions or positions. But we do know, and the particles do have the joint or shared property of total spin zero at all times. (The two-particle wave function Ψ₁₂ is rotationally symmetric, a symmetry that underlies the conservation of angular momentum, according to Emmy Noether.)

Specifically, conservation of momentum laws mean that just before the measurement, whatever the unknown spins and positions of the entangled particles, their spins are exactly opposite to one another, and whatever the unknown positions of the particles, they are equidistant from and on opposite sides of the initial entanglement, as Einstein described to Leon Rosenfeld in 1933 and the 1935 EPR authors called "elements of reality" and a "paradox."

Again, although we do not know those spins and directions, because they are ontologically indeterminate, the fact that at the instant of Alice's measurement Bob's particle must have exactly opposite properties is not because an "influence" travels from Alice to Bob faster than the speed of light (Einstein's "spooky action at a distance). It is that the particles must have opposite momenta when measured because the total is zero.

If without any external interaction the conservation of angular momentum law were violated at some moment, it would be a much great problem for physics than entanglement itself, however mysterious and puzzling it may seem.

Erwin Schrödinger, Discussion of Probability between Separated Systems (Entanglement Paper), Proceedings of the Cambridge Physical Society 1935, 31, issue 4, pp.555-563

David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I

David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II

David Bohm and Yakir Aharonov, Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky

John Bell, On the Einstein-Podolsky-Rosen Paradox

"Albert Einstein, On the Method of Theoretical Physics," The Herbert Spencer Lecture, Oxford, June 10, 1933, Ideas and Opinions, Bonanza Books, 1954, pp.270-276; original German in Mein Weltbild, Amsterdam, 1934, (PDF)

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