Comments about "EPR paradox" in Wikipedia
This document contains comments about the article "EPR paradox" in Wikipedia
 The text in italics is copied from that url
 Immediate followed by some comments
In the last paragraph I explain my own opinion.
Contents
Introduction
The article starts with the following sentence.

The EinsteinPodolskyRosen paradox or EPR paradox of 1935 is an influential thought experiment in quantum mechanics with which Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen ("EPR") claimed to demonstrate that the wave function does not provide a complete description of physical reality, and hence that the Copenhagen interpretation is unsatisfactory.

It is IMO impossible to discuss this issue about the physical reality with a thought experiment.

The essence of the paradox is that particles can interact in such a way that it is possible to measure both their position and their momentum more accurately than Heisenberg's uncertainty principle allows, unless measuring one particle instantaneously affects the other to prevent this accuracy, which would involve information being transmitted faster than light as forbidden by the theory of relativity ("spooky action at a distance").

This sentence discusses two claims:
 First the claim that it is impossible to measure of one particle both position and momemtum accurately.
IMO that claim is true because it is impossible to measure the position twice.
 Secondly that measuring one particle instantaneous affects the other
That claim is wrong.

This consequence had not previously been noticed and seemed unreasonable at the time; the phenomenon involved is now known as quantum entanglement.

Quantum entanglement means that the state of two parameters are correlated, but that has nothing to do with either the uncertainty principle nor with instantaneous communication.

While EPR felt that the paradox showed that quantum theory was incomplete and should be extended with hidden variables, the usual modern resolution is to say that due to the common preparation of the two particles (for example the creation of an electronpositron pair from a photon) the property we want to measure has a well defined meaning only when analyzed for the whole system while the same property for the parts individually remains undefined.

This is a very complex way to describe something simple.
The problem is if you want to understand a reaction than you have to measure the parameters you are interested in both before and after the reaction.
 Suppose you want to understand A > p1 + p2.

In that case you have to measure the parameters of p1 and p2.
If you are interesting in the direction than you need a circle of detectors which monitor the arrival.
In some special cases you will detect a correlation. That means if p1 escapes under an angle of phi degrees, than p2 escapes under an angle of phi+180 degrees. Nothing special.
It is also possible that the particles (photon's) are polarised. To detect that you need some beam splitter. 





1. History of EPR developments










2 Quantum mechanics and its interpretation

Since the early twentieth century, quantum theory has proved to be successful in describing accurately the physical reality of the mesoscopic and microscopic world, in multiple reproducible physics experiments.

I would refrash this sentence:
Since the early twentieth century, quantum theory has evolved to be successful in describing accurately the physical reality of the mesoscopic and microscopic world, as a result of multiple physical experiments.

Philosophical interpretations of quantum phenomena, however, are another matter: the question of how to interpret the mathematical formulation of quantum mechanics has given rise to a variety of different answers from people of different philosophical persuasions.

The issue is to what extend you can describe the results of experiments by means of a (strict) mathematical relation.
In general answer is that this depents, how exactly you can repeat a physical experiment.

According to the understanding of quantum mechanics known as the Copenhagen interpretation, measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable that was measured.

The whole issue IMO if this collapse is something more mathematical or physical (or reverse).
IMO if you perform two measurements in succession, the first one will influence the second.

He (Albert Einstein) presented a thought experiment in which electrons are introduced through a small hole in a sphere whose inner surface serves as a detection screen.

How is it possible to perform such an experiments as a thought experiment which should describe the physical reality?
The reader is advised to read the full text.

Einstein asks what makes each electron's wave front "collapse" at its respective location.

How do you know that there is a "collapse" when you perform a thought experiment.
Next we read:

Why do the electrons appear as single bright scintillations rather than as dim washes of energy across the surface? Why does any single electron appear at one point rather than some alternative point?

Again how do you know that by means of a thought experiment?

The behavior of the electrons gives the impression of some signal having been sent to all possible points of contact etc.









3. Einstein's opposition

The 1935 EPR paper condensed the philosophical discussion into a physical argument.

I would expect something different. You have a physical argument and to solve it you have a philosophical discussion with the intention to go deeper (more basic) into the subject.

The authors claim that given a specific experiment, in which the outcome of a measurement is known before the measurement takes place, there must exist something in the real world, an "element of reality", that determines the measurement outcome.

Within the context there are three possible experiments:
 experiments with only 1 possible outcome.
This type is physical not interesting.
 experiments with 1 particle with 2 possible outcomes
In this cathegory are the head and tail experiments. You have a 50% chance to guess correctly.
 experiments with 2 particles which each 2 possible outcomes.
This type is discussed more in detail.
When you start from a specific experiment than you know, as established by multiple experiments, what the possible outcomes are.
Generally speaking for type 3 there are three different possibilities:
 The outcome is completely random. The outcome is correlated. Something in between

In case the outcome is correlated and when there are two particles then if you measure the state of one than you also know the state of the other (without any measurement). This conclusion does require an "element of reality" etc.

They postulate that these elements of reality are local, in the sense that each belongs to a certain point in spacetime.

I do not think that all of this is relevent.






4. Description of the paradox

The original EPR paradox challenges the prediction of quantum mechanics that it is impossible to know both the position and the momentum of a quantum particle.

What is written is 100% true, except I doubt if this is the EPR paradox.






4.1 EPR paper

The original paper purports to describe what must happen to "two systems I and II, which we permit to interact ...", and, after some time, "we suppose that there is no longer any interaction between the two parts.

Okay. Let us continue this argument line by line.

As explained by Manjit Kumar, the EPR description involves "two particles, A and B, [which] interact briefly and then move off in opposite directions."

Okay

According to Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B exactly.

That is correct for both particle A and B

However, it is possible to measure the exact position of particle A.

The issue is how do you do that?
I expect placing a detector in the path of particle A and monitoring by means of a clock the arriving time.

By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known.

No this is not as simple as it seems.
First you must describe what this calculation is. In fact this calculation is the result of an experiment.
The easiest way to perform this experiment is as follow:
 The first step you place one detector in path of particle A and a second detector in the path of particle B, however in such away (and that is important) at a position that the arrival time for detector B is simultaneous with the arrival time of detector A.
However this also raises a serious issue: how accurate can you do this?
This step is called "calibration" and should be performed multiple times
The purpose is that when you measure the position of A at t0 you also know the position of B at t0.
 The second step is that you place detector B further away and measure in a new (multiple) experiment both the arriving time t0 of A and the arriving time t1 of B.
 The third step is mathematics. As a result of step 1 you know the position of B at t0. That means both the position of particle B is know at t0 and t1. With simple mathematics you can now calculate the speed of particle B.
You can also do the same exercise for particle A and calculate the speed.

Alternatively, the exact momentum of particle A can be measured, so the exact momentum of particle B can be worked out.

You can only calculate the momemtum of each particle. In order to calculate the momentum you have to know the mass and its speed.

Kumar writes: "EPR argued that they had proved that ... [particle] B can have simultaneously exact values of position and momentum. ... Particle B has a position that is real and a momentum that is real."

The most critical part of this whole exercise is how accurate can you perform the above two steps of the experiment. That means how accurate you can calculate the speed and secondly its momemtum assuming that the speed is constant. Nothing is proved.

EPR appeared to have contrived a means to establish the exact values of either the momentum or the position of B due to measurements made on particle A, without the slightest possibility of particle B being physically disturbed.

This is not true. What you have performed is to calculate the speed of particle B with disturbing both: particle A and particle B.


4.2 Measurements on an entangled state

According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet. The particles are thus said to be entangled.

This is not according to quantum mechanics. This is according to 1000 identical experiments

This can be viewed as a quantum superposition of two states, which we call state I and state II. In state I, the electron has spin pointing upward along the zaxis (+z) and the positron has spin pointing downward along the zaxis (z). In state II, the electron has spin z and the positron has spin +z. Because it is in a superposition of states it is impossible without measuring to know the definite state of spin of either particle in the spin singlet,

All of this is in accordance with 1000 experiments.
Of course in a specific experiment you don't know if you are in state I or state II. You have to at least perform one measurement to decide.

Alice now measures the spin along the zaxis. She can obtain one of two possible outcomes: +z or z. Suppose she gets +z. According to the Copenhagen interpretation of quantum mechanics, the quantum state of the system collapses into state I. The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the zaxis, there is 100% probability that he will obtain z. Similarly, if Alice gets z, Bob will get +z.

You do not know the Copenhagen interpretation to understand this. Its all in the 1000 experiments.

Whatever axis their spins are measured along, they are always found to be opposite. This can only be explained if the particles are linked in some way.

They are linked as part of the reaction in which they are created. They are created correlated and stay correlated.

Either they were created with a definite (opposite) spin about every axisa "hidden variable" argumentor they are linked so that one electron "feels" which axis the other is having its spin measured along, and becomes its opposite about that one axisan "entanglement" argument.

The "entanglement" argument is wrong. A measuremnet does not involve any sort of feeling.






4.3 Locality in the EPR experiment

The principle of locality states that physical processes occurring at one place should have no immediate effect on the elements of reality at another location.

Such an effect implies instantaneous action which is impossible.

At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of special relativity, which states that information can never be transmitted faster than the speed of light without violating causality.

It is unfortunate that an other concept is introduced i.e. causality which implies a clear definition.

It is generally believed that any theory which violates causality would also be internally inconsistent, and thus useless.

It would be more worthwhile to claim that any theory that is causal is also local

It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle of locality without violating causality.

Again a very unclear sentence.
The following sentences are clear (but not necessarily true)
All experiments using a quantum mechanical description do not violate locality or causality.
There are experiments using a classical description that do not violate locality but only violate causality.
There are also experiments using a classical description that violate both locality and causality.

Causality is preserved because there is no way for Alice to transmit messages (i.e., information) to Bob by manipulating her measurement axis.

This has nothing to do with causality. This has more to do with honesty versus cheating.
The whole idea behind this experiment is not to transmit information from A to B.




5 Resolving the paradox








5.1 Hidden variables








5.1.1 Bell's inequality

In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are significantly different from the predictions of a particular class of hidden variable theories (the local hidden variable theories).

It is very difficult to study science based on thought experiments.

Roughly speaking, quantum mechanics has a much stronger statistical correlation with measurement results performed on different axes than do these hidden variable theories.

Almost impossible to understand sentence.




5.2 Einstein's hope for a purely algebraic theory








5.3 "Acceptable theories" and the experiment








5.4 Implications for quantum mechanics
 Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is a "paradox" only because classical intuitions do not correspond to physical reality.

The whole issue is that a theory should describe the observations.






6 Mathematical formulation


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