Bell's Theorem - Easy explained

• Edgardo
In summary: But that is a far more radical and difficult assumption than anything that has actually been demonstrated in the experiments so far.In summary, the article discusses various paradoxes that come with quantum mechanics, and how Bell's theorem helps to solve them. It also mentions that although there is no violation of locality in Bell's theorem, there is still a paradox that arises from the fact that the particles are still affecting each other even though they are not in the same location.
Edgardo
Last week I had a blast with reading explanations on Bell's theorem. It was the first time that I've actually understood it. So I wanted to share some websites that explain the theorem in an easy way:

1) Spooky Action at a Distance – An Explanation of Bell’s Theorem
by Gary Felder

2) Does Bell’s Inequality rule out local theories of quantum mechanics?
Updated May 1996 by PEG (thanks to Colin Naturman).
Updated August 1993 by SIC.
Original by John Blanton.

3) Einstein-Podolsky-Rosen paradox and Bell’s inequalities
by Jan Schütz
A seminar report introducing the CHSH inequality that is used for experiments.

4) Bell’s Theorem explained
A post on the blog Skeptic’s Play that uses set theory to explain Bell’s theorem.

5) Bell’s theorem analogy
David M. Harrison uses a classroom analogy to derive Bell’s inequality.

6) Violation of Bell’s theorem
Lecture notes by Leonard Susskind also using a Venn diagram.

7) Lecture 17 – Einstein-Podolski-Rosen Experiment and Bell’s Inequality
An excellent lecture by Prof. James Binney. You can download the lecture notes here. This lecture assumes that you know some quantum mechanics, e.g. how to calculate probabilities using Dirac’s bra-ket notation.
Note: If you have wondered too (like me) about the probability density function $\rho(\lambda)$ read this wiki article on local hidden variables. It explains that $\rho(\lambda)$ describes the probability that the source emits entangled particles with the hidden variable $\lambda$.

8) Paradigms in Physics: Quantum Mechanics
This is an online textbook made available by the Department of Physics, Oregon State University. Have a look at chapter 4 (quantum spookiness). Although they don’t use the term probability density (see note above in 7) it becomes clear now what is meant with $\rho(\lambda)$ . The authors use populations instead.

9) Bell’s Theorem with Easy Math and Bell’s Theorem and Negative Probabilities
Two articles by David R. Schneider also known as our DrChinese.

10) John Bell himself presenting his theorem
This is a talk given by John Bell at CERN. The youtube video has captions and you can also view a transcript of the talk.

Thank you, must check it out :) Is there anyone specific link out of the above you suggest for the ones among us with little time?

I read them in the order listed above. After reading the first link I've already understood what Bell's theorem is about, so I recommend the first one if you have little time.

Oh I am not sure if you have posted the link of this or not but there's a website of one of the PF users {Dr. Chinese} , his article proved to help me greatly with the understanding of bell's theorem.

Edgardo said:
I read them in the order listed above. After reading the first link I've already understood what Bell's theorem is about, so I recommend the first one if you have little time.
I like that crystal-clear explanation a lot, and it does demonstrate why complete locality is untenable, but it does have one flaw-- it makes the common mistake of going overboard on the "action at a distance" or "nonlocal effects" picture, which unnecessarily gets people worried about faster-than-light causality violations. There is no such violation, however-- no information can ever be demonstrated to travel FTL even in a Bell-type experiment. The article concludes: "It's one thing to say the electrons must affect each other instantly, but you might still wonder how an electron here instantly knows what is happening millions of miles away. Moreover, in order to explain the results we got, we had to say that the measurement of one electron somehow changed the other one. Why should the electron, either one, be affected at all by my measuring it? My intent was simply to measure a property of the electron, not to change it. This result demonstrates one of the other strange results of modern physics, which is that the act of measuring a property always changes the system you are measuring. In this case the "system" apparently includes not only the electron you are measuring, but also the other one which isn't even there at the time. Physicists have been trying for over fifty years to understand these results, and there is no consensus on how to interpret them. There is clear agreement, however, that the results occur. Spooky action at a distance is part of nature." (my bold)

But although the article is in all other respects painstakingly careful not to assume more than it can demonstrate, it misses the fact that it is not a requirement of the results of that experiment that the particles must carry with them "instructions" in order for them to present consistent results in the detectors. Of course, if one does assume the presence of such carried instructions, then the nonlocality must appear as a kind of nonlocal "change" to those local instructions, so a kind of faster-than-light "influence". But this is overinterpreted. Better, in my view, is simply to drop the claim that there are any "local instructions" to change in the first place. After all, if we have established the system is nonlocal, who needs local instructions (and FTL "influences") in the first place! As we drop locality, it is much simpler to simply state that the "instructions" are also inherently nonlocal, and do away with any need for "influences" to propagate around. This makes it much clearer why there is in fact no violation of special relativistic notions of causality, as long as we drop locality. The article implies that such SR causality limitations are violated, and that's just not true.

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ibysaiyan said:
Oh I am not sure if you have posted the link of this or not but there's a website of one of the PF users {Dr. Chinese} , his article proved to help me greatly with the understanding of bell's theorem.

Here are several links to my Bell pages:

Original references:
http://www.drchinese.com/David/EPR_Bell_Aspect.htm

Negative probabilities:
http://drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm

http://www.drchinese.com/Bells_Theorem.htm

Easy Math:
http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

I forgot to say that if you post a link, please enumerate it and add a short description. So, the next link would be 11).

ibysaiyan said:
Oh I am not sure if you have posted the link of this or not but there's a website of one of the PF users {Dr. Chinese} , his article proved to help me greatly with the understanding of bell's theorem.

DrChinese's introduction is link number 9 in the list.

11) Nonlocal correlations between the Canary Islands
This is an excellent blogpost on the blog BackReaction. It discusses the role of nonlocality in Bell's theorem, in particular the locality loophole and the freedom-of-choice loophole.
Prerequisite: Understanding of Light cones. Here are some practice questions on light cones.

12) http://www.phys.washington.edu/users/schick/325A/mermin.pdf
This is a very intuitive article by David Mermin. He introduces machines that have 3 settings and two lamps (red and green) and shows that assigning 3 "real" properties to particles results in a contradiction.

13) Spooky Actions At A Distance?: Oppenheimer Lecture
A lecture by David Mermin on EPR and Bell's theorem. Here, he uses three entangled particles (see Greenberger–Horne–Zeilinger state) instead of two.
I recommend watching this lecture after you have read 12).

14) Bertlmann's socks and the nature of reality
by John Stewart Bell

Here, John Bell derives the d'Espagnat inequality by considering socks that may or may not survive one thousand washing cycles at 45°C, 90°C and 90°C.

The d'Espagnat inequality is: $N(A,notB) + N(B,notC) \geq N(A,notC)$
Bell mentions that this is trivial: Each member in $N(A,notC)$ on the right hand side either doesn't have property B and therefore is in $N(A,notB)$ or has property B and therefore is in $N(B,notC)$. Thus, the left hand side cannot be less than the right hand side, in other words the left hand side is greater or equal than the right hand side.

Note: When you read the document don't wonder about the figures. They are not missing but shown in the end.

----

By the way, Reinhold Bertlmann was a colleague of John Bell at CERN. He is a professor now and still seems to wear http://homepage.univie.ac.at/reinhold.bertlmann/socks.html.

Bell's original paper (see DrChinese's site) becomes much more understandable with this document.

Edgardo said:
8) Paradigms in Physics: Quantum Mechanics
This is an online textbook made available by the Department of Physics, Oregon State University. Have a look at chapter 4 (quantum spookiness). Although they don’t use the term probability density (see note above in 7) it becomes clear now what is meant with $\rho(\lambda)$ . The authors use populations instead.

How exactly do you get into each chapter to read it?

I would recommend Nick Herbert's explanation http://quantumtantra.com/bell2.html. The style of the proof is similar to Mermin's famous essay, but the logic is even simpler than Mermin's. The exact numbers used in Herbert's article, 0, 30, and 60, happen to be the ones used by Bell himself when he used to explain his theorem to popular audiences.

Moderator's Note: Thread reopened. Please keep on topic. Per our rules, posts pushing non-mainstream views will be deleted.

Edgardo said:
Last week I had a blast with reading explanations on Bell's theorem. It was the first time that I've actually understood it. So I wanted to share some websites that explain the theorem in an easy way:

I have also just finished reading Nick Herbert's A simple proof of Bell's Theorem.
I had questions about local and non-local quantum events, now it is clear to me like daylight.
Thanks for the links, I'll read each of them as I get free time.

Doc Al said:
Moderator's Note: Thread reopened. Please keep on topic. Per our rules, posts pushing non-mainstream views will be deleted.
Which posts in this thread are "pushing non-mainstream views"? Or, have they already been deleted? Just curious.

Neandethal00 said:
I have also just finished reading Nick Herbert's A simple proof of Bell's Theorem.
I had questions about local and non-local quantum events, now it is clear to me like daylight.
I'm curious how reading Nick Herbert's simple proof of Bell's theorem cleared you up on locality/nonlocality.

ThomasT said:
Which posts in this thread are "pushing non-mainstream views"? Or, have they already been deleted? Just curious.

ThomasT said:
I'm curious how reading Nick Herbert's simple proof of Bell's theorem cleared you up on locality/nonlocality.

Hey, I'm a newbie in quantum entanglement and Bell world. I didn't even know what they meant by quantum locality and non-locality.

Now I know an event in Anaheim (local) can be affected by an event in Baltimore (non-local). Even then I hoped, instead of showing percentage of mismatch, Herbert showed how measurements in Anaheim are changed by events in Baltimore.

Neandethal00 said:
Hey, I'm a newbie in quantum entanglement and Bell world. I didn't even know what they meant by quantum locality and non-locality.

Now I know an event in Anaheim (local) can be affected by an event in Baltimore (non-local). Even then I hoped, instead of showing percentage of mismatch, Herbert showed how measurements in Anaheim are changed by events in Baltimore.
I don't think that's what Herbert showed. What he showed was the simplest realization of Bell's theorem. The problem is that wrt Herbert's demonstration light would be expected, via a local realistic model, to behave in a way that light has never been observed to behave.

Suppose that our source emits only one type of electrons. The type that give green on 1 and 3, and red for 2. We agree that only she has the detector on her part and on my part the detector is inactive. Once I want to send an alarm to her, I activate my detector. So, she will start receiving 50% probability instead of 100% (or 0%) like did before.

Seems like this would be a way to send information instantly.

igorcov said:

Suppose that our source emits only one type of electrons. The type that give green on 1 and 3, and red for 2. We agree that only she has the detector on her part and on my part the detector is inactive. Once I want to send an alarm to her, I activate my detector. So, she will start receiving 50% probability instead of 100% (or 0%) like did before.

Seems like this would be a way to send information instantly.

Welcome to PhysicsForums, igorcov!

There are no such electrons possible. Entangled particles will not have a known spin (color in the example).

Thanks for answer. So if I understand correctly the spin changes randomly after each measurement. A reasonable question is in this case: Could the measurement tool change influence that?

Also, have this been confirmed with other properties than the spin?

igorcov said:
Thanks for answer. So if I understand correctly the spin changes randomly after each measurement. A reasonable question is in this case: Could the measurement tool change influence that?

Also, have this been confirmed with other properties than the spin?

Good questions!

Spin can change after a measurement, depends on the state prior to measurement: If unknown, then becomes known but non-commuting ones remain completely unknown and will yield a random value if subsequently measured. If it is H> and is measured on that basis again, it will yield H>. Other angles will follow the cos^2 rule.

Does the measurement tool influence an entangled particle? Sure you could say that, but you need to keep some constraints in mind. If you measure at the same angles, you get a known relative outcome every time. So whatever is "changing" changes the same way for both.

it is mathematically proved (assuming locality) that:

Probability of getting the same color from both detectors > 5/9
the actual results comes as 1/2
however if there was entanglement (non-local) then the result should be 1/3

so we have three (probabilistic) numbers:

1. >5/9 -- assuming locality
2. 1/3 -- assuming entanglement
3. 1/2 -- actual observed result

how do we reconcile 2 & 3 above?

are we assuming that the entanglement (effect) is partial, not full?

1. What is Bell's Theorem and why is it important?

Bell's Theorem is a concept in quantum mechanics that describes the relationship between entangled particles. It is important because it helps us understand the strange and counterintuitive behavior of particles at the quantum level.

2. How does Bell's Theorem challenge our understanding of reality?

Bell's Theorem challenges our understanding of reality by showing that there are non-local interactions between particles, meaning that the actions of one particle can affect the behavior of another particle regardless of the distance between them. This goes against our conventional understanding of cause and effect.

3. Can you explain Bell's Theorem in simple terms?

Bell's Theorem states that if two particles are entangled, measuring the state of one particle will instantaneously determine the state of the other particle, regardless of the distance between them. This implies that there is a hidden connection between the two particles that we cannot see or measure directly.

4. What are the implications of Bell's Theorem?

The implications of Bell's Theorem are wide-ranging and still being explored. It suggests that the universe is not as deterministic as we once thought and that there may be hidden variables at play that we cannot measure. It also has implications for quantum computing and communication.

5. How is Bell's Theorem tested and verified?

Bell's Theorem has been tested and verified through various experiments, including the famous Bell test experiments. These experiments involve measuring the correlation between the spin of two entangled particles in different directions. If the results of the measurements violate Bell's Inequality, then Bell's Theorem is confirmed.

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