Photon entanglement: why three angles?

[johana]

Here's something from DrChinese younger self.

https://www.physicsforums.com/showthread.php?t=39614&page=7

Despite what you (and others) might think, you don't need to change polarizer settings in flight or otherwise vary the angles to test Bell's Theorem. You only need to calculate the correlation percentages at three particular angle settings (these can be done fully independently). Then combine a la Bell.

Varying is only necessary if you are asserting that the measurement devices are (or might be) communicating with each other so as to affect the outcome of the correlation tests. We already know from Aspect that doesn't happen, because he did the experiments both ways and there was no difference in the outcomes! Even that should be a definitive conclusion of Aspect. Further regarding the varying issue:

a. If you are a local realist, I would assume that wouldn't be much of an issue to you since you think there are classical, intuitive explanations for everything anyway - strange new types of communication between measuring devices should not be an issue.
b. If, on the other hand, you follow the Copenhagen interpretation, varying also shouldn't matter as you don't isolate out communication with other parts of the measurement apparatus for any other type of experiment (such as double slit) either.
c. Also, if you believe the correlation is non-local then the varying analyzers are superfluous.
d. And finally, if you are a local non-realist like me :) then you already believe that the only "real" component being measured is the angle between the remote polarizers anyway i.e. the measurement is fundamental to the process.

This makes perfect sense. Switching angles is unnecessary and is not a substitute for placing detectors far apart. It's about some out of this world type of theory neither nonlocalists nor local realists care to imagine even in their wildest dreams. The only thing that doesn't make sense is "local non-realist". What in the world is that?

 

[stevendaryl]

This makes perfect sense. Switching angles is unnecessary and is not a substitute for placing detectors far apart. It's about some out of this world type of theory neither nonlocalists nor local realists care to imagine even in their wildest dreams. The only thing that doesn't make sense is "local non-realist". What in the world is that?

The quote from Dr. Chinese points out that switching angles in-flight is only unnecessary because people have ALREADY showed that it makes no difference. To demonstrate that there is no local, classical explanation for EPR, you have to check out the possibility that the filter settings affect the outcome in a slower-than-light way.

 

[johana]

The quote from Dr. Chinese points out that switching angles in-flight is only unnecessary because people have ALREADY showed that it makes no difference. To demonstrate that there is no local, classical explanation for EPR, you have to check out the possibility that the filter settings affect the outcome in a slower-than-light way.

Switching angles does not relate to any local or classical explanation, it's about nonlocal correlation between polarizers, not just photons. No one is proposing that, they are just messing up measurements with additional unnecessary complexity and randomness.


DrChinese explains redundancy of it well here:

a. If you are a local realist, I would assume that wouldn't be much of an issue to you since you think there are classical, intuitive explanations for everything anyway - strange new types of communication between measuring devices should not be an issue.

b. If, on the other hand, you follow the Copenhagen interpretation, varying also shouldn't matter as you don't isolate out communication with other parts of the measurement apparatus for any other type of experiment (such as double slit) either.

c. Also, if you believe the correlation is non-local then the varying analyzers are superfluous.

d. And finally, if you are a local non-realist like me :) then you already believe that the only "real" component being measured is the angle between the remote polarizers anyway i.e. the measurement is fundamental to the process.

 

[DrChinese]

The only thing that doesn't make sense is "local non-realist". What in the world is that?

There are a number of non-realistic interpretations. Not everyone will concur with my categorization, but this is an answer to your question.

We all know that the Bohmian group (de Broglie-Bohm, Bohmian Mechanics) and several others are explicitly non-local. So I call anything that is not EXPLICITLY non-local to be "non-realistic" by definition (to comply with Bell). That would then include: Many Worlds, Relational Blockworld (ask RUTA about that), Cramer's Absorber, Aharanov's Time Symmetric QM, and a few others.

You mentioned something in an earlier post about non-causal situations (future affects the past). Before you rule those out, you might take note: there are substantial experiments that demonstrate the future affects the past. These experiments are not a rock solid proof of same, but they are definitely powerful evidence. For example:

http://arxiv.org/abs/quant-ph/0201134
Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping
Thomas Jennewein, Gregor Weihs, Jian-Wei Pan, Anton Zeilinger
(Submitted on 29 Jan 2002)

"Quantum teleportation strikingly underlines the peculiar features of the quantum world. We present an experimental proof of its quantum nature, teleporting an entangled photon with such high quality that the nonlocal quantum correlations with its original partner photon are preserved. This procedure is also known as entanglement swapping. The nonlocality is confirmed by observing a violation of Bell's inequality by 4.5 standard deviations. Thus, by demonstrating quantum nonlocality for photons that never interacted our results directly confirm the quantum nature of teleportation. "

See page 5. All of this is garden QM, as time ordering is not critical in many setups using entangled systems.

 

[stevendaryl]

Switching angles does not relate to any local or classical explanation, it's about nonlocal correlation between polarizers, not just photons.

I'm not sure what is the root of our communication problems, but something is not getting communicated here. Switching angles at the last possible minute does relate to local or classical explanations in the sense that it proves that there are no such explanations. If you DON'T switch, then that leads to the possibility that the settings affect the hidden variable in a local way.

No one is proposing that, they are just messing up measurements with additional unnecessary complexity and randomness.

They're closing a possible loophole. That's all. I don't understand what it is that you don't understand about it.

 

[johana]

Switching angles at the last possible minute does relate to local or classical explanations in the sense that it proves that there are no such explanations. If you DON'T switch, then that leads to the possibility that the settings affect the hidden variable in a local way.

Varying is only necessary if you are asserting that the measurement devices are (or might be) communicating with each other so as to affect the outcome of the correlation tests.

Do you disagree with this? Are you suggesting supposed communication between distant measurement devices is a local theory?

 

[DrChinese]

Do you disagree with this? Are you suggesting supposed communication between distant measurement devices is a local theory?

When you say "distant measurement devices" in this context, we all know that means devices that are sufficiently far apart so that no signal could move from one to the other at a speed less than c so that Alice's setting could be transmitted to Bob (and vice versa). A typical batch of observations takes a few minutes. So that only helps if Alice and Bob are FAR removed from each other because after 1/20 of a second, there is no place on Earth that far removed. So you must randomly select and change settings very fast: fast switching.

To put things in context: a nanosecond is about a foot. Bell tests are usually done perhaps 5 to 500 feet apart (although much larger distances have been done too). And PDC pair production is on the magnitude of 1,000-10,000 per second.

The idea of fast switching was to PROVE there could be no communication between distant measuring devices via some sub-c mechanism which did not fit into any theory at all. If you can imagine, the local realist was trying to say: "There COULD be something local occurring that QM does not contemplate." Fast switching proves that wrong.

 

[johana]

We all know that the Bohmian group (de Broglie-Bohm, Bohmian Mechanics) and several others are explicitly non-local. So I call anything that is not EXPLICITLY non-local to be "non-realistic" by definition (to comply with Bell). That would then include: Many Worlds, Relational Blockworld (ask RUTA about that), Cramer's Absorber, Aharanov's Time Symmetric QM, and a few others.

I'm not sure "non-realistic" is adequate substitute for "non-local". How about Newton's gravity, would you say it's non-local just because interactions are instantaneous?

 

[DrChinese]

I'm not sure "non-realistic" is adequate substitute for "non-local". How about Newton's gravity, would you say it's non-local just because interactions are instantaneous?

"Non-realistic" is not a substitute for "non-local" in any sense I am aware.

Traditional Newtonian gravity is non-local, yes definitely.

 

[johana]

The idea of fast switching was to PROVE there could be no communication between distant measuring devices via some sub-c mechanism which did not fit into any theory at all. If you can imagine, the local realist was trying to say: "There COULD be something local occurring that QM does not contemplate." Fast switching proves that wrong.

The problem is that it's usually described as an essential part of the experiment, which makes it far from obvious it's just there to close some loophole no one even cares about.

Traditional Newtonian gravity is non-local, yes definitely.

Classical electromagnetism too. So the whole of classical physics was already non-local even before there was any QM. Why are we surprised then? What is different between QM non-locality and classical physics non-locality?

 

[stevendaryl]

Classical electromagnetism too. So the whole of classical physics was already non-local even before there was any QM. Why are we surprised then? What is different between QM non-locality and classical physics non-locality?

I wouldn't say that classical electromagnetism was nonlocal. Classical electromagnetism is the paradigm example of a local theory---no effect can propagate faster than the speed of light.

 

[johana]

I wouldn't say that classical electromagnetism was nonlocal. Classical electromagnetism is the paradigm example of a local theory---no effect can propagate faster than the speed of light.

I wouldn't call it non-local myself, but Coulomb and Lorentz force equations assume instantaneous interaction over distance just like Newton's law of gravity. Whether instantaneous action over distance indeed implies "non-local", I'm not sure, that's the question.

 

[billschnieder]

Fast random switching has nothing to do with non-locality. It is done *only* to eliminate the possibility that Alice and Bob have conspired in advance to manipulate the results. As to the topic question, "Why 3 angles". Actually almost no experiments measure 3 angles. They measure 4. And that is because nobody uses the original Bell's inequalities, they all use the CHSH version which is based on 4 angles. So the reason why they use 3 angles rather than just 2 is because they are not primarily trying to test the QM prediction, but also to test the inequality which was derived using 3 angles (Bell) or 4 angles (CHSH).

BTW: I do not share the believe that those inequalities actually apply to the experiments they are being used for, as has been discussed many times here already, and in the literature.

 

[Avodyne]

Coulomb and Lorentz force equations assume instantaneous interaction over distance just like Newton's law of gravity. Whether instantaneous action over distance indeed implies "non-local", I'm not sure, that's the question.

The complete EM interaction is not instantaneous. If two stationary charges are one light-year apart, and then you move one of them, the total force felt by the other charge does not change until one year later. The same is true of the gravitational force in general relativity.

What is different between QM non-locality and classical physics non-locality?

See this article by David Mermin from 1985: https://cp3.irmp.ucl.ac.be/~maltoni/PHY1222/mermin_moon.pdf
Begin with "A gedanken demonstration" on page 4.

 

[DrChinese]

What is different between QM non-locality and classical physics non-locality?

As a matter of convention, "classical physics" usually considers General Relativity rather than Newtonian gravity, making c fundamental. Further, causes precede effects and the observer does NOT have a fundamental role in defining reality ("the moon is there when no one looks").

We now realize that this neat and pretty picture of our universe is not accurate. Of course, beauty is in the eye of the beholder.

 

[johana]

Fast random switching has nothing to do with non-locality. It is done *only* to eliminate the possibility that Alice and Bob have conspired in advance to manipulate the results.

Is it about determinism, choice and free will? Is it actually a part of the inequality derivation?

BTW: I do not share the believe that those inequalities actually apply to the experiments they are being used for, as has been discussed many times here already, and in the literature.

There you said this:

In other words, as Alice or Bob rotates their polarizers, the coincidence counts change.

Is that coincidence count the same thing as the number of matching pairs? Isn't it supposed to change as Alice or Bob rotate their polarizers, what's your objection about?

 

[DrChinese]

Is that coincidence count the same thing as the number of matching pairs? Isn't it supposed to change as Alice or Bob rotate their polarizers, what's your objection about?

You are asking billschnieder's viewpoint. As he and I have had many discussions about this point, I will pass this on - and which you should take as a fair summary:

1. billschnieder is a local realist. Post-Bell, local realism is generally not considered viable.

2. Discussing the pro's and con's of local realism is outside of the scope of this thread. If you want to discuss his viewpoint and reasoning further, that should be in a new thread.

3. Even in a new thread, you should be aware that this is a moderated forum in which generally accepted science is discussed. No one has the right to put forth their own personal opinions when such viewpoint does not have suitable references to support same. You can check the forum guidelines for details, but that policy is enforced.

 

[billschnieder]

Is it about determinism, choice and free will? Is it actually a part of the inequality derivation?

Neither. It is only for avoiding a conspiracy in which Alice and Bob freely chose in advance to manipulate the results and fool everyone else.

Is that coincidence count the same thing as the number of matching pairs? Isn't it supposed to change as Alice or Bob rotate their polarizers, what's your objection about?

Yes, coincidence count means the same as number of matching pairs. Trick question: is coincidence a local result, Why can't you use coincidence to send information?

As to the other question, you can find my answer here.

 

[johana]

Neither. It is only for avoiding a conspiracy in which Alice and Bob freely chose in advance to manipulate the results and fool everyone else.

Why are there three angles in the derivation then?

Yes, coincidence count means the same as number of matching pairs. Trick question: is coincidence a local result, Why can't you use coincidence to send information?

Because each side receives only half of the whole information?

 

[stevendaryl]

Why are there three angles in the derivation then?

Because three works and two doesn't.

 

[billschnieder]

Why are there three angles in the derivation then?

Because the particles come in pairs and the "magic trick" requires talking about outcomes we did not measure but could have, so we need at least 3 angles.

Because each side receives only half of the whole information?

Yes, because you need information from both sides to determine coincidences (ie, coincidence is "nonlocal" information). Isn't that the same reason why you can't use entanglement or "nonlocality" to transmit information? I'll leave it up to the reader to figure out the implications.

 

[DrChinese]

Why are there three angles in the derivation then?

Please check out my post #6 in this thread. I explain in detail why 3 angles are needed (as stevendaryl states in the previous post). You can also do with 4 or more. Don't be confused about 3 angles vs 2 measurements. That is not an issue in the science of this.

Keep in mind the local realist position: particle attributes exist and are well defined at all times, immune from the changes of other particles at some distance. That means one photon has many predetermined "elements of reality" (to use EPR wording). They must be predetermined because they can be predicted with certainty, the logic goes, even though they cannot ALL be predicted with certainty simultaneously. Read EPR and you will see this explicitly stated.

When you compare the possible values of those elements of reality for 3 angles, you realize that no ensemble of them can reproduce the Malus relationship (we are still talking about a single stream of photons, not pairs). Since the reality of particle attributes must be subjective in some respect (dependent on the nature of measurements made and NOT fully predetermined), our premises fail.

 

[billschnieder]

Keep in mind the local realist position: particle attributes exist and are well defined at all times, immune from the changes of other particles at some distance.

Yes. Local realists make a clear distinction between particle attributes and observables in an experiment involving particles.

That means one photon has many predetermined "elements of reality" (to use EPR wording). They must be predetermined because they can be predicted with certainty

Yes, one photon has many real attributes, but the outcome if a measurement which also includes a measuring device can not be said to "belong" to the photon. It belongs to the whole experimental setup. It can not be said to exist before the experiment has been done, even if the particle attributes do exist before the experiment. This is the local realist view. We've discussed this previously here

When you compare the possible values of those elements of reality for 3 angles, you realize that no ensemble of them can reproduce the Malus relationship (we are still talking about a single stream of photons, not pairs).

You probably mean values of observables and not values of particle attributes. But what malus relationship for a single stream of unpaired photons at 3 angles ??

 

[DrChinese]

Yes, one photon has many real attributes, but the outcome if a measurement which also includes a measuring device can not be said to "belong" to the photon. It belongs to the whole experimental setup. It can not be said to exist before the experiment has been done, even if the particle attributes do exist before the experiment.

This is a cockamamie description of the EPR viewpoint. (And we do not need to hear your view, since it is not a generally accepted viewpoint.) The combo of the photon attributes AND the measuring device is an ELEMENT OF REALITY in the EPR local realist view. That is because the outcome of ANY measurement can be predicted with certainty PRIOR to actually performing that measurement. Further, EPR explicitly says that it is not reasonable to require all possible outcomes to be simultaneously predictable. Of course, that is their definition of realism. 1 angle, 2 angles, 3 angles, 360 angles, they all are pre-existing to EPR. The measurement device itself plays a role, sure, but that role must be very limited to get the same answer every time. If it added some element of randomness, we wouldn't be able to predict the outcome in advance with certainty.

Bill, stick with the straight historical interpretation of EPR/Bell/Aspect. Don't derail the thread with your pet ideas, or the outcome will be the same as the other times.

 

[Dale]

I wouldn't call it non-local myself, but Coulomb and Lorentz force equations assume instantaneous interaction over distance just like Newton's law of gravity.

Coulombs law is not a law of classical EM. The actual laws of classical EM differ from Coulombs law to prevent instantaneous action at a distance.

 

[DrChinese]

But what malus relationship for a single stream of unpaired photons at 3 angles ??

A single stream of photons, not pairs Bill. A stream from Alice alone. Those photons, according to the view espoused by EPR, have particle attributes independent of the act of observation. At all angles, say 0, 120 and 240 degrees. The polarization attributes of ANY photon stream polarized some way at 0 degrees has a % polarization relationship discovered by Malus at any other angle such as 120 or 240 degrees.

Pairs have nothing to do with this. The pairs vs triples thing is simply a ruse you execute to confuse others. Stop and instead, please assist others with the standard program. You have enough knowledge to help. Everyone has their own pet ideas, but they are not welcome here as you well know from past experience.

 

[billschnieder]

A single stream of photons, not pairs Bill. A stream from Alice alone. Those photons, according to the view espoused by EPR, have particle attributes independent of the act of observation. At all angles, say 0, 120 and 240 degrees. The polarization attributes of ANY photon stream polarized some way at 0 degrees has a % polarization relationship discovered by Malus at any other angle such as 120 or 240 degrees.

Pairs have nothing to do with this. The pairs vs triples thing is simply a ruse you execute to confuse others. Stop and instead, please assist others with the standard program. You have enough knowledge to help. Everyone has their own pet ideas, but they are not welcome here as you well know from past experience.

Please calm down and read my question again, I'm simply asking you to elaborate what you mean. You said

When you compare the possible values of those elements of reality for 3 angles, you realize that no ensemble of them can reproduce the Malus relationship (we are still talking about a single stream of photons, not pairs).

So I asked you what malus relationship are you talking about which involves a single stream of unpaired photons at 3 angles? Are you saying no ensemble of photons can reproduce the classical malus law? But malus law involves 2 angles not three so you will have to explain what you mean because it is not clear from your statement. No one other than you has mentioned pairs vs triples etc, and I'm not sure what alleged pet idea has you riled up. I'm talking pretty standard stuff here. Everyone knowledgeable in this field knows the difference between $\lambda$ and $A,B$, The former are the hidden variables which are claimed to exist prior to measurement, while the latter are the observables which only exist after measurement. Don't confuse the two as you appear to be doing.

 

[DrChinese]

So I asked you what malus relationship are you talking about which involves a single stream of unpaired photons at 3 angles? Are you saying no ensemble of photons can reproduce the classical malus law? But malus law involves 2 angles not three so you will have to explain what you mean because it is not clear from your statement.

Simple, and I am referring to EPR as a starting point. The following is not the view of QM.

Any set of Alice's photons (a stream) has polarization at all angles independent of the act of observation. That is because the polarization can be predicted in advance by looking at matching Bob (in the ideal case of course). Those angles would include the 3: 0/120/240 degrees.

1. According to Malus, the statistical match rate M() between any two of those angles (of Alice) is 25% (cos^2(theta or 120 degrees difference in this case). And further: M(0,120) = M(120,240) = M(0,240).

2. Since Alice and Bob are polarization clones (demonstrated by the perfect correlations), we can measure any element of Alice by measuring Bob. This allows us to accurately determine 2 simultaneous elements of Alice - one by measuring Bob, the other by measuring Alice. This would even allow us to know more than the HUP allows (this was the EPR reasoning).

3. So we now know Alice's match rate for any of the 3 pairs of angles of Alice. Since the nature of our observation, BY DEFINITION, cannot change the underlying reality, it does not matter which of the three match rates we choose to observe, M(0,120), M(120,240) or M(0,240).

But there is no underlying data set of values which will satisfy Malus at all three sets of angles for the Alice stream, as required by 3. Ergo, one of our assumptions must be wrong. The only one added for local realism is the requirement that Alice have simultaneous polarization values independent of the act of observations (realism). So that must be false. Or, as EPR points out, there is spooky action at a distance.

 

[billschnieder]

Simple, and I am referring to EPR as a starting point. The following is not the view of QM.

Any set of Alice's photons (a stream) has polarization at all angles independent of the act of observation. That is because the polarization can be predicted in advance by looking at matching Bob (in the ideal case of course). Those angles would include the 3: 0/120/240 degrees.

1. According to Malus, the statistical match rate M() between any two of those angles (of Alice) is 25% (cos^2(theta or 120 degrees difference in this case). And further: M(0,120) = M(120,240) = M(0,240).

2. Since Alice and Bob are polarization clones (demonstrated by the perfect correlations), we can measure any element of Alice by measuring Bob. This allows us to accurately determine 2 simultaneous elements of Alice - one by measuring Bob, the other by measuring Alice. This would even allow us to know more than the HUP allows (this was the EPR reasoning).

3. So we now know Alice's match rate for any of the 3 pairs of angles of Alice. Since the nature of our observation, BY DEFINITION, cannot change the underlying reality, it does not matter which of the three match rates we choose to observe, M(0,120), M(120,240) or M(0,240).

But there is no underlying data set of values which will satisfy Malus at all three sets of angles for the Alice stream, as required by 3. Ergo, one of our assumptions must be wrong. The only one added for local realism is the requirement that Alice have simultaneous polarization values independent of the act of observations (realism). So that must be false. Or, as EPR points out, there is spooky action at a distance.

You have a single stream of Alice's photons, there is no such thing as match rate for a single photon. What is matching what?

 

[johana]

Because three works and two doesn't.

Can you be more specific, work towards what goal? Are you talking about some equation, some law of physics, mathematics or logic?

 

[johana]

Because the particles come in pairs and the "magic trick" requires talking about outcomes we did not measure but could have, so we need at least 3 angles.

What is "magic trick", some equation? It requires 3 angles to achieve what goal?

 

[Nugatory]

Can you be more specific, work towards what goal? Are you talking about some equation, some law of physics, mathematics or logic?

From much earlier in this thread: https://www.physicsforums.com/showpost.php?p=4836252&postcount=3

We're trying to set up a situation in which the quantum mechanical prediction differs from any local hidden-variable theory that might have satisfied the EPR trio. That's the goal.

Bell's theorem shows that certain three-angle setups will work for that purpose.

 

[billschnieder]

What is "magic trick", some equation? It requires 3 angles to achieve what goal?

For 3 angles $a,b,c$ with outcomes $A,B,C$ each of which can be +1 or -1, you can do the following algebra
$AB - AC = A(B - C)$
Remembering that $BB = 1$
$A(B - C) = A(B - BBC) = AB(1 - BC)$
therefore
$AB - AC = AB(1 - BC)$
Taking absolute values
$|AB - AC| \leq |AB||(1 - BC)|$
since $|AB| = 1$ and $(1 - BC)$ is always positive anyway
$|AB - AC| \leq (1 - BC)$
and therefore
$|AB - AC| + BC \leq 1$
This is a Bell inequality.

Notice the absence of "locality" or "realism" in the above derivation. The "magic trick" is how we started with just AB and AC, and all of a sudden you have BC in the final expression. You can't do this trick without a third angle.

 

[DrChinese]

You have a single stream of Alice's photons, there is no such thing as match rate for a single photon. What is matching what?

Apparently you do not understand a basic application of Malus, circa 1809.

A stream of Alice photons polarized at 0 degrees as + will have a 25% chance of being polarized + at 120 degrees. A stream of Alice photons polarized at 120 degrees as + will have a 25% chance of being polarized + at 240 degrees. A stream of Alice photons polarized at 0 degrees as + will have a 25% chance of being polarized + at 240 degrees. So if it passes the polarizer, it is matched.

To the EPR local realist, a single photon has polarization properties at all angles which are definite at all times independent of the act of observation. The classical relationship between these values was determined long ago to be statistical in nature (a la Malus). That there is no possible dataset that could account for this was never considered because it was not clear that the polarization would be pre-determined at all possible angles. The advent of entanglement, as pointed out in EPR, to add this critical point.

Of course, EPR intended to provide a counter-example to the HUP to disprove the completeness of QM. They didn't realize that QM's observer dependent predictions would upset their apple cart, so to speak.

 

[DrChinese]

What is "magic trick", some equation? It requires 3 angles to achieve what goal?

You really need to read or understand Bell's Theorem, which reveals the "magic trick". You can find it here, although it is in a form which is a lot more difficult to follow than most lay derivations:

On the Einstein Podolsky Rosen paradox
http://www.drchinese.com/David/EPR_Bell_Aspect.htm

It explains everything, see his [14] where the third angle is introduced. Or see another of my Bell derivations that shows the impossibility of certain local realistic predictions (specifically a negative probability) using a modified form of the Bell reasoning:

Bell's Theorem and Negative Probabilities
http://www.drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm

You have been given the explanation in lay terms here. But there is no shortcut to the understanding of the 3 angles beyond what has been presented already. You must work it through at some point yourself.

 

[stevendaryl]

Can you be more specific, work towards what goal? Are you talking about some equation, some law of physics, mathematics or logic?

Okay, one version of the EPR experiment uses spin-1/2 particles: Through some process, an electron-positron pair is created and it is found that for any direction \vec{a}, if the electron is measured to be spin-up in direction \vec{a}, then the corresponding positron will measured to be spin-down in that direction. So the hypothesis is that for each electron produced, and for each possible direction \vec{a}, it is somehow pre-determined whether the electron is spin-up or spin-down in that direction.

What this hypothesis means is that associated with the n^{th} electron/positron pair, there is a function F_n(\vec{a}) that returns +1 if the electron has spin up in direction \vec{a} and returns -1 if the electron has spin-down in that direction. The corresponding function for the positron is just the negative of F_n.

What Bell's theorem shows is that there is no such function. Or rather, that no such function can possibly reproduce the predictions of quantum mechanics.

We can make the problem discrete by considering, not the full range of vectors \vec{a}, but some finite set of M possibilities: \vec{a}_1, \vec{a}_2, ..., \vec{a}_M. Let R_{i,j} be F_i(\vec{a}_j). So i refers to which electron/positron pair, and j refers to which direction its spin is measured with respect to.

Then the question of hidden variables becomes the question of whether it is possible to fill in the values R_{i,j} of a N\times M matrix such that:

1. For each i and j, R_{i,j} is either +1 or -1.
2. For a fixed j (that is, a fixed choice of direction \vec{a}_j), the average value of R_{i,j} over all possible i is 0. (Just as many spin-up as spin-down.)
3. For any pair of directions \vec{a}_j and \vec{a}_{j'}, the average over all i of R_{i,j} R_{i, j'} is the quantum prediction of \frac{1}{2}(cos^2(\frac{\theta_{j, j'}}{2}) - sin^2(\frac{\theta_{j,j'}}{2})), where \theta_{j,j'} is the angle between \vec{a}_j and \vec{a}_{j'}.

So the "one angle" versus "two angle" versus "three angle" is just this:

• It's always possible to fill in a one-column matrix (and satisfy the above rules)
• It's always possible to fill in a two-column matrix (and satisfy the above rules).
• For certain choices of directions \vec{a}_j, it is impossible to fill in a matrix with 3 or more columns (and satisfy the above rules).

 

[johana]

For 3 angles $a,b,c$ with outcomes $A,B,C$ each of which can be +1 or -1, you can do the following algebra
$AB - AC = A(B - C)$
Remembering that $BB = 1$
$A(B - C) = A(B - BBC) = AB(1 - BC)$
therefore
$AB - AC = AB(1 - BC)$
Taking absolute values
$|AB - AC| \leq |AB||(1 - BC)|$
since $|AB| = 1$ and $(1 - BC)$ is always positive anyway
$|AB - AC| \leq (1 - BC)$
and therefore
$|AB - AC| + BC \leq 1$
This is a Bell inequality.

Notice the absence of "locality" or "realism" in the above derivation. The "magic trick" is how we started with just AB and AC, and all of a sudden you have BC in the final expression. You can't do this trick without a third angle.

I don't see absence of locality, but I see it's general, so if it is indeed true it should not be violated regardless of whether data came from QM experiment, classical experiment, or from my dream.

Can you show an example QM dataset that can violate that inequality?

 

[stevendaryl]

I don't see absence of locality, but I see it's general, so if it is indeed true it should not be violated regardless of whether data came from QM experiment, classical experiment, or from my dream.

Can you show an example QM dataset that can violate that inequality?

In an EPR experiment, there are two particles produced, and for each particle, you get one opportunity to measure the spin relative to some angle. So in each "run" of the experiment, you only get the results of 2 angles.

So in terms of the matrix that I mentioned, that means that if you have 3 possible angles, then you have to fill in a 3-column matrix. But experimentally, you only test 2 values. So for each row, you only can fill in 2 of the three columns by experimental values. The third matrix element must be left blank.

Bell's inequality shows that there is no way to fill in the "blanks" by values in a way that satisfies the predictions of QM.

 

[stevendaryl]

In an EPR experiment, there are two particles produced, and for each particle, you get one opportunity to measure the spin relative to some angle. So in each "run" of the experiment, you only get the results of 2 angles.

So in terms of the matrix that I mentioned, that means that if you have 3 possible angles, then you have to fill in a 3-column matrix. But experimentally, you only test 2 values. So for each row, you only can fill in 2 of the three columns by experimental values. The third matrix element must be left blank.

Bell's inequality shows that there is no way to fill in the "blanks" by values in a way that satisfies the predictions of QM.

So in terms of Bill's notation, for every round of the EPR experiment, you can only learn the values of two of the three quantities A, B, C. So it's not really a dataset violating Bell's inequality. It's a partial dataset which cannot possibly be made complete.

 

[johana]

So in terms of Bill's notation, for every round of the EPR experiment, you can only learn the values of two of the three quantities A, B, C. So it's not really a dataset violating Bell's inequality. It's a partial dataset which cannot possibly be made complete.

I need to confirm what exactly is meant by "angle", "dataset", and "partial dataset". Say Alice and Bob can turn their polarizers to 0, 20, and 30 degrees, and we are testing for these three combinations:

a= (0,20) = 20°
b= (30,0) = 30°
c= (30,20) = 10°

With relative angle a = 20° we get for example this dataset A = --, +-, ++, -+, ++
With relative angle b = 30° we get for example this dataset B = +-, ++, -+, -+, +-
With relative angle c = 10° we get for example this dataset C = ++, -+, +-, -+, --

Correct? What partial dataset are you talking about?

 

[stevendaryl]

I need to confirm what exactly is meant by "angle", "dataset", and "partial dataset". Say Alice and Bob can turn their polarizers to 0, 20, and 30 degrees, and we are testing for these three combinations:

a= (0,20) = 20°
b= (30,0) = 30°
c= (30,20) = 10°

With relative angle a = 20° we get for example this dataset A = --, +-, ++, -+, ++
With relative angle b = 30° we get for example this dataset B = +-, ++, -+, -+, +-
With relative angle c = 10° we get for example this dataset C = ++, -+, +-, -+, --

Correct? What partial dataset are you talking about?

No, that's not what I mean. The assumption behind local hidden-variables theories is that each electron produced in EPR simultaneously has a spin component in EACH of the three directions a, b, and c. So associated with electron number i is a triple of numbers \langle R_{i,a}, R_{i,b}, R_{i,c} \rangle, where R_{i,a} is either +1 (to indicate spin-up in direction a) or -1 (to indicate spin-down). Analogously for R_{i,b} and R_{i,c}.

So a complete dataset for the hidden variables R_{i,j} would be a table consisting of one row for each electron produced, and each row would have three values, each of which is either +1 or -1.

Unfortunately, we can't measure the spin in more than one direction at a time. However, we can use the fact that in a twin-pair experiment, the spin of one particle in a particular direction is always the opposite of the spin of its twin in that direction. So that allows us to measure two of the three values for R_{i,j}. Alice can measure the spin in direction a for one of the particles, and Bob can measure the spin in direction b for the other particle. Since the two particles are anti-correlated, we just need to flip Bob's result to get the result that Alice would have measured if she had measured the spin in direction b. So we have two of the three angles covered. But we have no way to measure the spin in the third direction, c. So we leave that blank.

So suppose that in the first trial, Alice measures spin in the a direction and gets spin-up. Bob measures spin in the b direction and also gets spin-up, which means that Alice would[/itex] have gotten spin-down if she had measured in that direction. So the results of the first trial are written as the triple

\langle +, -, ? \rangle

In the second trial, Alice measures the spin in the a direction again, and gets spin-down. Bob measures the spin in direction c and gets spin-down, also, which means that Alice would have gotten spin-up. So the results of the second round are written as:

\langle -, ?, + \rangle

So the partial dataset might look like this:

\left( \begin{array}\\ + & - & ? \\ - & ? & +\\ + & ? & - \\ ... \end{array} \right)

 

[DrChinese]

Can you show an example QM dataset that can violate that inequality?

You have it backwards.

QM does not predict realism. The QM dataset consists of pairs in consonance with the predictions of QM, which violate the inequality BY DEFINITION. That is because the QM prediction is used to construct the inequality.

Please, stop and review the reference materials first. You are going around in circles. If nothing else, you are making me dizzy.

 

[billschnieder]

Apparently you do not understand a basic application of Malus, circa 1809.

A stream of Alice photons polarized at 0 degrees as + will have a 25% chance of being polarized + at 120 degrees. A stream of Alice photons polarized at 120 degrees as + will have a 25% chance of being polarized + at 240 degrees. A stream of Alice photons polarized at 0 degrees as + will have a 25% chance of being polarized + at 240 degrees. So if it passes the polarizer, it is matched.

It is matched with what? You need two things to do matching, don't you? You still haven't explained what you are matching the photon with, and what said matching has to do with malus at all.

 

[DrChinese]

It is matched with what? You need two things to do matching, don't you? You still haven't explained what you are matching the photon with, and what said matching has to do with malus at all.

This is off the subject of this thread, and I have already answered several times already.

For that matter, this question of this thread has been answered multiple times already, and I will summarize it:

The reason for 3 angles & entanglement to demonstrate why local realism fails: it traces back to Bell's Theorem. Multiple variations on this have been presented, as well as Bell's original paper. Any subsequent answer will simply be yet another version of the same. If you haven't followed what has been presented so far, READ THE REFERENCES instead of asking the same question a different way.

 

[johana]

No, that's not what I mean. The assumption behind local hidden-variables theories is that each electron produced in EPR simultaneously has a spin component in EACH of the three directions a, b, and c. So associated with electron number i is a triple of numbers \langle R_{i,a}, R_{i,b}, R_{i,c} \rangle, where R_{i,a} is either +1 (to indicate spin-up in direction a) or -1 (to indicate spin-down). Analogously for R_{i,b} and R_{i,c}.

So a complete dataset for the hidden variables R_{i,j} would be a table consisting of one row for each electron produced, and each row would have three values, each of which is either +1 or -1.

I don't see how three orthogonal measurement axis in electron case compare with anything in entangled photons experiment. Billschnieder says A, B, C are outcomes, you describe them as potential outcomes. Normally one would think the outcome refers to both Alice and Bob data for a single entangled pair, but the outcome you are talking about seems to be taken from three entangled pairs and only on one side for either Alice or Bob.

Unfortunately, we can't measure the spin in more than one direction at a time. However, we can use the fact that in a twin-pair experiment, the spin of one particle in a particular direction is always the opposite of the spin of its twin in that direction.

This also doesn't seem to compare with entangled photons experiment. For photons 100% match/mismatch is reserved only for 0 and 90 degrees relative angles. Can we stick with photons since the whole thread was about photons so far?

 

[stevendaryl]

Note: I edited my answer to make it about photons, rather than electrons. It really doesn't make any difference to the argument.

I don't see how three orthogonal measurement axis in electron case compare with anything in entangled photons experiment.

It's almost exactly the same. Instead of measuring spin-up or spin-down relative to an axis, Alice and Bob either observe that the photon passed the filter, or the photon did not pass the filter relative to an axis. In both experiments, Alice and Bob pick an orientation, then they perform a measurement that has two possible values. The argument works exactly the same.

Alice has three possible axes to measure a photon's polarization: a, b, c. Similarly, Bob has three possible axes that he can measure: a, b, c. We convince ourselves through experiment, or by looking at the QM predictions, that for a pair of entangled photons, if Alice and Bob both measure the polarizations of entangled photons using the same axis, then they ALWAYS get the same results. (or they always get opposite results, depending on how the entangled photons are produced; let's assume that they always get the same results).

Since Alice and Bob ALWAYS get the same results for the same filter orientations, that means that Bob, by measuring his photon, can learn something about Alice's photon.

To Einstein (and whoever P and R were), that means that there must be a deterministic answer to the question: "What would the result be if Alice measured her photon's polarization relative to axis a?" It must be a deterministic answer, because Bob can predict it with 100% certainty by measuring his photon's polarization relative to axis a. So to E, P, and R, there must be, associated with each photon, a triple of numbers \langle R_a, R_b, R_c\rangle telling whether Alice's photon will pass her filter or get blocked by her filter, should she set it at orientation a, b or c.

She can only actually measure one of those three numbers, but the EPR reasoning implies that the three numbers exist, whether she can measure them or not. Putting Alice's measurement together with Bob's, it's possible to figure out what two of the three numbers are. To figure out R_a and R_b, Alice measures polarization in direction a and Bob measures polarization in direction b. Then they have to leave the answer for direction c blank.

Billschnieder says A, B, C are outcomes, you describe them as potential outcomes.

Two of them are actual outcomes, and the third one is a "conterfactual": If Alice had oriented her filter at direction c, rather than a, her photon would have passed through (or would not have).

Normally one would think the outcome refers to both Alice and Bob data for a single entangled pair, but the outcome you are talking about seems to be taken from three entangled pairs and only on one side for either Alice or Bob.

No, it's not three entangled pairs. For each entangled pair, Alice and Bob measure two of three possible angles. So for each entangled pair, they produce a triple of values: One value is computed by Bob's result. The other value is computed by Alice's result, and the third value is left "?", because nobody measures that one. So you end up with a list of triples, where each triple has two values that are \pm 1 and one value that is "?".

This also doesn't seem to compare with entangled photons experiment.

No, it's almost exactly the same. Instead of measuring "spin-up in direction a", they measure "passes the filter when the filter is oriented at direction a". We pick three axes: a, b, c. Alice measures photon polarization relative to axis a, and Bob measures photon polarization of the twin photon relative to axis b. Nobody measures polarization relative to axis c, so that one would be left "?".

For photons 100% match/mismatch is reserved only for 0 and 90 degrees relative angles. Can we stick with photons since the whole thread was about photons so far?

It doesn't make any difference. The argument is exactly the same.

 

[johana]

Alice has three possible axes to measure a photon's polarization: a, b, c. Similarly, Bob has three possible axes that he can measure: a, b, c. We convince ourselves through experiment, or by looking at the QM predictions, that for a pair of entangled photons, if Alice and Bob both measure the polarizations of entangled photons using the same axis, then they ALWAYS get the same results. (or they always get opposite results, depending on how the entangled photons are produced; let's assume that they always get the same results).

Ok.

Since Alice and Bob ALWAYS get the same results for the same filter orientations, that means that Bob, by measuring his photon, can learn something about Alice's photon.

Same filter orientation, ok.

To Einstein (and whoever P and R were), that means that there must be a deterministic answer to the question: "What would the result be if Alice measured her photon's polarization relative to axis a?" It must be a deterministic answer, because Bob can predict it with 100% certainty by measuring his photon's polarization relative to axis a.

100% match/mismatch certainty is reserved only for 0 and 90 degrees relative angles. Overall, the answer is rather probabilistic.

So to E, P, and R, there must be, associated with each photon, a triple of numbers \langle R_a, R_b, R_c\rangle telling whether Alice's photon will pass her filter or get blocked by her filter, should she set it at orientation a, b or c.

It doesn't work with 100% certainty for any arbitrary relative angle. Your premise started based on Alice and Bob having the same filter polarization.

 

[stevendaryl]

100% match/mismatch certainty is reserved only for 0 and 90 degrees relative angles. Overall, the answer is rather probabilistic.

Okay, you still don't quite get the local hidden variables assumption. a, b and c are NOT relative angles. They are three different directions in space. For example, a might be the filter orientation in the x-y plane, with the filter slits running in the x-direction. b might be again the x-y plane, with the filter slits running in the y-direction. c might be again the x-y plane, with the filter slits running at a 45 degree angle relative to the x-direction. These are not relative angles.

The deterministic local hidden variables assumption is that there are 8 types of photons produced in the twin-pair experiment:

• Type 1: Passes through filters at orientations a, b or c.
• Type 2: Passes a and b, but blocked by c.
• Type 3: Passes a and c, but blocked by b.
• Type 4: Passes b and c, but blocked by a.
• Type 5: Blocked by a and b, but passes c.
• Type 6: Blocked by a and c, but passes b.
• Type 7: Blocked by b and c, but passes a.
• Type 8: Blocked by a, b or c

Since Alice and Bob always get the same answer to the same question, we assume that in every run of the experiment, Alice and Bob get photons of the same "type".

The assumption is that some unknown fraction of the time, call it P_1, type 1 photons are produced. Some other fraction of the time, P_2 type 2 photons are produced. Etc. So the probabilities, according to the hidden variables theory, don't come in the probability that a SPECIFIC photon will pass through a filter at a specific angle. The probabilities are assumed to be due to the fact that the type of photon, Type 1 through Type 8, is chosen randomly, according to a certain probability distribution.

So that's the hidden-variables theory: EACH photon has an associated "type". The type answers the question "Will this photon pass through a filter oriented at angle \alpha?" for each possible value for \alpha. It's assumed that in a twin-pair experiment, both Alice and Bob get the same type photon. If Alice's photon passes at angle a, and Bob's photon is blocked at angle b, then that means that their photons must have been Type 3 or Type 7 (according to the numbering above). If both photons pass, that means their photons must have been Type 1 or Type 2.

So we can reason as follows:

• Since 50% of the time when the filter is at setting a, the photon passes, we conclude that P_1 + P_2 + P_3 + P_7 = \frac{1}{2}. That's because if it passes through at angle a, then it must be a photon of type 1, 2, 3 or 7, according to the list above.
• Since the probability of passing a and also passing b is \frac{1}{2} cos^2(\theta_{a, b}), we conclude that P_1 + P_2 = \frac{1}{2} cos^2(\theta_{a, b}), where \theta_{a,b} is the angle between a and b
• etc.

That's the hidden-variables theory for twin-pair photons. The only problem with it is that the numbers don't work out. There are no solutions to the probabilities P_1 through P_8 that satisfy all the statistical predictions of quantum mechanics.

 

[RUTA]

This question of this thread has been answered multiple times already, and I will summarize it:

The reason for 3 angles & entanglement to demonstrate why local realism fails: it traces back to Bell's Theorem. Multiple variations on this have been presented, as well as Bell's original paper. Any subsequent answer will simply be yet another version of the same. If you haven't followed what has been presented so far, READ THE REFERENCES instead of asking the same question a different way.

Haha, I was just about to commend you for being so patient and answering the same questions over and over. Someone posted a link to Mermin's 1985 paper "Is the moon there when nobody looks?" https://cp3.irmp.ucl.ac.be/~maltoni/PHY1222/mermin_moon.pdf that answers the original question directly, yet I've seen that same question asked afterwards. Here is an excerpt from p 9

"Alas, this explanation –the only one, I maintain, that someone not steeped in quantum mechanics will ever be able to come up with (though it is an entertaining game to challenge people to try)- is untenable. It is inconsistent with the second feature of the data: There is no conceivable way to assign such instruction sets to the particles from one run to the next that can account for the fact that in all runs taken together, without regard to how the switches are set, the same colors flash half the time. Pause to note that we are about to show that “something one cannot know anything about” –the third entry in an instruction set- cannot exist. For even if instruction sets did exist, one could never learn more than two of the three entries (revealed in those runs where the switches ended up with two different settings)."

He then goes on to give the argument. Note that the title of the paper is making exactly this point, i.e., the third entry -- the one that doesn't get measured (not looked at) -- "cannot exist." So the answer to the title question is "The moon is not there when nobody looks," where "when nobody looks" means "not interacting with anything else in the universe." If someone reads that paper and still doesn't see the answer to the OP, I'm not sure you can help them here, despite your heroic efforts

 

[johana]

Please, stop and review the reference materials first. You are going around in circles. If nothing else, you are making me dizzy.

The origin of the three angles within inequality derivation seems to be a different question than the original question which was about experiments, but I did think they are the same question. Maybe I should open a new thread about the derivation?

I listened to your advice, but at the end I found what I was looking for in Wikipedia.

http://en.wikipedia.org/wiki/CHSH_inequality

The usual form of the CHSH inequality is:

(1) − 2 ≤ S ≤ 2,

where
(2) S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′).

a and a′ are detector settings on side A, b and b′ on side B, the four combinations being tested in separate subexperiments.

This is it, no partial datasets or imaginary outcomes, and it actually applies to photon entanglement experiments we are talking about. With this beautiful definition my question becomes very simple and straight forward:

S = E(a,b)
S = 0

There it is equality QM violates all the way from -1 to 1, while according to standard local reality prediction S can not be different than zero. Only one relative arbitrary angle required, so what for do we need any more?