Photon entanglement: why three angles?

[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?