Did rotating polarizer show violations of Bell's Inequality?

[Ian J Miller]

The Bell inequality requires three conditions, A, B and C that can have two values (pass or fail, say). In the Aspect experiment A defines a plane, B a plane of A rotated by 22.5 degrees, while C is A rotated by 45 degrees. We take joint probabilities, and two are A+.B-, and B+.C-, and from the sine squared relation (Bell, J. S. 1981. Bertlmann's socks and the nature of reality. Journal de Physique (Paris) Colloque C2 (suppl. Au numero 3) 42, 41-61) Bell showed these two are both given values of 0.146 (assuming the plus term is 1, the photons involves in the test. Bell actually gave the values half of this, to include all photons, including the half that were not counted in his calculation.) This leads to a clear violation of the inequality iff all terms are independently valid.

My problem is that both minus terms are referenced to the plus terms, but A and B are not the same so there is not a common frame of reference. That means that B- is actually a subset of A, and hence is not a B. Further, because the source is rotationally symmetric, from Noether's theorem B+.C- is simply a repeat of what was called A+.B-. There is no derivation that says 2A+.B- ≥ A+.C-

Further, suppose we have an external frame of reference, say from a polarized light source. Now if we assume the Malus law, the calculated outcome complies with Bell's inequality. So the question now is, why is the Noether theorem ignored, and why do we assume the inequality was violated? And why do you get different answers from a set of photons with common polarization compared with a set of photons with mixed polarization?

 

[vanhees71]

I don't understand the question: The measurements are done in a well-defined reference frame, the lab frame where the measurement apparatus etc. are at rest.

 

[DrChinese]

1. My problem is that both minus terms are referenced to the plus terms, but A and B are not the same so there is not a common frame of reference.

2. That means that B- is actually a subset of A, and hence is not a B. Further, because the source is rotationally symmetric, from Noether's theorem B+.C- is simply a repeat of what was called A+.B-. There is no derivation that says 2A+.B- ≥ A+.C-

Further, suppose we have an external frame of reference, say from a polarized light source. Now if we assume the Malus law, the calculated outcome complies with Bell's inequality. So the question now is, why is the Noether theorem ignored, and why do we assume the inequality was violated?

3. And why do you get different answers from a set of photons with common polarization compared with a set of photons with mixed polarization?

A few key points:

1. Reference frame does not figure into a Bell test. The quantum expectation is dependent only on the difference between Alice and Bob's measurement settings. As vanhees71 says, a lab frame is used.2. The hidden variable assumption is present in the local realistic program, but not (as you point out) in the quantum program. That's the reason there are only 2 quantities measured at a time, but 3 terms appear.

Noether's Theorem is not really relevant here. Also, although the quantum expectation looks (and essentially acts) nearly the same as Malus' Law, it is derived separately.3. Entangled photons are not considered to have "common" polarization but are considered to be in a "superposition" of polarizations. Bell tests demonstrate that they are not given well-defined polarization upon creation. Typically, entangled photon pairs are created by shining a laser into a BBo nonlinear crystal. Most of the photons go straight through the crystal without anything special happening. Occasionally 2 come out which are entangled (sometimes called a biphoton).

"Mixed" light sources do not produce any pairs that can be identified as entangled. Neither do sources such as lasers (by themselves, without a BBo crystal) where all light is of the same polarization - there are no identifiable entangled photon pairs.

 

[vanhees71]

3. Entangled photons are not considered to have "common" polarization but are considered to be in a "superposition" of polarizations. Bell tests demonstrate that they are not given well-defined polarization upon creation. Typically, entangled photon pairs are created by shining a laser into a BBo nonlinear crystal. Most of the photons go straight through the crystal without anything special happening. Occasionally 2 come out which are entangled (sometimes called a biphoton).

The important point is that for Bell states, which are pure two-photon states, the polarization of the photons is not a pure state and thus also not "a superposition" but a mixed state, and here it's even one of maximum entropy, i.e., both photons are ideally unpolarized.

As an example take the "singlet ket",
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|HV \rangle-|VH \rangle),$$
where $H$ and $V$ are linear-polarization states with respect to an arbitrary direction. The two-photon state is thus
$$\hat{\rho}=|\Psi \rangle \langle |\Psi \rangle,$$
i.e., a projection operator and thus a pure state.

The statistical operator for the polarization of photon 1 is given by the reduced state
$$\hat{\rho}_1 = \mathrm{Tr}_2 \hat{\rho} = \sum_{\alpha_1,\alpha_2,\beta \in \{H,V \}} |\alpha_1 \rangle \langle \alpha_1,\beta|\Psi \rangle \langle \Psi|\alpha_2,\beta \rangle \langle \alpha_2 \rangle=\frac{1}{2} (|H \rangle \langle H| + |V \rangle \langle V|) = \frac{1}{2} \hat{1}.$$

 

[.Scott]

This leads to a clear violation of the inequality iff all terms are independently valid.

I'm not sure that I catch the gist of what you mean by "independent".
The point of the Bell argument is only that the measurement results do not reflect a state that is independent of the measurement decisions.

The opposite assumption would be that the measurement results are predetermined at the time that the entangled particles are created. So, for any pair "p", there is a predetermined $A_p$, $B_p$, and $C_p$ waiting to be measured at each station.
In order to sideline the issue with the plus and minus, I will describe the experiment a bit differently. I will talk about the two measurements "agreeing" with each other or not.
So if one detector is measuring "+", then the "agreeing" measurement will be "-" from the other detector. Or, if you prefer, to adjust one of the sensors to invert it result (for photons, I think that means rotate the measurement by 90 degrees), so now the readings would be considered "agreeing" if they are the same.
That "predetermined" assumption holds up as long as both stations measure the same value (both A, both B, or both C). In those situations, the results will always agree (always inverted from each other). This "predetermined" assumption also withstands a single measurement, since any combination of (A and B, or A and C, or B and C) is allowed.
The problem comes when many measurements are made - and we notice that the A and C measurement disagreements are more numerous than the total of A and B measurement disagreements and B and C measurement disagreements combined.
That's like saying that three people took a test, two scored 20 points different than each other but each is only 5 points different than the the third. You can't even cheat and get those results. It means that when you were comparing scores between A and B, then between A and C, and then between B and C, you were not using the same scores in each of the three comparisons.

 

[Ian J Miller]

I don't understand the question: The measurements are done in a well-defined reference frame, the lab frame where the measurement apparatus etc. are at rest.

Surely the whole point of a reference frame is to allocate unique values of coordinates. Let us take the form of the inequality A+.B- + B+C- ≥ A+.C- . Now, from Bell, the joint probability comes from sin squared θ, θ being the difference in the amount the two detectors are rotated from the original position, which can be set in the frame of reference of the lab, but everything else depends on the difference between the angles of the detrectors. If we take Bell's figures, the + probability is 1 (Bell subsequently divided by 2, but the sin squared term is calculated first) and 1 seems a better value because the second detector only counts photons that arrive within a specific time of the first detector (to ensure we only count entangled pairs). Now, take B+ and B-; these are at right angles to each other, so the detector at B- should detect zero photons. But when Bell put it into the inequality (p10 of the reference) he came up with the sin squared 22.5, or 0.146. If the wall provides a proper frame of reference AND the detectors are actually referenced to it, the terms should be constant. The same problem arises with C-. The Bell figures show it is either 0.146 or 0.5, dependning on which time you look at it. That is my problem with the internalo frame of reference. Each second detector is referenced to the first, but thqat means the terms are not constant. If you derive the inequality with set theory, the terms are required to be constant, i.e. a B- has one and only one value.

 

[PeterDonis]

Let us take the form of the inequality A+.B- + B+C- ≥ A+.C- .

What do your dots mean? What do + and - mean? Are you using some standard notation that is in a reference you can give, or are you just making up your own?

 

[Ian J Miller]

A few key points:

1. Reference frame does not figure into a Bell test. The quantum expectation is dependent only on the difference between Alice and Bob's measurement settings. As vanhees71 says, a lab frame is used.2. The hidden variable assumption is present in the local realistic program, but not (as you point out) in the quantum program. That's the reason there are only 2 quantities measured at a time, but 3 terms appear.

Noether's Theorem is not really relevant here. Also, although the quantum expectation looks (and essentially acts) nearly the same as Malus' Law, it is derived separately.3. Entangled photons are not considered to have "common" polarization but are considered to be in a "superposition" of polarizations. Bell tests demonstrate that they are not given well-defined polarization upon creation. Typically, entangled photon pairs are created by shining a laser into a BBo nonlinear crystal. Most of the photons go straight through the crystal without anything special happening. Occasionally 2 come out which are entangled (sometimes called a biphoton).

"Mixed" light sources do not produce any pairs that can be identified as entangled. Neither do sources such as lasers (by themselves, without a BBo crystal) where all light is of the same polarization - there are no identifiable entangled photon pairs.

1. A reference frame is needed because otherwise terms can vary their value. See my repsonse to vanhees71. Note that if you have a polarized source, the plane of that source used as a reference leads to Bell's inequality being complied with, at least using the Malus law.

2 a. In the Aspect test, you have the detectors and you measure clicks. The three terms appear because you make three series of measurements, surely.

2b. The quantum expectation can be derived separately, but if you use enough photons you enter into the classical realm, and surely the Malus law must apply. The Malus law is simply a geometric law that follows from applying the conservation of energy to a polarized wave, and the polarised light source is a polarized wave, or at least it can be treated as such statistically, surely. Further, to test the inequality, surely you have to use enough photons to get around statistical variation, so surely the Malus law must apply to enough photons to remove statistical variation. In any case, both give the same sin squared function in the way this Bell test is done.

2c. I don't understand why Noether's theorem is not relevant. You expect that if you move the experiment to the other end of the bench you get the same result; why don't you by turning the equipment to a new angle, given the rotational symmetry of the source? Note that the A+.B- does in fact give the same result as the B+.C-, at least according to Bell.

3. Why are entangled photons not considered to have a common polarization? Further, I do not see how you can say that photons in Bell tests are not given well-defined polarization when (a) if they were you would get these results provided the entangled pairs were randomly polarized, and (b) it seems wrong to me to use the conclusions of the violations of the inequality to refute the claim that the inequality is not violated because the terms are not used properly. I am aware that they are generally regarded as being in a superposition, but you do not actually know that.

As for entanglement in the Aspect test, as I understand it, the entanglement arose through the conservation of angular momentum due to two paired electron in excited state calcium decaying to ground state paired electrons. Of course, there is no reason other means of entanglement cannot be used, but how you do it will cause possible differences in the relation of the polarization.

Finally, thank you for the effort you put into answering my question.I appreciate it, as it helps clarify my thoughts.

 

[Ian J Miller]

The important point is that for Bell states, which are pure two-photon states, the polarization of the photons is not a pure state and thus also not "a superposition" but a mixed state, and here it's even one of maximum entropy, i.e., both photons are ideally unpolarized.

As an example take the "singlet ket",
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|HV \rangle-|VH \rangle),$$
where $H$ and $V$ are linear-polarization states with respect to an arbitrary direction. The two-photon state is thus
$$\hat{\rho}=|\Psi \rangle \langle |\Psi \rangle,$$
i.e., a projection operator and thus a pure state.

The statistical operator for the polarization of photon 1 is given by the reduced state
$$\hat{\rho}_1 = \mathrm{Tr}_2 \hat{\rho} = \sum_{\alpha_1,\alpha_2,\beta \in \{H,V \}} |\alpha_1 \rangle \langle \alpha_1,\beta|\Psi \rangle \langle \Psi|\alpha_2,\beta \rangle \langle \alpha_2 \rangle=\frac{1}{2} (|H \rangle \langle H| + |V \rangle \langle V|) = \frac{1}{2} \hat{1}.$$

The Bell inequality can be derived simply from set theory provided that you have three conditions that you label A, B and C that can have + or - values assigned to them. Why the values are what they are is beside the point. The reference I gave above explained this in terms of washing socks. All Bell needed was three different temperatures and a pass/fail requirement such as socks shrunk/did not shrink. There was also the condition that the sum of passes and fails equals 1, the condition that a sock either shrinks or does not shrink, but cannot do both.

 

[Ian J Miller]

I'm not sure that I catch the gist of what you mean by "independent".
The point of the Bell argument is only that the measurement results do not reflect a state that is independent of the measurement decisions.

The opposite assumption would be that the measurement results are predetermined at the time that the entangled particles are created. So, for any pair "p", there is a predetermined $A_p$, $B_p$, and $C_p$ waiting to be measured at each station.
In order to sideline the issue with the plus and minus, I will describe the experiment a bit differently. I will talk about the two measurements "agreeing" with each other or not.
So if one detector is measuring "+", then the "agreeing" measurement will be "-" from the other detector. Or, if you prefer, to adjust one of the sensors to invert it result (for photons, I think that means rotate the measurement by 90 degrees), so now the readings would be considered "agreeing" if they are the same.
That "predetermined" assumption holds up as long as both stations measure the same value (both A, both B, or both C). In those situations, the results will always agree (always inverted from each other). This "predetermined" assumption also withstands a single measurement, since any combination of (A and B, or A and C, or B and C) is allowed.
The problem comes when many measurements are made - and we notice that the A and C measurement disagreements are more numerous than the total of A and B measurement disagreements and B and C measurement disagreements combined.
That's like saying that three people took a test, two scored 20 points different than each other but each is only 5 points different than the the third. You can't even cheat and get those results. It means that when you were comparing scores between A and B, then between A and C, and then between B and C, you were not using the same scores in each of the three comparisons.

What I meant by "independent", and I take your point that it might have been a poor choice of words, is that when you obtain a probability for, say, B-, the value stays the same irrespective of how the other detector is oriented. The reason for saying that is that when you derive the inequality, you have to use each term more than once, and it would be poor mathematics to have a term with a value changing in an undefined way.

I fully agree with what you say in your second last paragraph, and hence for your last one. The issue I have is because of the internal referencing, and hence the terms not having a constant value if we did all possible tests, they do not qualify to be put into the Bell inequality. I am not disputing any resulot, or any calculation other than the decision to use the inequality.

My question is, is not the violations we see a consequence of not properly using the inequality rather than some unexpected physics that somehow seems to violate mathematics?

 

[Ian J Miller]

What do your dots mean? What do + and - mean? Are you using some standard notation that is in a reference you can give, or are you just making up your own?

The + and - simply mean "pass the test" or "fail the test respectively. The dots are simply a multiplication operator to get joint probabilities from two singly determined probabilities. I m not making up a nomenclature, but others may well represent it better. I have had this discussion elsewhere on the web, although this is the first time I have had a useful response, and since the other sites did not use LaTeX I have had to simplify. Sorry about that.

 

[PeterDonis]

The + and - simply mean "pass the test" or "fail the test respectively.

What test? What does "pass" or "fail" mean?

The dots are simply a multiplication operator to get joint probabilities from two singly determined probabilities.

What does this mean?

I m not making up a nomenclature

Then please give a specific reference for where you are getting it from. If it's Bell's "Bertlmann's socks" paper, referring to some specific equations in that paper would be helpful.

 

[PeterDonis]

The Bell inequality can be derived simply from set theory provided that you have three conditions that you label A, B and C that can have + or - values assigned to them. Why the values are what they are is beside the point. The reference I gave above explained this in terms of washing socks.

Do you understand that the whole point of Bell's "Bertlmann's socks" argument was to show why the type of hidden variable model you are describing does not and cannot make the same experimental predictions as QM? Which means that, since we now know the experimental predictions of QM for these experiments are correct, that the type of hidden variable model you are describing cannot be correct?

is not the violations we see a consequence of not properly using the inequality rather than some unexpected physics that somehow seems to violate mathematics?

The physics of QM does not "violate mathematics". It just violates the Bell inequalities. The reason it does is that the physics of QM violates the assumptions that are used to derive the Bell inequalities. A "Bertlmann's socks" model, by contrast, does not violate those assumptions, so such a model predicts that the Bell inequalities will not be violated in experiments. But they are.

 

[PeroK]

My question is, is not the violations we see a consequence of not properly using the inequality rather than some unexpected physics that somehow seems to violate mathematics?

Bell's theorem is a result from classical probability theory. QM works on complex probability amplitudes. Bell's insight was to devise a thought experiment where the outcome would distinguish between these two. I.e. distinguish between the two mathematical models. There is nothing unmathematical about QM. Quite the reverse, the core of QM is the mathematical model.

 

[Ian J Miller]

What test? What does "pass" or "fail" mean?What does this mean?Then please give a specific reference for where you are getting it from. If it's Bell's "Bertlmann's socks" paper, referring to some specific equations in that paper would be helpful.

Reference? Baggot, in "Beyond Measure" Appendix 19 starts off with the nomenclature, and elsewhere explains what pass/fail means (basically there are two options only to a test, so one is labelled pass, the other fail. They could have been labelled anything. In his case, the + and - signs are as subscripts. Because I have been also raising this issue elsewhere where subscripts are difficult to be inserted. This question I raised is whether the assignment of the probabilities complies with the requirements of the inequality. I would hope that any enlightenment for me is not going to be thwarted by whether I used a subscript. The question's problems should be visible with the nomenclature I am using.

 

[PeterDonis]

This question I raised is whether the assignment of the probabilities complies with the requirements of the inequality.

I don't know what you mean by this. What "assignment of the probabilities" is taking place, and what "requirements of the inequality" do these have to be checked against?

In Bell's own "Bertlmann's socks" paper that you reference, as well as his other papers on the topic, he explicitly derives his form of the inequalities from explicitly stated mathematical assumptions. In the CHSH paper that derives their form of the inequalities, they do the same. All of this is perfectly self-consistent math and it makes perfectly definite predictions: that if the actual physics of our world satisfies the mathematical assumptions of the theorems, then experiments will not show correlations that violate the inequalities. But experiments do show correlations that violate the inequalities: therefore, at least one of the mathematical assumptions of the theorems must be violated by the actual physics of our world.

Are you disputing the above?

 

[Ian J Miller]

Do you understand that the whole point of Bell's "Bertlmann's socks" argument was to show why the type of hidden variable model you are describing does not and cannot make the same experimental predictions as QM? Which means that, since we now know the experimental predictions of QM for these experiments are correct, that the type of hidden variable model you are describing cannot be correct?The physics of QM does not "violate mathematics". It just violates the Bell inequalities. The reason it does is that the physics of QM violates the assumptions that are used to derive the Bell inequalities. A "Bertlmann's socks" model, by contrast, does not violate those assumptions, so such a model predicts that the Bell inequalities will not be violated in experiments. But they are.

I have not mentioned a hidden variable. What I have stated is the source Aspect used has rotational symmetry, which I believe was demonstrated by rotating a detector and getting constant results. I then used the count at each detector, and the angle between the detector planes. These are all observed variables, controlled by the person running the tests.

As I said above, the "minus" terms, or if you prefer, the probabilities obtained from the second detector vary according to the orientation with the first detector. But the derivation of the inequality requires them to be constant, i.e. say with Bertlmann's socks, a B- meant x socks shrunk when washed at 35 degrees, and x always has the same value. You say "the physics of QM violates the assumptions used to derive the Bell inequalities" If that is so then surely you should not use the inequality? This is similar to my argument: the "violations" arise because the data is used in a way that violates the derivation of the inequality, but surely if the conditions of a mathematical relationship are not met, the relationship should not be used?

 

[PeterDonis]

You say "the physics of QM violates the assumptions used to derive the Bell inequalities" If that is so then surely you should not use the inequality?

Nobody uses the inequalities to make actual predictions. We use QM for that.

The inequalities are used to demonstrate that the actual physics of our world must violate at least one of the assumptions that were used to derive the inequalities, since actual experimental results violate them.

 

[Ian J Miller]

Bell's theorem is a result from classical probability theory. QM works on complex probability amplitudes. Bell's insight was to devise a thought experiment where the outcome would distinguish between these two. I.e. distinguish between the two mathematical models. There is nothing unmathematical about QM. Quite the reverse, the core of QM is the mathematical model.

My argument is that Bell's inequality is simply derived from the associative laws of sets, and hence is a purely mathematical relationship. The question I am raising is simply about the use of the data. The only inputs are probabilities obtained from the count at each detector, and the angle between the detector planes. I am not denying the mathematics of quantum mechanics; I am questioning the use or misuse of simple numbers in a pure mathematical relationship, namely where terms in the derivation are constant, the way they are used in this application, they are no longer constant. What is puzzling me at the moment is that nobody seems to be addressing the issue I raised. Am I totally obscure in some way? Have I not phrased the question properly?

 

[PeterDonis]

the "minus" terms, or if you prefer, the probabilities obtained from the second detector vary according to the orientation with the first detector. But the derivation of the inequality requires them to be constant, i.e. say with Bertlmann's socks, a B- meant x socks shrunk when washed at 35 degrees, and x always has the same value.

This is a possible source of differences between the actual observed correlations and the correlations that would be predicted by a "Bertlmann's socks" model, yes. But in order to justify your claim that this source of differences alone can explain the entire differences between the actual observed correlations and the correlations that would be predicted by a "Bertlmann's socks" model--i.e., that your claimed source of differences is sufficient to account for the actual observed violations of the Bell inequalities, all by itself--you can't just wave your hands and say so. You need to actually calculate how large the error bars are in the relative orientations of the detectors in actual experiments, and how large the resulting error bars are in the correlations.

The experimenters who did the actual experiments did those things in their experimental papers, and showed that those sources of error were not sufficient, by themselves, to account for the observed Bell inequality violations. In other words, your claimed explanation for the observed Bell inequality violations has been ruled out by experiment.

 

[Ian J Miller]

I must be explaining my problem very badly. Let me try to explain what the Aspect experiment involves another way. We start off by aligning detector 1 in some orientation, say vertical, and assign that as A+. We then align the second detector normal to that, i.el. horizontal, and call that A-. The second detector detects the same number of clicks as the first, but when we apply the requirement that we only count clicks that arrive within, say, 19 ns of the first, which is the limit of when an entangled partner could arrive in this experiment, the count of those that arrive within that time limit is zero. Leaving aside possible shortening of the 19 ns limit, do you agree that is what the Aspect experiment should produce?

The next step (2) is that both detectors are rotated by 22.5 degrees, are labelled B+ and B- respectively, and the above is repeated with the same outcome. Then both detectors are rotated by 22.5 degrees, are labelled C+ and C- respectively, and the above is repeated with the same outcome. All rotations are the same direction. Do you agree that should be observed?

The next step (3) is that the second detector is rotated by 22.5 degrees. This configuration produces the term that was labelled A+.B- (using the same nomenclature as above), and the joint probability is calculated as sin squared 22.5, or 0.146. Do you agree that happened?

The next step (4) is that the first detector is rotated by 22.5 degrees, and the second by a further 22.5 degrees (bringing its total to 45 degrees), and the joint probability is calculated as sin squared 22.5, or 0.146. This is labelled B+.C-. Do you agree that happened?

The final step (5) is the first detector is returned to its original position and the joint probability is calculated as sin squared 45, or 0.5. This is labelled B+.C-. Do you agree that happened?

The above is all theory and the calculations of the joint probabilities of those configurations are consistent with QM, namely the sin squared relationship, which is derivable from quantum mechanics. Do you agree with that?

Finally, the Aspect results were in good agreement with those calculations. However, my question revolves around the appropriateness of even using the theoretical results in Bell's inequality. The question has nothing to do with experimental accuracy provided the experiment is not carried out very badly.

I apologize if this seems overly lengthy, but the set of concerns that make up my question appears to be essentially disjoint with the set of answers so far, so I feel I have to try and locate where I have gone off at a tangent to everyone else.

 

[PeterDonis]

I must be explaining my problem very badly.

No, you are simply not understanding the actual logic of the theoretical predictions.

In both of the theoretical models in question, the "Bertlmann's socks" type model that is used to derive the Bell or CHSH inequalities, and the QM model that is used to predict what actual results we expect to see in these experiments, the predicted correlations are continuous functions of the angle differences between the detectors. The models don't suddenly stop working just because the actual angle difference is not exactly 22.5 degrees, or 45 degrees, or whatever value was used in the theoretical calculation. The fact that, in real life, we can never exactly set the angle differences to those values just means that there are small error bars around the theoretical predictions.

For example, the theoretical prediction for the CHSH inequality is that the CHSH number that is calculated from the data must be less than or equal to $1$. The effect of the small variation in the actual angles from the theoretical angles is that, to satisfy the CHSH inequality (which the "Bertlmann's socks" model predicts will happen), the CHSH number must now be less than or equal to $1$ plus some small error $\epsilon$. It doesn't mean the CHSH inequality suddenly becomes invalid or that the prediction of the "Bertlmann's socks" model can suddenly be anything whatever.

Similarly, the QM prediction for the actual experimental correlation at appropriately chosen angles is that the CHSH number calculated from the data will be $\sqrt{2}$. The effect of the small variation in the actual angles from the theoretical angles is that, to satisfy the QM prediction, the CHSH number must now be $\sqrt{2}$ minus some small error $\delta$. It doesn't mean QM suddenly stops working or that the QM model's prediction can suddenly be anything whatever.

In actual experiments, the actual correlations are within the small error $\delta$ of the QM predictions, and they are many, many times the small error $\epsilon$ away from the "Bertlmann's socks" predictions. That is why physicists can confidently state that the QM predictions are confirmed and that "Bertlmann's socks" models are ruled out.

Nothing you have posted in this thread casts any of the above into doubt.

my question revolves around the appropriateness of even using the theoretical results in Bell's inequality.

And your argument is that you don't think the theoretical models (or at least the "Bertlmann's socks" model) are applicable at all unless the actual angles used in experiments are exactly the same as the theoretical angles used in the calculations. And that is wrong. See above.

 

[Ian J Miller]

No, you are simply not understanding the actual logic of the theoretical predictions.

In both of the theoretical models in question, the "Bertlmann's socks" type model that is used to derive the Bell or CHSH inequalities, and the QM model that is used to predict what actual results we expect to see in these experiments, the predicted correlations are continuous functions of the angle differences between the detectors. The models don't suddenly stop working just because the actual angle difference is not exactly 22.5 degrees, or 45 degrees, or whatever value was used in the theoretical calculation. The fact that, in real life, we can never exactly set the angle differences to those values just means that there are small error bars around the theoretical predictions.

For example, the theoretical prediction for the CHSH inequality is that the CHSH number that is calculated from the data must be less than or equal to $1$. The effect of the small variation in the actual angles from the theoretical angles is that, to satisfy the CHSH inequality (which the "Bertlmann's socks" model predicts will happen), the CHSH number must now be less than or equal to $1$ plus some small error $\epsilon$. It doesn't mean the CHSH inequality suddenly becomes invalid or that the prediction of the "Bertlmann's socks" model can suddenly be anything whatever.

Similarly, the QM prediction for the actual experimental correlation at appropriately chosen angles is that the CHSH number calculated from the data will be $\sqrt{2}$. The effect of the small variation in the actual angles from the theoretical angles is that, to satisfy the QM prediction, the CHSH number must now be $\sqrt{2}$ minus some small error $\delta$. It doesn't mean QM suddenly stops working or that the QM model's prediction can suddenly be anything whatever.

In actual experiments, the actual correlations are within the small error $\delta$ of the QM predictions, and they are many, many times the small error $\epsilon$ away from the "Bertlmann's socks" predictions. That is why physicists can confidently state that the QM predictions are confirmed and that "Bertlmann's socks" models are ruled out.

Nothing you have posted in this thread casts any of the above into doubt.And your argument is that you don't think the theoretical models (or at least the "Bertlmann's socks" model) are applicable at all unless the actual angles used in experiments are exactly the same as the theoretical angles used in the calculations. And that is wrong. See above.

No, I never said the angle had to be precise or it would stop working. I used the sin squared function, and produced the same numbers for this wave application that Bell did (see p 10 of the reference.) Of course there can be errors, and of course the experiment will have errors. The point was that for the purposes of focusing on the important point I ignored the issue of experimental error in setting up the equipment, but that does not mean I did not appreciate it could happen. When it does, the probabilities will be a bit different, but since the question involves the applicability of the use of the terms as labelled by the experimenters, this is nothing more than the inability of experiment to ever agree exactly with theory. Sorry, but I should have emphasized that the question assumes ideal experimental conditions, but automatically realizes that observations almost never exactly matches theory, and if it does, it is accidental.

In the above where I asked "Do you agree with that?" it should be taken to mean that the angles are set to the best of ability, but it is not critical if they are in error. I await to see if anyone disagrees with the procedural steps.

 

[PeterDonis]

I should have emphasized that the question assumes ideal experimental conditions

Then I have no idea what your issue is. I have read every post you have made and I still don't see it.

Has anyone in the literature raised the issue you are raising? Is there a paper you can reference that discusses it?

 

[PeterDonis]

I await to see if anyone disagrees with the procedural steps.

The "procedural steps" you describe look OK to me, but I have not checked them against any of the experimental papers that describe the actual experiments.

I don't think anyone is disputing that the actual experimental results agree with the QM predictions, or that both of those things violate the Bell inequalities (or the CHSH inequalities, whichever are applicable to the particular experiment).

What I can't figure out is what issue you are trying to describe or why you think it's an issue.

 

[PeterDonis]

I used the sin squared function

What do you think the sin squared function represents?

 

[PeroK]

My argument is that Bell's inequality is simply derived from the associative laws of sets, and hence is a purely mathematical relationship.

No. It's probability theory - using classical probabilities.

The question I am raising is simply about the use of the data. The only inputs are probabilities obtained from the count at each detector, and the angle between the detector planes. I am not denying the mathematics of quantum mechanics; I am questioning the use or misuse of simple numbers in a pure mathematical relationship, namely where terms in the derivation are constant, the way they are used in this application, they are no longer constant. What is puzzling me at the moment is that nobody seems to be addressing the issue I raised. Am I totally obscure in some way? Have I not phrased the question properly?

I don't understand what you mean by this.

 

[PeroK]

@Ian J Miller you could try this, to get an insight into how probabilities in QM are calculated differently than in the classical case:

https://www.scottaaronson.com/democritus/lec9.html

 

[Ian J Miller]

Then I have no idea what your issue is. I have read every post you have made and I still don't see it.

Has anyone in the literature raised the issue you are raising? Is there a paper you can reference that discusses it?

Then consider the following derivation. Now I am going to use brackets to separate terms, which removes the need for dots that confused some. The test requires that each entangled pair either passes or fails, but not both, which, for the probabilities at A give
(A+) + (A-) = 1 ..... (1)
and so on for B and C. Now we can write
(A+)(B-) = (A+)(B-)[(C+) + (C-)] (2)
and
(B+)(C-) = [(A+) + (A- )](B+)(C-) (3)
Add the right hand sides of (2) and (3) and we get
(A+)(C-)[(B+) + (B-)] + (A)(B-)(C+) + (A-)(B+)(C-) (4)
Again, the square brackets equals 1, and since the last two terms must be positive numbers or zero, we have
(A+)(B-) + (B+)(C-) ≥ (A+)(C-) (5) i.e. a Bell inequality.

Now, from Bell (pp 9 - 10 of the reference above) the initial position is the plus and minus detectors at right angles to each other. The joint probability is given sin squared θ, where θ is the difference between the movement of the two detectors from the initial position. Bell counts that as sin squared 22.5 degrees, which is approximately 0.146. But consider the array where B+ and B- are counted. The count at B- is restricted to photons that arrive within x ns of B+, and from rotational symmetry, B+ must be a positive number. But the joint probability is zero because sin squared zero is zero. (In practice this won't be exact.) But if the joint probability is zero, B- must be zero (expected from the Malus law). But if B- is zero in the test that was for B alone, unless it is a function of the position of the other detector, it cannot give any result other than 0 elsewhere.

Because of the rotational symmetry of the Aspect source, A+ = B+ = C+, hence A- = B- = C- and if the apparatus is working correctly the latter three must be zero, which puts the first three as equal to 1. Further, since the second photon is only counted AFTER the first one is counted, and is referenced to it by the x ns requirement, the first probability is 1, because it already happened for the second count to occur. That gives
0 + 0 ≥ 0, and we note the two extra terms in (4) above contain at least one "fail" so they also equal 0 and we can sum and say 0 = 0, which is true, if not very interesting.

So my problem is that the term B- (and similarly C-) is not constant. If we accept that the plus terms are 1, then C- is either 0.146 or 0.5. But the derivation above requires the terms to have a constant value for a given set of tests. Thus with the sock washing, if one temperature uses woollen socks, another nylon socks, and another paper socks, you can violate easily by putting the paper socks in the appropriate run. Then problem, as I started out, was that the reference frame is varying depending on where the first detector is, and the second detector is referenced to the first, and not to some common reference.

Let us provide an external frame of reference. Assume a source that provides entangled photons, all of which are polarized in one plane. Align the A+ detector with this plane, and, as with Aspect, assign B as a rotation of 22.5 degrees and C as a rotation of 45 degrees. If so, A+ has a probability of 1 (assuming everything is perfect)- B- a probability of sin squared 22.5 and C- sin squared 45 degrees. In short, (A+)(B-) and (A+)(C-) are now the same as calculated for the Aspect experiment, however, (B+) now has a probability of cos squared 22.5 degrees, and C- is the same as above. Inserting these values into our derived inequality and we get
1 x 0.146 + 0.8536 x 0.5 should be ≥ 0.5.
which comes out to 0.573 ≥ 0.5,
which is true. So, provided the photons are counted properly in this last thought experiment, the inequality is NOT violated. So how can we say that there are violations when the reference frames are varied but not when there is an external reference frame?

There is no reference to this. I hope I am allowed to think for myself. And if i have made a mistake somewhere, I would appreciate someone showing me where.

 

[PeterDonis]

There is no reference to this. I hope I am allowed to think for myself.

It depends on what you mean. If by "think for myself" you mean "develop a personal theory that claims that everyone in QM is wrong about what Bell inequality violation experiments show", then no, that's not permissible here. Personal theories are off limits at PF.

If, OTOH, by "think for myself" you mean "ask questions about a standard result in QM that I don't understand", then you shouldn't be making up your own derivations and your own notations. You should be giving us a reference that contains a derivation of the standard result you have questions about, and then asking questions about that reference, using the standard notation in that reference. For example, if the "Berltmann's socks" paper by Bell is your chosen reference, which statements or equations in that paper do you have questions about?

Your multiple walls of text in this thread are incomprehensible to me, and, I gather, to the other participants in this thread as well. You would be much better served by doing what I suggested in the last paragraph.

 

[PeterDonis]

The joint probability is given sin squared

As I have already asked once: what do you think the sin squared function represents? It's a simple question and you should be able to give a simple answer without another wall of text.

 

[PeterDonis]

from Bell (pp 9 - 10 of the reference above) the initial position is the plus and minus detectors at right angles to each other.

I am unable to recognize this as a reference to anything in Bell's "Bertlmann's socks" paper.

 

[Ian J Miller]

For an entangled pair, where one photon gives a click on the first detector, the sin squared function surely represents the probability that the partner photon will give a click on its detector, the term θ being the difference between rotations of the two detectors from the initial position.

The reason for "the walls of text" is that when I initially asked the question, none of the responses addressed what I was asking so I felt I had to elaborate.

This is NOT a personal theory. It is questioning how a term that should have a single value in the derivation of a relationship appears to have multiple values when these are used to argue that said mathematical relationship is violated. It is a simple question. Either a term such as B- should be required to have one value or it should not, and if not there should be some sort of derivation that shows what its limits are.

I am sorry you do not get what I see as a problem, but I have done all I can to explain it. There is probably no point in continuing this because it appears I cannot seem to make any progress.

 

[PeterDonis]

For an entangled pair, where one photon gives a click on the first detector, the sin squared function surely represents the probability that the partner photon will give a click on its detector

Probability according to which theoretical model?

It is questioning how a term that should have a single value in the derivation of a relationship appears to have multiple values when these are used to argue that said mathematical relationship is violated.

The reason you are having difficulty getting helpful responses is that nobody but you can understand how you are even getting to this belief in the first place. That's why I think it would be much better if, instead of trying to inundate us with your personal derivations, you would point us to explicit quotations and equations from whatever reference you are using that you think justify this belief of yours.

 

[PeterDonis]

Probability according to which theoretical model?

explicit quotations and equations

In the spirit of taking my own advice, I am going to give the answer to this question by quoting from Bell's Bertlmann's socks paper. I am using the PDF version that can be found on CERN's website [1].

Equation (4) in that paper gives probabilities in terms of the sin squared function. As the paper says, these are probabilities according to quantum mechanics, i.e., that is the theoretical model used to derive them. For simplicity I'll just give the probability of a match (up/up or down/down):

$$
P_\text{QM} = \frac{1}{2} \sin^2 \frac{\theta}{2}
$$

where $\theta$ is the difference in angles, called $a - b$ in the paper.

Equation (3) in that paper gives probabilities as they are derived from a different theoretical model, a "Bertlmann's socks" type of model:

$$
P_\text{socks} = \frac{| \theta |}{\pi}
$$

Bell's argument is then a simple one: $P_\text{socks}$ does not match $P_\text{QM}$ except at an isolated set of values of $\theta$, namely, $0$, $\pi / 2$, and $\pi$. So no "Bertlmann's socks" type of model can match the predictions of QM over the entire range of possible angles.

@Ian J Miller you talk of "multiple values". Where are the multiple values here? For any given actual run of an actual experiment, $\theta$ will have one value. And we can compare the two probabilities, using the above formulas, at whatever that value is, and see that they don't match unless that value is one of the three isolated ones I gave above. That is what all of the experimental tests end up boiling down to.

[1] https://cds.cern.ch/record/142461/files/198009299.pdf