EvC Forum: Bells, Bells, Bells, Bells ...

This article is an offshoot of the discussion on the {Quantum Entanglement - what is it?} Thread
EvC Forum: Quantum Entanglement - what is it?

Proposed location: Columnists Corner
EvC Forum: Columnist

I've been asked to be a little more specific, more complete, in stating my position, so if you thought some of other posts were long ...

Bell's Theorem

Most of the sites that talk about "Bell's Theorem" do not even state the theorem, but talk about the "Inequality Experiment," so this is the best I could find:

From Wikipedia:
(Bell's theorem - Wikipedia)

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

(Information copied from above site on 26MAY2006)

Kind of hard to argue with that eh? (of course it doesn't matter whether those predictions are true or not for this to hold). I'd use observed behavior instead of predictions, but then I can't seem to find any original statement of Bell's Theorem ...

Note that my beef is not with this theorem but with the application of it to the three sensor photon experiment and the concept of "Inequality," which we'll come to later. Much later. We have groundwork to cover first. I want anyone to be able to understand this so I will try to keep it simple. And accessible.

Now page down the wikipedia article (which is not really that accessible eh?) until you come to:

Bell's thought experiment

This sets the stage for the Bell's Inequality Test.

Bell considered a setup in which two observers, Alice and Bob, perform independent measurements on a system S prepared in some fixed state. Each observer has a detector with which to make measurements. On each trial, Alice and Bob can independently choose between various detector settings. Alice can choose a detector setting a to obtain a measurement A(a) and Bob can choose a detector setting b to measure B(b). After repeated trials Alice and Bob collect statistics on their measurements and correlate the results.

There are two key assumptions in Bell's analysis: (1) each measurement reveals an objective physical property of the system (2) a measurement taken by one observer has no effect on the measurement taken by the other.

In the language of probability theory, repeated measurements of system properties can be regarded as repeated sampling of random variables. One might expect measurements by Alice and Bob to be somehow correlated with each other: the random variables are assumed not to be independent, but linked in some way. Nonetheless, there is a limit to the amount of correlation one might expect to see. This is what the Bell inequality expresses.

(ibid)

This last paragraph is where I think things start to go awry... and I'll tell you why:

" ... the random variables are assumed not to be independent, but linked in some way."

This is bad thinking, imh(ysa(atm))o, for several reasons. Not only that, the analysis later will be based on assuming their being independent, but we'll come to that later. For now lets assume a system, just to see how this "any old randomly chosen three (or four or whatever number we like) measurements will do" approach works:

RAZD's thought experiment

Let's say I want to find when the high to low tidal change in the Bay of Fundy is the greatest and when it is the least.

(1) One measurement I need is the tidal levels in the Bay of Fundy. We'll call this TL.

Then we can assume that the orientation of the earth, the position of the moon and the position of the sun are variables, so I can measure:

(2) The orientation of the earth we'll measure as the angle of the Greenwich Meridian to the red heart of Scorpio (Antares, the "rival of mars"). We'll call this EA.

(3) Next we'll measure the position of the moon relative to the line from the earth to Sirius (the "Dog" star, the brightest star in the sky). We'll call this MS.

(4) We'll measure the position of the sun relative to the line from the earth to Regulus, the heart of the Lion and the bottom of the "question mark" that is the signature of Leo. We'll call this SR.

Now Bob and Carol and Ted and Alice can each go and take independent measurements, choosing at random which ones they measure, even choosing whether they measure two or three on a given day.

We'll let them record data for several years with the only stipulation that they all make the measurements at the same time on any given day -- we'll let some random buzzer tell them when to make the measurements -- so that the results won't be artificially related to the time of day, which is one of the variables.

We'll also let some poor mathematician try to figure out what functions\relationships.TL = f(EA) + f(MS) + f(SR)
Can be derived from the data. Note that Bob and Carol and Ted and Alice will happily always get the same measurement for the same variable on the same measurement date\time. We can assume this validates the measurements eh?

Remember that we are interested in when the maximum or minimum changes between tides occur and not when the tides change.

Of course to compare the data at all our poor mathematician needs to reduce the data because we also have:

(a) MS = f(angle between the Earth Greenwich Meridian to Sirius, and the position of the Moon), and
(b) SR = f(angle between the Earth Greenwich Meridian to Regulus, and the position of the Sun)

And we'd have to worry about whether those relationships changed over time, whether we had a long enough record to measure the variations possible.

Now we'll simplify the system. Greatly. We'll throw out one of the "variables" above and reduce the system to:

(1) TL as before
(2) Sun position in relation to the Greenwich Meridian, SGM, and
(3) Moon position in relation to the Greenwich Meridian, MGM

Again we have Bob and Carol and Ted and Alice each go and take independent measurements, choosing at random which two they measure, (if we measured all the variables at once we might be able to figure it out quicker, but that's not the point of this exercise is it?)

We'll also let our somewhat happier mathematician figure out what functions\relationships.TL = f(SGM) + f(MGM)
Can be derived from the data. Note that Bob and Carol and Ted and Alice will happily always get the same measurement for the same variable on the same measurement date\time. We can assume this validates the measurements eh?

At this point some physicists will make comparisons to a certain experiment of theirs and claim that this proves that:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

(ibid)

But now we'll simplify the system. Again. We'll throw out another of the "variables" above and reduce the system to:

(1) TL as before
(2) The angle between the Sun, with the Earth as the vertex, and the Moon, SEM.

Again we have Bob and Carol and Ted and Alice each go and take independent measurements, choosing at random which one they measure, (if we measured both at once it would almost be a no-brainer eh?)

We'll also let our almost happy mathematician figure out what functions\relationships.TL = f(SEM)
Can be derived from the data. Note that Bob and Carol and Ted and Alice will happily always get the same measurement for the same variable on the same measurement date\time. We can assume this validates the measurements eh?

Of course our poor mathematician needs to still figure out that there is some hidden variable that is not being measured, but which makes it impossible to figure out the relationship without a lot of data to normalize and reduce ... he'll need to use probability analysis to figure out what the maximum tidal changes and minimal tidal changes are from tables and tables of data on what the Tide Level was for all similar measurements of the other variables.

So, now we'll simplify the system. One More Time. We'll throw out the "hidden variable" above and measure the information we are actually interested in -- the difference between High Tide and Low Tide on the day of data measurements -- and finally reduce the system to:

(1) TL_{high tide} - TL_{low tide} = DTL
(2) The angle between the Sun, with the Earth as the vertex, and the Moon, SEM.

Again we have Bob and Carol and Ted and Alice each go and take independent measurements, choosing at random which one they measure, (if we measured both at once it would be a no-brainer eh?)

We'll also let our almost delirious mathematician figure out what functions\relationships.
DTL = f(SEM)

Can be derived from the data. In fact I would be surprised if Bob and Carol and Ted and Alice didn't figure it out even though they only make one measurement a day. Please note that they will still happily always get the same measurement for the same variable on the same measurement date\time. We can assume this validates the measurements eh?

We've gone from TL = f(EA) + f(MS=f(Moon,Sirius,Greenwich)) + f(SR=f(Sun,Regulus,Greenwich))
to
DTL = f(SEM)
And (relatively) easily seeing that the actual equation is of the form

DTL = ktb*cos(SEM) + tbb
(where ktb and tbb are some constants related to the tidal basin behavior, if you want to get more technical than necessary).

I'll call this a demonstration of "RAZD's Rule of Irreducible Simplicity ("RIS")" - which can be stated:

If you can reduce the number of variables and still fully describe the function, then the {system} described is NOT irreducibly simple, and the variables that can be omitted don't add information, but if you cannot reduce the number of variables without losing some description of the function, then your variables have reached the state of "Irreducible Simplicity" (IS) and each variable adds some information about the system.

It IS or it ISn't. Simple, eh?

I would also say that if you haven't gone through the process to determine

whether you have reduced the number of possible variables to the minimum needed to describe the behavior, or
whether the observed behavior is really related to the variable being measured, that
then you don't know what you are measuring

The first we have discussed above. For the second consider measuring the wind direction at the mouth of the Bay of Fundy and whether it is blowing in or out.

The data is recordable, and the results, when the same variable is measured by different people for the same measurement date\time is that they will still happily always get the same measurement. We can thus no longer assume this validates the measurements, because the wind direction has no bearing on the {DTL = TL_high - TL_low} data. Both of these situations are eliminated by using "RAZD's Rule of Irreducible Simplicity ("RIS")"

Of course some people will recognize this as the parsimony principle, or (more commonly) as Occam's (Bloody) Razor.

Bell's Inequality

Now it gets a little more interesting.

From wikipedia again:

Original Bell's inequality

The original inequality that Bell derived (Bell, 1964) was:1 + C(b,c) >or= |C(a,b) - C(a,c)|where C is the "correlation" of the particle pairs and a, b and c settings of the apparatus.

(ibid)

And we're already talking about the experimental setup ...

To understand this concept better we go back to Bell's thought experiment and repeat the essential elements (to pick up where we were before my little detour into developing my Rule of Irreducible Simplicity):

Each observer has a detector with which to make measurements. On each trial, Alice and Bob can independently choose between various detector settings. Alice can choose a detector setting a to obtain a measurement A(a) and Bob can choose a detector setting b to measure B(b).

(ibid)

And we add one other element: the stability and repeatability of the measured properties. This next bit is from "Spooky Action at a Distance, An Explanation of Bell's Theorem" by Gary Felder, but similar statements can be found in a number of articles, and this has been repeated in other posts on this topic (on the original thread): (Teaching and Learning STEM)

Just to give it a name which shouldn't in any way prejudice us as to what is actually being measured I will refer to this property as the electron's "happiness". For example I might try sending the same electron into two of these detectors (assumed to be identical) one right after another. If I find that the two detectors always give the same result when they measure the same electron, then I can conclude that the electron's happiness is stable over time; in other words it doesn't change as it moves between the two detectors.

(Information copied from above site on 26MAY2006)

I'm happy to use {Felder's Happiness}, or "{FH}ness", for the property under consideration, to denote this condition of stability over time.

This is important for a couple of reasons, not least of which is that you can depend on your results to refer to the same data base when you finish as when you started.

The most important reason though, is that this demonstrates that the property does not depend on any other variable (that is included in the test anyway).

Now if you remember, Bob and Carol and Ted and Alice also determined that they could always get the same results for the same measurements, in all their various attempts to figure out the relationship of earth, moon, and sun to tidal extremes, so what does this mean?

Lets start with measurement {X}. Whenever I measure the (binary) {X}ness of the system I get either {X>0 (+)} or {X<0 (-)} and whenever I repeat the measurement I get the same {X>0 (+)} or {X<0 (-)} result.

I can say that {RESULT} = f(X) - the result is some function of {X}ness.

I can think of the {Object Being Measured ("OBM")} as having some location on an {X} axis that makes it either {X>0 (+)} or {X<0 (-)} .

Pretty simple: if {X} is positive (+) we get a green light, And if {X} is negative (-) we get a red light.

Now I add measurement {Y}. Whenever I measure the (binary) {Y}ness of the system I get either {Y>0 (+)} or {Y<0 (-)} and whenever I repeat the measurement I get the same {Y>0 (+)} or {Y<0 (-)} result.

I can say that {RESULT} = f(Y) - the result is some function of {Y}ness.

I can think of the {OBM} as having some location on a {Y} axis that makes it either {Y>0 (+)} or {Y<0 (-)}.

Still pretty simple: if {Y} is positive (+) we get a green light, And if {Y} is negative (-) we get a red light.

And when I look at both these measurements together I can conclude

That they both measure the same thing, with Y = f(X) (as in Y = aX+b or Y = aX²+bX+c, etc), OR
That they measure aspects of two completely different things (ie - tidal fluctuations in the Bay of Fundy and temperature changes in Afghanistan), OR
That they measure two different aspects of the same thing

If (and only if) we assume (or better still, demonstrate) that we are measuring two different aspects of the same thing, do we have data that relates to anything meaningful, and then we can get a sense of interaction between the two variables:{RESULT} = f(X,Y)Where the result is some function of {X}ness and {Y}ness.

And I can think of the {OBM} as having some location on a {X} axis that makes it either {X>0 (+)} or {X<0 (-)} PLUS some location on a {Y} axis that makes it either {Y>0 (+)} or {Y<0 (-)}.

We have defined a 2D space where {X}ness is independent of {Y}ness (and {X's FH}ness is preserved), and {Y}ness is independent of {X}ness (and {Y's FH}ness is preserved).

This gives me a 2x2 grid of four regions:

------------------
| +Y -X  | +Y +X |
------------------
| -Y -X  | -Y +X |
------------------

A standard 2D grid.

We could be measuring a position in {X,Y} space, or the head or tail of a vector orientation in {X,Y} space, but it is a measurement that would be repeated by any other measurement of the {OBM} in any other {X,Y} space.

Now for some arcane reason we'll go back to the dysfunctional Bob and Carol and Ted and Alice that can only measure one variable at a time, with random sampling of {X}ness and {Y}ness, and we want to know when the results are the same (S) and when they are different (D).

We ship the results off to our overworked mathematician who determines the results are:

  Detector 1
| X | Y |
-----------
X | S | D |
----------- Detector 2
Y | D | S |
-----------

Where Same (S) = ++ OR -- and Different (D) = +- OR -+

I've mentioned this before, with {A} and {B} in place of {1} and {2} and {1} and {2} in place of {X} and {Y}. This is exactly the same 2x2 grid.

Now I add measurement {Z}. Whenever I measure the (binary) {Z}ness of the system I get either {Z>0 (+)} or {Z<0 (-)} and whenever I repeat the measurement I get the same {Z>0 (+)} or {Z<0 (-)} result.

I can say that {RESULT} = f(Z) - the result is some function of {Z}ness.

I can think of the {OBM} as having some location on a {Z} axis that makes it either {Z>0 (+)} or {Z<0 (-)}.

Still pretty simple: if {Z} is positive (+) we get a green light, And if {Z} is negative (-) we get a red light.

XYZ

And when I look at all three of these measurements together, we now have a three way combination of the above conclusions about what is being measured, and I can conclude

All three measure measure something entirely different (ie - tidal fluctuations in the Bay of Fundy, temperature changes in Afghanistan and stock market prices in Tokyo), OR
Any two measure the same thing, and the third measures something different, OR
All three measure the same thing {Y = f(X) & Z = g(X)}, OR
Any two measure the same thing, and the third measures a different aspect of the same thing, OR
All three measure three different aspects of the same thing

If (and only if) we assume (or better still, demonstrate) that we are measuring three different aspects of the same thing, do we have data that relates to anything meaningful, and then we can get a sense of interaction between the three variables:{RESULT} = f(X,Y,Z)Where the result is some function of {X}ness, {Y}ness and {Z}ness.

And I can think of the {OBM} as having some location on a {X} axis that makes it either {X>0 (+)} or {X<0 (-)} PLUS some location on a {Y} axis that makes it either {Y>0 (+)} or {Y<0 (-)} PLUS some location on a {Z} axis that makes it either {Z>0 (+)} or {Z<0 (-)}.

We have defined a 3D space where {X}ness is independent of {Y}ness AND {Z}ness (so {X's FH}ness is preserved), and {Y}ness is independent of {X}ness AND {Z}ness (so {Y's FH}ness is preserved), and {Z}ness is independent of {X}ness AND {Y}ness (so {Z's FH}ness is preserved).

This gives me a 2x2x2 grid of eight regions, with:

------------------
| +Y -X  | +Y +X |
------------------ +Z
| -Y -X  | -Y +X |
------------------

over

------------------
| +Y -X  | +Y +X |
------------------ -Z
| -Y -X  | -Y +X |
------------------

Two stacked standard 2D grids for the two different Z values.

------------------------
| +Z +Y -X  | +Z +Y +X |
------------------------
| +Z -Y -X  | +Z -Y +X |
------------------------

over

------------------------
| -Z +Y -X  | -Z +Y +X |
------------------------
| -Z -Y -X  | -Z -Y +X |
------------------------

These 8 regions can be represented as:

(X,Y,Z) --- --+ -+- -++ +-- +-+ ++- +++
Or (RG) RRR RRG RGR RGG GRR GRG GGR GGG
OR digitally as 000 001 010 100 101 110 111

We could be measuring a position in {X,Y,Z} space, or the head or tail of a vector orientation in {X,Y,Z} space, but it is a measurement that would be repeated by any other measurement of the {OBM} in any other {X,Y,Z} space.

Now for some arcane reason we'll go back to the half-way house heavyweights, Bob and Carol and Ted and Alice, that can still only measure one variable at a time, with random sampling of {X}ness and {Y}ness, and we want to know when the results are the same (S) and when they are different (D).

We ship the results off to our overstressed and underpaid mathematician who determines the results are:

(1) Whenever the measurement is {X}ness and {Y}ness:

------------------    -----------    ---------
| +Y -X  | +Y +X |    | GR | GG |    | D | S |
------------------ OR ----------- OR ---------
| -Y -X  | -Y +X |    | RR | RG |    | S | D |
------------------    -----------    ---------

With {Z}ness undetermined

(2) Whenever the measurement is {X}ness and {Z}ness:

------------------    -----------    ---------
| +Z -X  | +Z +X |    | GR | GG |    | D | S |
------------------ OR ----------- OR ---------
| -Z -X  | -Z +X |    | RR | RG |    | S | D |
------------------    -----------    ---------

With {Y}ness undetermined

(3) Whenever the measurement is {Y}ness and {Z}ness:

------------------    -----------    ---------
| +Z -Y  | +Z +Y |    | GR | GG |    | D | S |
------------------ OR ----------- OR ---------
| -Z -Y  | -Z +Y |    | RR | RG |    | S | D |
------------------    -----------    ---------

With {X}ness undetermined

(4) Whenever the measurement is {X}ness alone:

------------------    -----------    ---------
| -X -X  | +X +X | OR | RR | GG | OR | S | S |
------------------    -----------    ---------

{X's FH}ness is preserved with {Y}ness and {Z}ness undetermined.

(5) Whenever the measurement is {Y}ness alone:

------------------    -----------    ---------
| -Y -Y  | +Y +Y | OR | RR | GG | OR | S | S |
------------------    -----------    ---------

{Y's FH}ness is preserved with {X}ness and {Z}ness undetermined.

(6) Whenever the measurement is {Z}ness alone:

------------------    -----------    ---------
| -Z -Z  | +Z +Z | OR | RR | GG | OR | S | S |
------------------    -----------    ---------

{Z's FH}ness is preserved with {X}ness and {Y}ness undetermined.

He also notes that (1) goes with (6), (2) goes with (5) and (3) goes with (4).

He sends his report off and is about to go sailing on the Bay of Fundy when the Head Of Indeterminate Experimentation {HOIE} comes storming into his office, and exclaims: "Can't you do better than this? Here, try this Grid:

   Detector  A
|  X |  Y |  Z |
------------------
X | %S | %D | %D |
------------------
Y | %D | %S | %D | Detector B
------------------
Z | %D | %D | %S |
------------------

Because we know this becomes:

   Detector  A
|   X   |   Y   |   Z   |
---------------------------
X | 100%S |  %D   |  %D   |
---------------------------
Y |  %D   | 100%S |  %D   | Detector B
---------------------------
Z |  %D   |  %D   | 100%S |
---------------------------

In order for '{FH}ness' to be preserved, so there must be some inequality in the data between {S}ness and {D}ness for classical objects if this is the case."

And our poor misunderstood mathematician looks at it and chuckles.

"You have a couple of errors there. For starters, you are trying to represent a 2³ space with a 3² surface. Your data actually shows 50:50 distribution between {S}ness and {D}ness."

"Of course, says the {HOIE}, but that's because we are measuring {Strange, Mysterious Objects (SMO)s} and not normal ones."

"No," says the mathematician, "it's because you are ignoring some of the data in your 3x3 grid. You are projecting your theoretical 3D space onto a 2D surface and only selecting part of the data to project."

"When you have {X}ness vs {X}ness (your 100%S condition) you also have {Y}ness vs {Z}ness (which has 50%S and 50%D), and vice versa, and you are also not considering the {S}ness and {D}ness of {X} vs {Y} or {X} vs {Z} in that condition yet."

"What do you think you are measuring?" he asks.

Our earnest quartet of quantitative qualifiers, Bob and Carol and Ted and Alice, are standing there with their mouths open, astounded that anyone would contradict or question the {HOIE}.

The Experiment

At long last. That was one long introduction, eh? So I'll spare you the ongoing theatrics between the long suffering mathematician, the overconfident {HOIE} and the peanut gallery fearsome foursome, Bob and Carol and Ted and Alice, and get to the meat of the argument.

The experiment involves sending pairs of photons through two separate polarized filters.

One detector, the "A" detector, measures the {Pass\Fail} passage of one particle, the "A" particle, through the "A" detector, and the other detector, the "B" detector, measures the {Pass\Fail} passage of the other particle, the "B" particle, through the "B" detector. A light turns GREEN if the particle passes and RED if it fails to pass on each detector.

The filters in each detector are arranged at 120^o intervals, as in:

The pairs of photons are created by a process that preserves angular momentum from a source with zero angular momentum, thus particle A has the same angular momentum value but opposite direction as particle B. Thus detector B has to be turned over compared to detector A to end up with:

Detector A receives particle A with angular {momentum\orientation} represented by {Q}

and

Detector B receives particle B with angular {momentum\orientation} represented by {Q}

We are measuring the same {Green/Red}ness of {Q} at detector {A} and detector {B}. what we change is the angle at which the {Green/Red}ness of {Q} is measured.

The reason Bob and Carol and Ted and Alice can only measure one orientation at a time is that we only have two particles, one to establish a baseline and the other to measure the change in {Green/Red}ness for a change in detector angle.

What we really have is not three filters at 120^o but three filters at 60^o that repeat their measurements at 180^o intervals:

The Reality

Like the {sun\earth\moon\tide} experiment, we don't have 3 or 4 different variables, but three different measures of the same variable, such that I can use:

{Green/Red}ness₁ = f(Q)

{Green/Red}ness₂ = f(Q+n)

{Green/Red}ness₃ = f(Q+m)

Furthermore, we know from other photon experiments that the probability of getting a {Green\Pass} result or getting a {Red\Fail} result can be calculated as:

{Green}ness₁ = cos²(Q)
and
{Red}ness₁ = sin²(Q)

Where Q is the angle of orientation of the particle to filter #1 (the 12 O'clock position).

The probability of {Green\Pass} plus the probability of {Red\Fail} == 1 for all angles {Q} and for any orientation of the detecting filter. We will either get {Green\Pass} or {Red\Fail} for any particle.

Now I can write:

{Green}ness₂ = cos²(Q+120^o)
and
{Red}ness₂ = sin²(Q+120^o)

and

{Green}ness₃ = cos²(Q+240^o)
and
{Red}ness₃ = sin²(Q+240^o)

which is the same as

{Green}ness₃ = cos²(Q+60^o)
and
{Red}ness₃ = sin²(Q+60^o)

You can think of Detector {A} as determining what the {Green\Red} value is for Particle {A} and Detector {B} as measuring the phase shifted probability of the mate particle {B}.

Predictions

(1) That for all photons where f(Q) = {GREEN} at detector {A} the distribution of probabilities at detector {B} are:

{Green}ness₁ = cos²(Q)_(Q=0,180) = 1
and
{Red}ness₁ = sin²(Q)_(Q=0,180) = 0

and

{Green}ness₂ = cos²(Q+120)_(Q=0,180) = 1/4
and
{Red}ness₂ = sin²(Q+120)_(Q=0,180) = 3/4

and

{Green}ness₃ = cos²(Q+60)_(Q=0,180) = 1/4
and
{Red}ness₃ = sin²(Q+60)_(Q=0,180) = 3/4

(2) That for all photons where f(Q) = {RED} at detector {A} the distribution of probabilities at detector {B} are:

{Green}ness₁ = cos²(Q)_(Q=90,270) = 0
and
{Red}ness₁ = sin²(Q)_(Q=90,270) = 1

and

{Green}ness₂ = cos²(Q+120)_(Q=90,270) = 3/4
and
{Red}ness₂ = sin²(Q+120)_(Q=90,270) = 1/4

and

{Green}ness₃ = cos²(Q+60)_(Q=90,270) = 3/4
and
{Red}ness₃ = sin²(Q+60)_(Q=90,270) = 1/4

(3) That I get the same results for (2) and (3) as I do for (1).

(4) That the final grid of "Inequality" is:

         Detector  A
|   1   |   2   |   3   |
---------------------------
1 | 100%S |  25%S |  25%S |
|   0%D |  75%D |  75%D |
---------------------------
2 |  25%S | 100%S |  25%S | Detector B
|  75%D |   0%D |  75%D |
---------------------------
3 |  25%S |  25%S | 100%S |
|  75%D |  75%D |   0%D |
---------------------------

(5) That I can repeat the experiment with these added features:

Two identical setups, one with Bob and Alice and one with Carol and Ted.
Move one detector in each further from the source so that there is always a first signal at the closer one ({A} and {C}) and a second signal at the farther one ({B} and {T}). This can still be done with enough separation between {A} and {B} (and between {C} and {T}) that a signal cannot travel from {A} to {B} at the speed of light before the detector records the results.
Use the second signal ({B} and {T}) to sort the data into {GREEN} and {RED} portions and then compare the results between {C} and {T}.

Note: this removes any effect from "entanglement" in the data. It also allows data to compare {2} and {3} based on {1}=Green and {1}=Red, etc.

(6) That the results will still be the same, and thus are independent of any effect from "entanglement" between the photon pairs.

(7) That other angles of phase shift can be measured, and that the results will still match the predicted values from the phase shift equations.

Conclusions

The "inequality" is not a true representation of the reality, and is based on either a misunderstanding or a misrepresentation of what is really being measured. Or both: with the misrepresentation being of the 3 "variables" being measured as independent rather than the same variable with a phase shift, and with the misunderstanding being that what is being measured is the change in {Green\Red} response as a result of phase shifting of the detectors

Another way to look at it is that the so-called "inequality" is a 2D grid in 120 degree increments:

           Detector  A
|   0   |  120  |  240  |
-----------------------------
0 | 100%S |  25%S |  25%S |
|   0%D |  75%D |  75%D |
-----------------------------
120 |  25%S | 100%S |  25%S | Detector B
|  75%D |   0%D |  75%D |
-----------------------------
240 |  25%S |  25%S | 100%S |
|  75%D |  75%D |   0%D |
-----------------------------

and see that it is just an excerpt from a more complete table, such as one using 5 degree increments:

                       Detector  A
|   0   |   5   |   10  |   15  |  ...  |  360  |
-----------------------------------------------------
0 | 100%S |  98%S |  94%S |  87%S | ...%S | 100%S |
|   0%D |   2%D |   6%D |  13%D | ...%D |   0%D |
-----------------------------------------------------
5 |  98%S | 100%S |  98%S |  94%S | ...%S |  98%S |
|   2%D |   0%D |   2%D |   6%D | ...%D |   2%D |
-----------------------------------------------------
10 |  94%S |  98%S | 100%S |  98%S | ...%S |  94%S |
|  13%D |   2%D |   0%D |   2%D | ...%D |   6%D |
----------------------------------------------------- Detector B
15 |  87%S |  94%S |  98%S | 100%S | ...%S |  87%S |
|  13%D |   6%D |   2%D |   0%D | ...%D |  13%D |
-----------------------------------------------------
... | ...%S | ...%S | ...%S | ...%S | 100%S | ...%S |
| ...%D | ...%D | ...%D | ...%D |   0%D | ...%D |
-----------------------------------------------------
360 | 100%S |  98%S |  94%S |  87%S | ...%S | 100%S |
|   0%D |   2%D |   6%D |  13%D | ...%D |   0%D |
-----------------------------------------------------

There is nothing magical about getting the same result for the same phase shift, nor about getting different results for different phase shifts. There is nothing magical about getting results based on phase shift and no other variable, "entangled" or hidden.

Certainly I do not need to invoke "entanglement" to describe or predict the results, and thus I see no reason to be impressed with either the "Inequality" concept or the experimental results.

Enjoy.

Edited by RAZD, : No reason given.

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