Bell’s inequality in pictures ...and other observations

Mike Ulrey   Wolfram Technology Conference   October 17, 2018

1. What is this presentation about?

“Spooky action at a distance”, i.e non-locality --

Einstein's concern that led him to claim that quantum mechanics in its present form must be incomplete [Einstein & Podolsky & Rosen 1935].

Bell’s theorem [Bell 1964]: No local model of reality can underlie the quantum facts. Reality must be non-local.

Bell’s Theorem  is one of the most important results in the area of quantum theory foundations.

Also one of the most controversial.

2. Why am I interested?

Ringing the Bell: The ongoing debate

Why is there still so much activity (papers, conferences, debates) surrounding a paper written in 1964, and going back to ideas from the 1930’s?

Google Scholar citations -- almost 6000 citations of John Bell’s papers in 2017 alone!

https://scholar.google.com/citations?user=Bd9KX0oAAAAJ&hl=en #d=gsc_md _hist&p=&u=

dates= Table[k,{k,1981,2018}];
citations={974,921,929,1065,959,1020,923,1121,1250,1424,1340,1457,1436,1601,1787,1773,1807,1908,2159,2173,2403,2693,2581,2841,3364,3384,3581,3838,4155,4157,4496,5200,5675,5755,5644,5975,5908,4178};BarChart[citations,ChartLabels->dates,BarOrigin->Left,AxesLabel->{"Year","Citations"},PlotLabel->"Figure 1. Google Citations for John Bell",ImageSize->1200]

The Vaxjo conferences -- held every year since 2000 -- still continues to exhibit fierce disagreements among experts concerning the foundations of QM!

Tim Maudlin, “What Bell Did”  [Maudlin 2014 (1)], [Maudlin 2014 (2)] https://www.youtube.com/watch?v=Vg5z_zeZP60

Maintains that most mainstream physicists misunderstand BOTH what Einstein was concerned about and what Bell actually proved in his theorem

Not randomness or the role of probability in QM, but instead, the implications for non-locality

Physics World video illustrates the common (mistaken) beliefs about Einstein’s concern:  https://www.youtube.com/watch?v=hXb6jJ1lQ_o

3. Who am I to dare offer my two cents?

Openly admit to being a novice in quantum mechanics

PhD in mathematics (THE Ohio State Univ) -- thesis in Information Theory

Career (mostly) as an engineer at NCR, Raytheon, but mostly Boeing (Seattle 30 years) -- flight controls, software, computer simulation, safety and reliability, air traffic control

Used Mathematica in aerospace applications from 1992 to 2016.

Have used Mathematica as a physics hobbyist since my retirement in 2016.

Love Mathematica.

Fascinated by quantum mechanics for 30 years, especially  the mysteries of quantum foundations, ever since reading Nick Herbert’s book Quantum Reality.

In my fumbling attempts to understand Bell’s Theorem and the controversy surrounding it, I have developed a model-based approach with a geometric flavor that greatly enhanced my understanding

Maybe it will be helpful to other interested amateurs (or maybe even professionals?)

4. Goals and objectives

Develop a framework around Bell’s Theorem based on probability models with parameters

Show how these models relate to each other -- with a heavy emphasis on diagrams and pictures

Shed some light on the relationship between the concepts of “realism” and “locality” as defined by Bell

5. Summary of results: Partially-ordered lattice of five models

Figure 2.  Relationship among the 5 models in this presentation: Quantum mechanics (QM), Bell Real, Bell local, Maximal, and Minimal.
The order relation is set inclusion and the comparison is between achievable sets of correlations.

s0="Minimal \nSatisfies CHSH inequalities \nMax s =0 \nMin  s=0 ";
s1="Bell local \nSatisfies CHSH inequalities \nMax s=2 \nMin s=-2 ";
s2="Bell real \nSatisfies CHSH inequalities \nMax s=2 \nMin s=-2 ";
s3="QM \nMax s=2(Sqrt[2]) \nMin s=-2(Sqrt[2])";
s4="Maximal \nMax s=4 \nMin s=-4";
s5="Satisfies CHSH inequalities";

{s0,s1,s2,s3,s4}=Map[Style[#,Bold,Red]&,{s0,s1,s2,s3,s4}];

lgp={s0->s1,s0->s3,s1->s2,s2->s4,s3->s4};
LayeredGraphPlot[lgp,Bottom,VertexLabeling->True,AspectRatio-> 1/2,PlotRangePadding->0.05,ImageSize->800]

In this presentation, it is shown that, with appropriate definitions of “Bell real” and “Bell local” models,

A probability model is Bell real if and only if all 8 CHSH* inequalities hold.

*CHSH = Clauser, Horne, Shimony, and Holt developed a generalization of Bell’s inequality [Clauser & Horne & Shimony & Holt 1969].

If a probability model is Bell local, then all 8 CHSH inequalities hold. On the other hand, if all 8 CHSH inequalities hold for a given probability model, it MIGHT BE Bell local, but not necessarily.

The set of Bell local models forms a proper subset of the set of Bell real models. That is, there are Bell real models that are NOT Bell local.

The set of QM models and the set of Bell real models have a non-empty intersection. But neither is a subset of the other.

The set of QM models and the set of Bell local models have a non-empty intersection. But neither is a subset of the other.

Remarks

Number (1) is not new, as it is simply a combination of Bell's Theorem (when properly dissected) and Fines' Theorem [Fine 1982].

Arthur Fine: 
University of Washington
Professor Emeritus of Philosophy, Adjunct Professor of History, Adjunct Professor of Physics

Or more generally the work of Vorobev on consistent families of measures [Vorobev 1962].

Nikolai Nikolayevich Vorobyov (also Vorobiev) (Russian):

Николай Николаевич Воробьёв, 18 September 1925, Leningrad — July 14, 1995) was a Soviet and Russian mathematician, an expert in the field of abstract algebra, mathematical logic and probability theory, the founder of the Soviet school of game theory. He is an author of two textbooks, three monographs, a large number of mathematical articles and a number of popular science books. He supervised over 30 kandidat and doktor dissertations.

Numbers (2) - (6) are subtleties which don’t seem to be generally recognized in the literature, or at least not emphasized.

6. Let’s get to the pictures: Selected sets of correlations for each of the 5 models

Table 1. Pictures of 3D -slices (fixed w) through the 4D-sets of correlations (w,x,y,z) for the five models, where the first correlation w in each case is set to :  w=-1,-1/2,0,1/2,1 from left to right

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Notes

Each of the 5 rows of model pictures in Table 1  shows five 3D “slices” (hyperplanes) of the corresponding full set of achievable correlations for that model. In each row, the slices correspond to w=-1, -(1/2),0,(1/2), 1, from left to right. 
 In other words, each “slice” is a plot of the points (Subscript[ℐ, w])={(x,y,z) ∈(R^3) | -1≤(({{x}, {y}, {z}}))≤1 and  (({{w}, {x}, {y}, {z}})) is an achievable set of correlations for the model} for w=-1, -(1/2),0,(1/2),1.

First row: 3D QM contours.  {(cos(2 ((Subscript[θ, 1])-(Subscript[θ, 3])-(Subscript[θ, 4]))),cos(2 (Subscript[θ, 3])),cos(2 (Subscript[θ, 4])))|(Subscript[θ, 1])=(1/2)arccos(w),-90°≤(Subscript[θ, 3]),(Subscript[θ, 4])≤90°}. These are 3D surfaces representing slices through the 4D parametric surface {(cos(2 (Subscript[θ, 1])),cos(2(Subscript[θ, 2])),cos(2 (Subscript[θ, 3])),cos(2 (Subscript[θ, 4])))|(Subscript[θ, 1])=(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4]),-90°≤(Subscript[θ, 2]),(Subscript[θ, 3]),(Subscript[θ, 4])≤90°} where cos(2(Subscript[θ, 1])) is fixed at 5 different values, namely, (Subscript[θ, 1])=(1/2)arccos(w) for w=-1,-0.5,0,0.5,1.

Second row: 3D Bell real regions. (Subscript[ℐ, w])={(x,y,z) ∈(R^3) |-2≤ (({{-w+x+y+z}, {w-x+y+z}, {w+x-y+z}, {w+x+y-z}}))≤2and -1≤(({{x}, {y}, {z}}))≤1} for w∈{-1,-0.5,0,0.5,1}. The 5 polyhedra represent 3D slices (hyperplanes) through the 4D “Bell real region”, that is the set ℐ={(w,x,y,z) ∈(R^4) |-2≤ (({{-w+x+y+z}, {w-x+y+z}, {w+x-y+z}, {w+x+y-z}}))≤2and -1≤(({{w}, {x}, {y}, {z}}))≤1} of 4-tuples that satisfy all 8 CHSH inequalities. Each slice is a convex polyhedron. The vertices are indicated by red dots. Each polyhedron corresponds to a fixed value for the first correlation w. In this visualization, w=-1,-0.5,0,0.5,1 from left to right. The special cases on the end are tetrahedrons corresponding to w=-1 (on the left) and w=+1 (on the right). The tetrahedrons have 4 vertices and 4 triangular faces. The three polyhedra in the middle have 12 vertices and 6 rectangular faces and 8 triangular faces. The one in the middle (w=0) is “semi-regular” as the rectangles are congruent squares  and the triangles are congruent equilateral triangles. This is called a cuboctahedron. 

Third row: 3D Bell local contours. (Subscript[ℐ, w])={(x,y,z) ∈(R^3) |(({{w}, {x}, {y}, {z}}))=(({{((Subscript[α, 1])-(Subscript[α, 3])) ((Subscript[β, 1])-(Subscript[β, 3]))}, {((Subscript[α, 1])-(Subscript[α, 3])) ((Subscript[β, 2])-(Subscript[β, 4]))}, {((Subscript[α, 2])-(Subscript[α, 4])) ((Subscript[β, 1])-(Subscript[β, 3]))}, {((Subscript[α, 2])-(Subscript[α, 4])) ((Subscript[β, 2])-(Subscript[β, 4]))}}))  where (Subscript[α, k])≥0,(Subscript[β, k])≥0  &(Subscript[α, 1])+(Subscript[α, 3])=(Subscript[β, 1])+(Subscript[β, 3])=1  & (Subscript[α, 2])+(Subscript[α, 4])=(Subscript[β, 2])+(Subscript[β, 4])=1 and -1≤(({{x}, {y}, {z}}))≤1} for w∈{-1,-0.5,0,0.5,1}. These shapes are hyperbolic paraboloids (“Pringles”), although the middle case (w=0) is a degenerate case of four triangles and a vertical line extending from (0,0,-1) to (0,0,1), all meeting at the origin.

Fourth row: The maximal model, being completely unconstrained, is of course the same for any value of w, namely the entire cube [-1,1] ×[-1,1] ×[-1,1].

Fifth row: The minimal model, being completely constrained to (0,0,0,0), can only show up for w=0, hence appears only in the middle plot -- as the point (0,0,0) of course.

7. Selected combinations of the sets of correlations of the five models

Table 2. Some combinations of 3D slices (fixed w) of 4D sets of correlations (w,x,y,z) for the five different probabilitity models.  w=-1,-1/2,0,1/2,1 from left to right

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Notes

Each of the first 3 rows of pairs of models in Table 2 shows five 3D “slices” (hyperplanes) of the corresponding full set of achievable correlations of two different models. The last row, minimal model and Bell local model, shows only one combination since the minimal model only exists for w=0.

First row: 3D QM contours & Bell local contours.  In all five cases, there is a non-empty intersection between the Bell local contour and  the corresponding QM contour. In the end cases (w=-1 and w=1) the Bell local “Pringle” lies completely inside the contour of the QM “pillow”, but intersecting only at the four corners of each shape. These corners are also the vertices of the Bell real tetrahedra -- see the third row below, end cases.

Second row: 3D QM contours & Bell real regions. In the end cases (w=-1 and w=1), the Bell real region is contained entirely within the QM contour (a “pillow”-shaped surface), but they intersect only at the 4 vertices of the tetrahedra. The middle three are more complicated, and  the Bell region can be both inside and outside the QM contour. In other words, these models are incomparable (in the sense of set inclusion of correlations) and is the reason these two models are on different “paths” in the PO lattice of Fig. 1.

Third row: 3D Bell real regions and Bell local contours. Note that the Bell local contours are contained entirely within the Bell real regions in all 5 cases. This is true in general and is the reason that “Bell local” sits below “Bell real” in the PO lattice of Fig. 1.

Fourth row: Minimal model (singleton point (0,0,0,0) and the Bell local models with w=0. Note the minimal correlations (red dot (0,0,0)) is on the Bell local surface, at the intersection of the four triangles and the vertical. This is the reason that the “Minimal” model lies below the  “Bell local” model in the PO lattice of Fig. 6.

Blowup of the QM and Bell local comparison for w=0

Red lines show the intersection of the QM and Bell local contours. That is, there exist (an infinite number of) parameter settings (Alice and Bob’s measurement settings only!) where the two models are exactly the same as far as the correlations (that go into the Bell-CHSH inequality) are concerned.

Figure 3. Red lines indicate (some)sets of correlations common to both Bell local and QM models (case w=0 only)

8. Whirlwind tour of the history: Einstein thought experiment, Bell’s Theorem, and Aspect’s experiment

9. What bothered Einstein

Einstein: “It seems hard to sneak a look at God’s cards. But that he plays dice and uses ‘telepathic’ methods (as the present quantum theory requires of him) is something that I cannot believe for a moment.” [Dukas & Hoffman 1979]

Tim Maudlin: “Note the second part of Einstein’s concern: not merely that God plays dice but that he ‘uses ‘telepathic’ methods’. This is, of course, the ‘spukhafte Fernwirkung’ (‘spooky action at-a-distance’) that Einstein is also known to have railed against. A careful reading of Einstein makes clear that it is the spooky action-at-a-distance, i.e. the non-locality, implicit in the standard account of quantum theory that bothered him, not the indeterminism per se. Einstein did not look for a deterministic underpinning of quantum mechanical predictions because he was wedded to determinism, he did so because he was wedded to locality, and he was the first to recognize that in quantum theory indeterminism can further imply non-locality.”

What Bell Did  [Maudlin 2014 (1)], [Maudlin 2014 (2)] https://www.youtube.com/watch?v=Vg5z_zeZP60

10. Bell’s work pointed the way to a real lab experiment

One version: Photon polarization experiment [Aspect 1982]

If the experiment shows that Bell’s inequality can be violated, then quantum mechanics (QM) is a better description of nature than a deterministic, local description

Figure 5. The photon polarization version of an Aspect-like experiment

Figure 6. Data generation and data collection in an Aspect-like exepriment

11. Experimental outcomes and statistics of interest

The experiment is performed a large number  of times N. Table 3 below shows an imagined set of outcomes and some computed statistics.

Table 3. Imaginary record of the outcomes from an idealized Aspect-like experiment and the statistics of interest

sPad

Bell’s inequality says, that under the assumptions of realism (hidden variables) and locality, it MUST be the case that -2≤(OverHat[s])≤2.

Quantum mechanics predictions says only that -2(Sqrt[2])≤(OverHat[s])≤2(Sqrt[2]) and experimental outcomes seem to confirm this (e.g., [Aspect 2002] performed in 1982, plus many later experiments).

Bell’s conclusion is that one must pick any 2 out of 3 (at least one must be abandoned):  Realism (hidden variables), locality, or quantum mechanics.

This is the essence of Bell's Theorem.

12. Bell’s Theorem in a nutshell: Two models and some test functions

Figure 7. Bell's  theorem  in a nutshell: picture version

{comparoPlot3,diagram67P5}

If (Subscript[θ, 2])=(Subscript[θ, 3])=(Subscript[θ, 4])=22.5° and (Subscript[θ, 1])=67.5°, then
 (Subscript[S, 1])=-(Subscript[Q, 11])+(Subscript[Q, 12])+(Subscript[Q, 21])+(Subscript[Q, 22])=-cos(2*67.5°)+cos(2*22.5°)+cos(2*22.5°)+cos(2*22.5°)=2(Sqrt[2])>2, 
 contradicting the Bell inequality (or more accurately, one of the CHSH inequalities)

Table 4. Bell's  theorem  in a nutshell: table version

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13. The five probability models

Definition: A pmf (probability mass function) is a finite ordered set of real numbers ((Subscript[q, 1]),(Subscript[q, 2]),...,(Subscript[q, n])) such that (Subscript[q, k])≥0 all k and (∑_(k=1)^n)(Subscript[q, k])=1.

Definition: A probability model is a set of outcomes together with an associated pmf.

Table 5. The five models:Maximal, QM, Bell real, Bell local,and  Minimal

modelsTable

Notation and definitions

A = a random variable representing Alice’s outcome, (Subscript[D, A]) = a random variable representing Alice’s measurement setting

B = a random variable representing Bob's outcome, (Subscript[D, B]) = a random variable representing Bob’s measurement setting

(Subscript[D, A])=-1 means “Alice chooses measurement angle a” 
(Subscript[D, A])=+1 means “Alice chooses measurement angle a'” 
(Subscript[D, B])=-1 means “Bob chooses measurement angle b” 
(Subscript[D, B])=+1 means “Bob chooses measurement angle b'”

As for the probabilities --
 (Subscript[δ, a])≐P((Subscript[D, A])=-1)
 (Subscript[δ, a'])≐P((Subscript[D, A])=+1)
 (Subscript[δ, b])≐P((Subscript[D, B])=-1)
 (Subscript[δ, b'])≐P((Subscript[D, B])=+1)

For the QM model, the reason for the constraint that (Subscript[θ, 1])=(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4]) can be seen in the following diagram. Although the linear relation among the (Subscript[θ, k])’s may turn out differently depending on labeling choices, the arguments to follow can all be adjusted to accommodate them without changing the conclusions.

It is standard in the literature to let a and a' stand for Alice’s two measurement angle choices and b and b' stand for Bob’s two measurement angle choices. In this diagram and throughout this presentation, the difference angles will be denoted as follows:
(Subscript[θ, 1])=a-b'
(Subscript[θ, 2])=a-b
(Subscript[θ, 3])=b-a'  and
(Subscript[θ, 4])=a'-b'

Figure 8. Alice and Bob's measurement angles (a,a’,b,b’)and their differences ((Subscript[θ, 1]),(Subscript[θ, 2]),(Subscript[θ, 3]),(Subscript[θ, 4]))

diagramGeneric

14. Correlations for the five models

Table 6. Correlations in terms of the model parameters

Grid[Append[Prepend[Transpose[{{"w","x","y","z"},correlationsMaximal,correlationsQM,correlationsBellReal,correlationsBellLocal,correlationsMinimal}],Text@Style[#,Bold,Blue]&/@{"Correlation","Maximal","QM","Bell real","Bell local","Minimal"}],{"Constraints","(Subscript[γ, k])≥0 & ∑(Subscript[γ, k])=1 \nDenominators≠0","(Subscript[θ, 1])=(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4])","(Subscript[ρ, k])≥0 & ∑(Subscript[ρ, k])=1","(Subscript[α, k])≥0,(Subscript[β, k])≥0 \n(Subscript[α, 1])+(Subscript[α, 3])=(Subscript[β, 1])+(Subscript[β, 3])=1 \n(Subscript[α, 2])+(Subscript[α, 4])=(Subscript[β, 2])+(Subscript[β, 4])=1","No constraints"}],Frame->All]

15. Maxima and minima for the s-functions

Table 7. s-functions for the five models together with their maxes and mins

Grid[Append[Append[Prepend[Transpose[{{"(Subscript[s, 1])","(Subscript[s, 2])","(Subscript[s, 3])","(Subscript[s, 4])"},sBell.correlationsMaximal,sBell.correlationsQM,sBell.correlationsBellReal,sBell.correlationsBellLocal,sBell.correlationsMinimal}],Text@Style[#,Bold,Blue]&/@{"s-function","Maximal","QM","Bell real","Bell local","Minimal"}],{"Constraints","(Subscript[γ, k])≥0 & ∑(Subscript[γ, k])=1 \nDenominators≠0","(Subscript[θ, 1])=(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4])","(Subscript[ρ, k])≥0 & ∑(Subscript[ρ, k])=1","(Subscript[α, k])≥0,(Subscript[β, k])≥0 \n(Subscript[α, 1])+(Subscript[α, 3])=(Subscript[β, 1])+(Subscript[β, 3])=1 \n(Subscript[α, 2])+(Subscript[α, 4])=(Subscript[β, 2])+(Subscript[β, 4])=1","No constraints"}],{"s-function Min and Max","-4 and +4","-2(Sqrt[2]) and +2(Sqrt[2])","-2 and +2","-2 and +2","0 and 0"}],Frame->All]

Proof: Maximal model s-function max and min  = ±4

See “Maximal” column from Table 7 above. Claim that each entry of this column is between -4 and 4. First of all, the possibility of denominators equal to zero must be addressed. For easy reference, recall the pmf for the Maximal model:

pmfMaximalTable

Consider for example the first denominator (Subscript[γ, 1])+(Subscript[γ, 5])+(Subscript[γ, 9])+(Subscript[γ, 13]) in Table 7. In the Maximal model table just above, note that (Subscript[γ, 1]),(Subscript[γ, 5]),(Subscript[γ, 9]),(Subscript[γ, 13]) correspond to outcomes where ((Subscript[D, A]),(Subscript[D, B]))=(-1,-1) always. And (A,B) runs through all possibilities (-1,-1),(-1,1),(1,-1),(1,1) in the outcomes corresponding to (Subscript[γ, 1]),(Subscript[γ, 5]),(Subscript[γ, 9]),(Subscript[γ, 13]). Hence
 (Subscript[γ, 1])+(Subscript[γ, 5])+(Subscript[γ, 9])+(Subscript[γ, 13])=P((A,B)=(-1,-1),((Subscript[D, A]),(Subscript[D, B]))=(-1,-1)) +P((A,B)=(-1,1),((Subscript[D, A]),(Subscript[D, B]))=(-1,-1)) +P((A,B)=(1,-1),((Subscript[D, A]),(Subscript[D, B]))=(-1,-1)) +P((A,B)=(1,1),((Subscript[D, A]),(Subscript[D, B]))=(-1,-1)) =P(((Subscript[D, A]),(Subscript[D, B]))=(-1,-1)), 
 that is
(Subscript[γ, 1])+(Subscript[γ, 5])+(Subscript[γ, 9])+(Subscript[γ, 13])=P(((Subscript[D, A]),(Subscript[D, B]))=(-1,-1)).
By a similar argument, the other denominators are
  (Subscript[γ, 2])+(Subscript[γ, 6])+(Subscript[γ, 10])+(Subscript[γ, 14])=P(((Subscript[D, A]),(Subscript[D, B]))=(-1,1)) ,  (Subscript[γ, 3])+(Subscript[γ, 7])+(Subscript[γ, 11])+(Subscript[γ, 15])=P(((Subscript[D, A]),(Subscript[D, B]))=(1,-1)) ,and  (Subscript[γ, 4])+(Subscript[γ, 8])+(Subscript[γ, 12])+(Subscript[γ, 16])=P(((Subscript[D, A]),(Subscript[D, B]))=(1,1)), respectively. 
 So no denominator is zero if it is assumed that Alice and Bob always have a positive probability of choosing any pair of detector angle settings. In other words, rule out the situation where Alice and Bob together never use one of the detector angle setting pairs.

Since the (Subscript[γ, k])’s in each fraction are all non-negative (and at least one is positive), and the numerator is the same as the denominator except for signs, it is clear that each fraction must be between -1 and 1.  Next notice that the fractions in each row neatly partition the (Subscript[γ, k])’s into mutually exclusive subsets whose union is the entire set {(Subscript[γ, 1]),(Subscript[γ, 2]),(Subscript[γ, 3]),..., (Subscript[γ, 15]),(Subscript[γ, 16])}, so each fraction can independently be assigned values between -1 and +1 (subject to the (Subscript[γ, k])’s summing to 1 of course). Finally then, it is clear that
 -4≤(({{-(((Subscript[γ, 1])-(Subscript[γ, 5])-(Subscript[γ, 9])+(Subscript[γ, 13]))/((Subscript[γ, 1])+(Subscript[γ, 5])+(Subscript[γ, 9])+(Subscript[γ, 13])))+(((Subscript[γ, 2])-(Subscript[γ, 6])-(Subscript[γ, 10])+(Subscript[γ, 14]))/((Subscript[γ, 2])+(Subscript[γ, 6])+(Subscript[γ, 10])+(Subscript[γ, 14])))+(((Subscript[γ, 3])-(Subscript[γ, 7])-(Subscript[γ, 11])+(Subscript[γ, 15]))/((Subscript[γ, 3])+(Subscript[γ, 7])+(Subscript[γ, 11])+(Subscript[γ, 15])))+(((Subscript[γ, 4])-(Subscript[γ, 8])-(Subscript[γ, 12])+(Subscript[γ, 16]))/((Subscript[γ, 4])+(Subscript[γ, 8])+(Subscript[γ, 12])+(Subscript[γ, 16])))}, {(((Subscript[γ, 1])-(Subscript[γ, 5])-(Subscript[γ, 9])+(Subscript[γ, 13]))/((Subscript[γ, 1])+(Subscript[γ, 5])+(Subscript[γ, 9])+(Subscript[γ, 13])))-(((Subscript[γ, 2])-(Subscript[γ, 6])-(Subscript[γ, 10])+(Subscript[γ, 14]))/((Subscript[γ, 2])+(Subscript[γ, 6])+(Subscript[γ, 10])+(Subscript[γ, 14])))+(((Subscript[γ, 3])-(Subscript[γ, 7])-(Subscript[γ, 11])+(Subscript[γ, 15]))/((Subscript[γ, 3])+(Subscript[γ, 7])+(Subscript[γ, 11])+(Subscript[γ, 15])))+(((Subscript[γ, 4])-(Subscript[γ, 8])-(Subscript[γ, 12])+(Subscript[γ, 16]))/((Subscript[γ, 4])+(Subscript[γ, 8])+(Subscript[γ, 12])+(Subscript[γ, 16])))}, {(((Subscript[γ, 1])-(Subscript[γ, 5])-(Subscript[γ, 9])+(Subscript[γ, 13]))/((Subscript[γ, 1])+(Subscript[γ, 5])+(Subscript[γ, 9])+(Subscript[γ, 13])))+(((Subscript[γ, 2])-(Subscript[γ, 6])-(Subscript[γ, 10])+(Subscript[γ, 14]))/((Subscript[γ, 2])+(Subscript[γ, 6])+(Subscript[γ, 10])+(Subscript[γ, 14])))-(((Subscript[γ, 3])-(Subscript[γ, 7])-(Subscript[γ, 11])+(Subscript[γ, 15]))/((Subscript[γ, 3])+(Subscript[γ, 7])+(Subscript[γ, 11])+(Subscript[γ, 15])))+(((Subscript[γ, 4])-(Subscript[γ, 8])-(Subscript[γ, 12])+(Subscript[γ, 16]))/((Subscript[γ, 4])+(Subscript[γ, 8])+(Subscript[γ, 12])+(Subscript[γ, 16])))}, {(((Subscript[γ, 1])-(Subscript[γ, 5])-(Subscript[γ, 9])+(Subscript[γ, 13]))/((Subscript[γ, 1])+(Subscript[γ, 5])+(Subscript[γ, 9])+(Subscript[γ, 13])))+(((Subscript[γ, 2])-(Subscript[γ, 6])-(Subscript[γ, 10])+(Subscript[γ, 14]))/((Subscript[γ, 2])+(Subscript[γ, 6])+(Subscript[γ, 10])+(Subscript[γ, 14])))+(((Subscript[γ, 3])-(Subscript[γ, 7])-(Subscript[γ, 11])+(Subscript[γ, 15]))/((Subscript[γ, 3])+(Subscript[γ, 7])+(Subscript[γ, 11])+(Subscript[γ, 15])))-(((Subscript[γ, 4])-(Subscript[γ, 8])-(Subscript[γ, 12])+(Subscript[γ, 16]))/((Subscript[γ, 4])+(Subscript[γ, 8])+(Subscript[γ, 12])+(Subscript[γ, 16])))}}))≤4 , 
 where the inequalities are understood to apply component-wise to the rows. 
 As an example, the first row can be made equal to +4 by the assignment (Subscript[γ, 5])=(Subscript[γ, 2])=(Subscript[γ, 3])=(Subscript[γ, 4])=(1/4)and all other (Subscript[γ, k])=0, k≠2,3,4,5. It can be made equal to -4 by the assignment (Subscript[γ, 1])=(Subscript[γ, 6])=(Subscript[γ, 7])=(Subscript[γ, 8])=(1/4) and all other (Subscript[γ, k])=0, k≠1,6,7,8.

Proof: QM model s-function max and min =  ±2(Sqrt[2])

See “QM” column from Table 5 above. Claim that each entry of this column is between -2(Sqrt[2]) and 2(Sqrt[2]).

Since the QM s-functions max and min are well-known, and in the interest of avoiding a lengthy distraction, the straightforward but tedious derivation is omitted, to be included in an upcoming paper.

Proof: Bell real model s-function max and min = ±2

See “Bell real” column from Table 5 above. Claim that each entry of this column is between -2 and 2. Consider for example the first row 2 (Subscript[ρ, 1])-2 (Subscript[ρ, 2])+2 (Subscript[ρ, 3])-2 (Subscript[ρ, 4])-2 (Subscript[ρ, 5])-2 (Subscript[ρ, 6])+2 (Subscript[ρ, 7])+2 (Subscript[ρ, 8])+2 (Subscript[ρ, 9])+2 (Subscript[ρ, 10])-2 (Subscript[ρ, 11])-2 (Subscript[ρ, 12])-2 (Subscript[ρ, 13])+2 (Subscript[ρ, 14])-2 (Subscript[ρ, 15])+2 (Subscript[ρ, 16]). Since the (Subscript[ρ, k])’s form a pmf, it is obvious that this expression must be between -2 and +2, inclusive. Similarly for the other three rows. 
Note: This was an easy derivation of the CHSH inequalities (a generalization of Bell’s inequality) without even trying hard!

Proof: Bell local model s-function max and min =  ±2

See “Bell local” column from Table 5 above. Claim that each entry of this column is between -2 and 2.
Note that the conditional probabilities (Subscript[α, k]) and (Subscript[β, k]) (k=1,2,3,4) are non-negative and they satisfy  (Subscript[α, 1])+(Subscript[α, 3])=1,(Subscript[α, 2])+(Subscript[α, 4])=1,(Subscript[β, 1])+(Subscript[β, 3])=1,(Subscript[β, 2])+(Subscript[β, 4])=1, hence the terms (Subscript[α, 1])-(Subscript[α, 3]),(Subscript[α, 2])-(Subscript[α, 4]),(Subscript[β, 1])-(Subscript[β, 3]),(Subscript[β, 2])-(Subscript[β, 4]) must all be between -1 and 1 inclusive. Hence terms such as ±((Subscript[α, 1])-(Subscript[α, 3])) ((Subscript[β, 1])-(Subscript[β, 3])) must be between -1 and +1 inclusive, also.
Now consider the first s-function term: 
 -((Subscript[α, 1])-(Subscript[α, 3])) ((Subscript[β, 1])-(Subscript[β, 3]))+((Subscript[α, 2])-(Subscript[α, 4])) ((Subscript[β, 1])-(Subscript[β, 3]))+((Subscript[α, 1])-(Subscript[α, 3])) ((Subscript[β, 2])-(Subscript[β, 4]))+((Subscript[α, 2])-(Subscript[α, 4])) ((Subscript[β, 2])-(Subscript[β, 4]))= ((Subscript[β, 1])-(Subscript[β, 3]))(-(Subscript[α, 1])+(Subscript[α, 2])+(Subscript[α, 3])-(Subscript[α, 4]))+((Subscript[β, 2])-(Subscript[β, 4]))((Subscript[α, 1])+(Subscript[α, 2])-(Subscript[α, 3])-(Subscript[α, 4])).
Show that this last expression on the right
 ((Subscript[β, 1])-(Subscript[β, 3]))(-(Subscript[α, 1])+(Subscript[α, 2])+(Subscript[α, 3])-(Subscript[α, 4]))+((Subscript[β, 2])-(Subscript[β, 4]))((Subscript[α, 1])+(Subscript[α, 2])-(Subscript[α, 3])-(Subscript[α, 4])) 
 must be between -2 and 2. 
 Work on the maximum first. Clearly the α’s and β’s can be manipulated independently, so set (Subscript[β, 1])=(Subscript[β, 2])=1 and (Subscript[β, 3])=(Subscript[β, 4])=0. As for the α’s, one optimal choice is to take (Subscript[α, 2])=1 and (Subscript[α, 4])=0. Since the ±((Subscript[α, 1])-(Subscript[α, 3])) term appears in each with opposite signs,  it is arbitrary which one to make 1 and which to make 0. So set (Subscript[α, 1])=1 and (Subscript[α, 3])=0. This then yields (1-0) x (-1+1+0-0) + (1-0) x (1+1-0-0) = 2. A similar argument works to obtain the minimum -2. Similarly for the other rows.

Use Mathematica’s Maximize and Minimize to confirm the reasoning above. The following calls to Maximize and Minimize find the max and min (and one set of parameters each that yields the max or min) of the first s-function (Subscript[s, 1])(w,x,y,z)=-w+x+y+z, all subject to the Bell local model parameter constraints.

solMax=Maximize[{sBellLocal[[1]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]]
solMin=Minimize[{sBellLocal[[1]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]]

Mathematica gets a maximum of 2 and a minimum of -2, which confirms the reasoning above. 
Now check that the solution arguments calculated by Mathematica produce the desired results. See calculations below. Two out of 4 s-functions reach their max (or min) in each case, and the others are between -2 and 2 (1 and -1 in fact).
 Note: A different pattern of maxes and mins (and other values between -2 and +2) would have been obtained by maximizing or minimizing one of the other three s-functions, but still all between -2 and +2 of course.

(sBell.correlationsBellLocal)/.solMax[[2]]
(sBell.correlationsBellLocal)/.solMin[[2]]

Note: Again, similar to the Bell real case, this was a relatively painless derivation of the CHSH inequalities (a generalization of Bell’s inequality)!

Proof: Minimal model s-function max and min = 0 and 0

See “Minimal” column from Table 5 above. Since all correlations are 0, it is obvious that all s-functions are also 0, hence max and min are 0.

16. Summary and future work

Summary

Showed how to re-frame the concepts of “Bell realism” and “Bell locality” in terms of probability models with variable parameters.

Illustrated in geometric pictures the relationships among the correlations (and functions of correlations) of the QM, real, and local models.

In particular, showed explicitly that

"Bell locality" is the more restrictive condition, in that it implies "Bell reality" without being the same. Note that this can be viewed as the contra-positive of Einstein's observation that indeterminism implies non-locality.

The sets of correlations of both the Bell local and Bell real models have a non-empty intersection with the set correlations of QM models, but neither is a subset of the other.

Future work

Refine the “Bell local” category into the subsets of “Remote context independence” and “Remote outcome independence” and insert into the appropriate position of the partially-ordered lattice of models (Fig. 2)

Classify ALL of the intersections of Bell local, Bell real, and QM models. See Fig. 3 as a start.

Closer examination of EPR, Bell, CHSH, and current related papers to pin down exactly how their language corresponds to the language in this presentation.

May help to resolve some of the current controversy over what Bell actually “proved”.

Questions

Is “Bell locality” the same as “Einstein locality”?

What about the quantum theory “No communication theorem”, which says that information cannot be transmitted faster than the speed of light?

Note that Alice and Bob eventually have to compare their lengthy sequences of results, which requires a quite ordinary speed-limited process.

And the BIG question: Is there an underlying deeper “pre-quantum” theory which simultaneously

Recovers all of the current QM predictions, and

Is either “local” OR explains how non-locality “works”?

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[Hess  & De Raedt & Michielsen 2016 (1)] The digital computer as a metaphor for the perfect laboratory experiment:Loophole-freeBell experiments, Hans DeRaedta, Kristel Michielsen, Karl Hess, Computer Physics Communications, August 2016.

[Hess & De Raedt & Michielsen 2016 (2)] From Boole to Leggett-Garg: Epistemology of Bell-type Inequalities, Karl Hess, Hans De Raedt, Kristel Michielsen, arXiv.org > quant-ph > arXiv:1605.04887v1

[Hess & De Raedt  & Michielsen 2010] Extended Boole-Bell inequalities applicable to quantum theory, de Raedt, Hess, Michielsen, arXiv.org>quant-ph>arXiv:0901.2546, May 2010, https://arxiv.org/abs/0901.2546

[Khrennikov 2014] Beyond Quantum, Andrei Khrennikov, CRC Press, 2014. https://www.amazon.com/Beyond-Quantum-Andrei-Khrennikov/dp/9814411736

[Maudlin 2014 (1)] What Bell Did, Tim Maudlin, J. Phys. A: Math. Theor. 47 (2014) 424010 (24pp),  doi:10.1088/1751-8113/47/42/424010

[Maudlin 2014 (2)] What Bell Did, Tim Maudlin, YouTube, https://www.youtube.com/watch?v=Vg5z_zeZP60

[Myrvold 2003?] A loophole in Bell’s Theorem? Parameter dependence in the Hess-Philipp model, Wayne Myrvold, University of Western Ontario, date???

[Myrvold 2002] Title??, Myvold, W. C., Proceedings of 18th Biennial Meeting of the Philosophy of Science Association, to be held Nov. 7–10, 2002, Milwaukee, WI.

[Muller & Placek 2001] Against a minimalist reading of Bell’s Theorem: Lessons from Fine, Konstanzer Online-Publikations-System (KOPS)
URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-271372

[Pitowsky 1989]  Quantum Probability – Quantum Logic, Pitowsky, I., Lecture Notes in Physics 321, Springer, Berlin, 1989 .

[Schell 1955] Proc. 2-nd Symp. Lin. Programming, Washington, Vol. 2, 615-642, E. D. Schell,  (1955).

[Rosinger 2004] George Boole and the Bell Inequalities, Rosinger, arXiv.org>quant-ph/0406004v1, Jun 2004, https://arxiv.org/abs/quant-ph/0406004

[Schumacher & Westmoreland 2010]  Quantum Processes and Information, Schumacher and Westmoreland, Cambridge University Press, 2010.

[Shimony 2009] Bell’s Theorem, Abner Shimony, The Stanford Encyclopedia of Philosophy, First published Wed Jul 21, 2004; substantive revision Thu Jun 11, 2009  https://plato.stanford.edu/entries/qt-epr/#QuanMechDescPhysRealConsComp

[Simon 2011] Extreme Points and the Krein-Milman Theorem, Barry Simon textbook, CIT, Chapter 8, Cambridge University Press.

[Velozo & Nogales & Araujo] The Clauser-Horne-Shimony-Holt inequality in the context of a broad random variable, arXiv:1805.11200v2 [quant-ph] 30 May 2018

[Vorob’ev 1962]  Consistent Families of Measures and their Extensions, N. N. Vorob’ev (Translated by N. Greenleaf), Theory of Probability and Its Applications, Vol II, No 1, 1962

[Web Link Euler ] Euler's Formula in Higher Dimensions, https://people.math.osu.edu/fiedorowicz.1/math655/HyperEuler.html

[Web Link 5-cell] 5-cell, https://en.wikipedia.org/wiki/5-cell

Backup slides

18. Definitions and terminology

Definition: A pmf (probability mass function) is a finite ordered set of real numbers ((Subscript[q, 1]),(Subscript[q, 2]),...,(Subscript[q, n])) such that (Subscript[q, k])≥0 all k and (∑_(k=1)^n)(Subscript[q, k])=1.

Notation: (w,x,y,z) where -1≤w,x,y,z≤1 will denote a set of generic (raw uncentered) correlations. In this presentation, typically this will occur in the context of four random variables ((Subscript[X, 1]),(Subscript[X, 2]),(Subscript[X, 3]),(Subscript[X, 4])) each taking values in {-1,1}. Then we might have for example (w,x,y,z)=(E (Subscript[X, 1])(Subscript[X, 3]),E (Subscript[X, 1])(Subscript[X, 4]),E (Subscript[X, 2])(Subscript[X, 3]),E (Subscript[X, 2])(Subscript[X, 4])), where the four expectation operators may be conditioned on possibly different values for other random variables.

Definition: The s-test functions are the 4 components of the linear map s=((Subscript[s, 1]),(Subscript[s, 2]),(Subscript[s, 3]),(Subscript[s, 4])):(R^4)->(R^4) represented by the matrix S=(({{-1, 1, 1, 1}, {1, -1, 1, 1}, {1, 1, -1, 1}, {1, 1, 1, -1}})) relative to the standard Cartesian basis. For example, (Subscript[s, 2])(w,x,y,z) = w-x+y+z and similarly for the other three.

Definition: The CHSH inequalities [Clauser & Horne & Shimony & Holt 1969] are -2≤S(({{w}, {x}, {y}, {z}})) =(({{-w+x+y+z}, {w-x+y+z}, {w+x-y+z}, {w+x+y-z}}))≤2, where the inequalities are to be taken component-wise. Thus there are 8 (4 pairs of 2) CHSH inequalities.

Note: In this presentation, the more general CHSH inequalities will be used instead of Bell’s original simpler inequality {Bell 1964].

19. The Consistency Theorem

Consider a random vector ((Subscript[X, 1]),(Subscript[X, 2]),(Subscript[X, 3]),(Subscript[X, 4])) whose outcome space is  (({-1,1})^4) and associated pmf is  ρ=((Subscript[ρ, 1]),(Subscript[ρ, 2]),...,(Subscript[ρ, 16])), as summarized in Table 8.

Table 8. Outcome space and pmf for ((Subscript[X, 1]),(Subscript[X, 2]),(Subscript[X, 3]),(Subscript[X, 4]))

pmfConsistencyTable

Definition (CHSH inequalities): Denote the four correlations (Subscript[c, 1])=⟨(Subscript[X, 1])(Subscript[X, 3])⟩, (Subscript[c, 2])=⟨(Subscript[X, 1])(Subscript[X, 4])⟩, (Subscript[c, 3])=⟨(Subscript[X, 2])(Subscript[X, 3])⟩, (Subscript[c, 4])=⟨(Subscript[X, 2])(Subscript[X, 4])⟩ . Then the following 8 inequalities (4 pairs) are called the CHSH inequalities. ([Clauser Horne Shimony Holt 1969])

-2<=(({{-1, 1, 1, 1}, {1, -1, 1, 1}, {1, 1, -1, 1}, {1, 1, 1, -1}}))(({{(Subscript[c, 1])}, {(Subscript[c, 2])}, {(Subscript[c, 3])}, {(Subscript[c, 4])}}))≤2

Definition (consistency): Let ((Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3]),(Subscript[c, 4])) be an arbitrary 4-tuple of real numbers between -1 and 1. Then the  pmf ρ=((Subscript[ρ, 1]),(Subscript[ρ, 2]),...,(Subscript[ρ, 16])) is said to be consistent with  ((Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3]),(Subscript[c, 4])) if (⟨(Subscript[X, 1])(Subscript[X, 3])⟩,⟨(Subscript[X, 1])(Subscript[X, 4])⟩,⟨(Subscript[X, 2])(Subscript[X, 3])⟩,⟨(Subscript[X, 2])(Subscript[X, 4])⟩)=((Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3]),(Subscript[c, 4])).

The Consistency Theorem: The following two conditions are equivalent.
I. There exists a pmf ρ= ((Subscript[ρ, 1]),(Subscript[ρ, 2]),...,(Subscript[ρ, 16])) on the outcome space (({-1,1})^4) of ((Subscript[X, 1]),(Subscript[X, 2]),(Subscript[X, 3]),(Subscript[X, 4])) that is consistent with ((Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3]),(Subscript[c, 4])). 
II.  ((Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3]),(Subscript[c, 4])) satisfy all 8 of the CHSH inequalities (Eq 1).

Proof: The backward direction (I⟸II) is long and involved, so it is omitted. See remarks below for proof references.
On the other hand, the forward direction (I⟹II) is almost trivial. Assume condition I. A straightforward calculation of correlations yields

 (({{⟨(Subscript[X, 1]) (Subscript[X, 3])⟩}, {⟨(Subscript[X, 1]) (Subscript[X, 4])⟩}, {⟨(Subscript[X, 2]) (Subscript[X, 3])⟩}, {⟨(Subscript[X, 2]) (Subscript[X, 4])⟩}}))=(({{(Subscript[ρ, 1])+(Subscript[ρ, 2])-(Subscript[ρ, 3])-(Subscript[ρ, 4])+(Subscript[ρ, 5])+(Subscript[ρ, 6])-(Subscript[ρ, 7])-(Subscript[ρ, 8])-(Subscript[ρ, 9])-(Subscript[ρ, 10])+(Subscript[ρ, 11])+(Subscript[ρ, 12])-(Subscript[ρ, 13])-(Subscript[ρ, 14])+(Subscript[ρ, 15])+(Subscript[ρ, 16])}, {(Subscript[ρ, 1])-(Subscript[ρ, 2])+(Subscript[ρ, 3])-(Subscript[ρ, 4])+(Subscript[ρ, 5])-(Subscript[ρ, 6])+(Subscript[ρ, 7])-(Subscript[ρ, 8])-(Subscript[ρ, 9])+(Subscript[ρ, 10])-(Subscript[ρ, 11])+(Subscript[ρ, 12])-(Subscript[ρ, 13])+(Subscript[ρ, 14])-(Subscript[ρ, 15])+(Subscript[ρ, 16])}, {(Subscript[ρ, 1])+(Subscript[ρ, 2])-(Subscript[ρ, 3])-(Subscript[ρ, 4])-(Subscript[ρ, 5])-(Subscript[ρ, 6])+(Subscript[ρ, 7])+(Subscript[ρ, 8])+(Subscript[ρ, 9])+(Subscript[ρ, 10])-(Subscript[ρ, 11])-(Subscript[ρ, 12])-(Subscript[ρ, 13])-(Subscript[ρ, 14])+(Subscript[ρ, 15])+(Subscript[ρ, 16])}, {(Subscript[ρ, 1])-(Subscript[ρ, 2])+(Subscript[ρ, 3])-(Subscript[ρ, 4])-(Subscript[ρ, 5])+(Subscript[ρ, 6])-(Subscript[ρ, 7])+(Subscript[ρ, 8])+(Subscript[ρ, 9])-(Subscript[ρ, 10])+(Subscript[ρ, 11])-(Subscript[ρ, 12])-(Subscript[ρ, 13])+(Subscript[ρ, 14])-(Subscript[ρ, 15])+(Subscript[ρ, 16])}})). Since consistency with ((Subscript[c, 1]),(Subscript[c, 2]),(Subscript[c, 3]),(Subscript[c, 4])) is assumed, i.e., (({{⟨(Subscript[X, 1]) (Subscript[X, 3])⟩}, {⟨(Subscript[X, 1]) (Subscript[X, 4])⟩}, {⟨(Subscript[X, 2]) (Subscript[X, 3])⟩}, {⟨(Subscript[X, 2]) (Subscript[X, 4])⟩}}))=(({{(Subscript[c, 1])}, {(Subscript[c, 2])}, {(Subscript[c, 3])}, {(Subscript[c, 4])}})), then
 
  (({{-1, 1, 1, 1}, {1, -1, 1, 1}, {1, 1, -1, 1}, {1, 1, 1, -1}}))(({{(Subscript[c, 1])}, {(Subscript[c, 2])}, {(Subscript[c, 3])}, {(Subscript[c, 4])}}))=(({{-1, 1, 1, 1}, {1, -1, 1, 1}, {1, 1, -1, 1}, {1, 1, 1, -1}}))(({{⟨(Subscript[X, 1]) (Subscript[X, 3])⟩}, {⟨(Subscript[X, 1]) (Subscript[X, 4])⟩}, {⟨(Subscript[X, 2]) (Subscript[X, 3])⟩}, {⟨(Subscript[X, 2]) (Subscript[X, 4])⟩}}))=(({{2 (Subscript[ρ, 1])-2 (Subscript[ρ, 2])+2 (Subscript[ρ, 3])-2 (Subscript[ρ, 4])-2 (Subscript[ρ, 5])-2 (Subscript[ρ, 6])+2 (Subscript[ρ, 7])+2 (Subscript[ρ, 8])+2 (Subscript[ρ, 9])+2 (Subscript[ρ, 10])-2 (Subscript[ρ, 11])-2 (Subscript[ρ, 12])-2 (Subscript[ρ, 13])+2 (Subscript[ρ, 14])-2 (Subscript[ρ, 15])+2 (Subscript[ρ, 16])}, {2 (Subscript[ρ, 1])+2 (Subscript[ρ, 2])-2 (Subscript[ρ, 3])-2 (Subscript[ρ, 4])-2 (Subscript[ρ, 5])+2 (Subscript[ρ, 6])-2 (Subscript[ρ, 7])+2 (Subscript[ρ, 8])+2 (Subscript[ρ, 9])-2 (Subscript[ρ, 10])+2 (Subscript[ρ, 11])-2 (Subscript[ρ, 12])-2 (Subscript[ρ, 13])-2 (Subscript[ρ, 14])+2 (Subscript[ρ, 15])+2 (Subscript[ρ, 16])}, {2 (Subscript[ρ, 1])-2 (Subscript[ρ, 2])+2 (Subscript[ρ, 3])-2 (Subscript[ρ, 4])+2 (Subscript[ρ, 5])+2 (Subscript[ρ, 6])-2 (Subscript[ρ, 7])-2 (Subscript[ρ, 8])-2 (Subscript[ρ, 9])-2 (Subscript[ρ, 10])+2 (Subscript[ρ, 11])+2 (Subscript[ρ, 12])-2 (Subscript[ρ, 13])+2 (Subscript[ρ, 14])-2 (Subscript[ρ, 15])+2 (Subscript[ρ, 16])}, {2 (Subscript[ρ, 1])+2 (Subscript[ρ, 2])-2 (Subscript[ρ, 3])-2 (Subscript[ρ, 4])+2 (Subscript[ρ, 5])-2 (Subscript[ρ, 6])+2 (Subscript[ρ, 7])-2 (Subscript[ρ, 8])-2 (Subscript[ρ, 9])+2 (Subscript[ρ, 10])-2 (Subscript[ρ, 11])+2 (Subscript[ρ, 12])-2 (Subscript[ρ, 13])-2 (Subscript[ρ, 14])+2 (Subscript[ρ, 15])+2 (Subscript[ρ, 16])}})).
 
 Since ρ= ((Subscript[ρ, 1]),(Subscript[ρ, 2]),...,(Subscript[ρ, 16])) is a pmf, it is obvious that every row of the last matrix (i.e., entry of the column vector) is between -2 and +2, hence all 8 CHSH inequalities are satisfied. Thus I ⟹II.

Remarks

The suggestive name  “Consistency Theorem” was made up by the author for the purposes of this presentation only. It is NOT a new result. Much more general results have already been proved by Nikolai Nikolayevich Vorobev (sometimes spelled Vorobyov or Vorob’ev or Vorobedv or Vorobiev) in his extensive work [Vorobev 1962]. In that paper he introduced the idea of cyclicity, which is applicable to the derivation of Bell-type inequalities. As a trivial example, consider the expression a b +a c +b c. If a, b, and c take only the values ±1, then we must have  a b +a c +b c≥-1 (for example). Once the values of the first two product terms are given, the third is already determined. Despite the fact that Vorobev’s work preceded Bell’s famous 1964 paper, it was probably not known to him or other quantum theory foundations researchers for many years afterwards. As far as this author can tell,  it has  not been cited in the quantum foundations literature until recently. Such an example is [Hess Philipp 2005], although there may be earlier ones. [Hess Philipp 2005] also mention the papers [Bass 1955] and [Schell 1955] as having derived similar but more specific results to Vorobev’s.

Minimize[a b+a c +b c,(a==-1||a==1)&&(b==-1||b==1)&&(c==-1||c==1),{a,b,c}]
Maximize[a b+a c +b c,(a==-1||a==1)&&(b==-1||b==1)&&(c==-1||c==1),{a,b,c}]

(Nikolai Nikolayevich Vorobyov (also Vorobiev) (Russian: Николай Николаевич Воробьёв, 18 September 1925, Leningrad — July 14, 1995) was a Soviet and Russian mathematician, an expert in the field of abstract algebra, mathematical logic and probability theory, the founder of the Soviet school of game theory. He is an author of two textbooks, three monographs, a large number of mathematical articles and a number of popular science books. He supervised over 30 kandidat and doktor dissertations.)

Of course, Bell (or followers like [Clauser Horne Shimony Holt 1969]) did not explicitly invoke condition I above as an assumption, only “hidden variables” and “locality”. However, Bell (and his followers, such as [Clauser Horne Shimony Holt 1969]) implicitly assumed it, thus implying the Bell inequality (or more generally the CHSH inequalities)  straight away, without any further assumptions, according to the Consistency Theorem. If one considers the “realism” alternative to the “hidden variables” assumption, that is, the existence of 4 definite source-prepared values for the potential measurement outcomes (aka “counterfactual definiteness”),  one could argue that this implies condition I above. But there seems to be no need for the locality assumption at all. In fact, this presentation shows that locality is stronger than condition I above and it is a sufficient condition for the CHSH inequalities to hold, but not necessary.

In the context of quantum theory foundations, the backwards direction of the Consistency Theorem (I ⟸ II) is essentially Fines’ Theorem [Fine 1982 (1)]. Alternate proofs have also appeared more recently, for example in  [Halliwell 2014]. A different proof by the author will be given in an upcoming paper, together with visualizations that lend some intuition.

A simple variation of Bell’s inequality was already stated by George Boole [Boole 1854] about a century before Bell. Hence the name ‘Boole-Bell inequality’ is often given to Bell’s inequality. This fact further exposes the purely logical/probabilistic nature of the result, since of course quantum theory only began to develop more than a half century later.

20. Source of the Bell real model

Bell real model in the literature

Consider the Bell real model in Table 9 below. This corresponds to the standard textbook presentation of the CHSH derivation of Bell’s inequality [Schumacher & Westmoreland 2010]. Note that the r.v.’s are ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])), not (A,B,(Subscript[D, A]),(Subscript[D, B])). This is intended to be a probabilistic representation of the “realism” scenario in which the entangled pair of photons is “prepared” at the source in such a way that it is “encoded” with a quadruple ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2]))∈(({-1,1})^4), where ((Subscript[A, 1]),(Subscript[A, 2])) is sent to Alice and ((Subscript[B, 1]),(Subscript[B, 2])) is sent to Bob. If Alice chooses measurement setting (Subscript[D, A])=-1 (that is angle a), then her measurement outcome will be (Subscript[A, 1]).  If Alice chooses measurement setting (Subscript[D, A])=+1 (that is angle a'), then her measurement outcome will be (Subscript[A, 2]). Similarly for Bob. In the analyses to follow,  the numbers (Subscript[ρ, 1]),(Subscript[ρ, 2]),...,(Subscript[ρ, 16]) will sometimes be assumed to represent a pmf, but in other cases the goal will be to show that such a pmf exists (or not).

Table 9. The Bell real model in terms of the random vector ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2]))

pmfBellRealismTable

Translating the Bell real model in terms of ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) (Table 9) to the Bell real model in terms of ((Subscript[D, A]),(Subscript[D, B]),A,B) (Table 5)

The Bell realism models from Tables 5 and 9 are similar, but not directly comparable, since the random vectors (A,B,(Subscript[D, A]),(Subscript[D, B]))  and ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) are obviously different. In order to compare them, a translation from one to the other is required. In order to do this, an intermediate combined table construction is helpful. The rows have been numbered for easy reference. Note that row 50 is highlighted in red.

Table 10. The "combined" Bell real model

pmfBellCombinedNumberedTable

How to interpret rows 1-64 of this table?
Consider row number 50 (in red):   ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2]))=(1,1,-1,-1)  and (A,B,(Subscript[D, A]),(Subscript[D, B]))=(1,-1,-1,1).  The first four numbers represent the prepared definite outcomes, awaiting only Alice and Bob’s measurement setting choices. 
Now consider the second four numbers and notice that ((Subscript[D, A]),(Subscript[D, B]))=(-1,1), which means that Alice uses her first measurement setting and Bob uses his second. Therefore, Alice gets the measurement outcome (Subscript[A, 1]) and Bob gets the measurement outcome (Subscript[B, 2]). Hence their joint measurement outcome is (A,B)=(1,-1), so finally (A,B,(Subscript[D, A]),(Subscript[D, B]))=(1,-1,-1,1). Thus the full row 50 is (1,1,-1,-1,1,-1,-1,1) . Similarly for the other 63 rows (exercise for the reader to check!).
The probability associated with row number 50 comes from the fact that ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2]))=(1,1,-1,-1) is in the 13th row of the Bell real model table (see Table 9), hence has probability (Subscript[ρ, 13]), and the probability of Alice choosing her first measurement setting is (Subscript[δ, a]) and Bob choosing his second measurement setting is (Subscript[δ, b']). This yields an overall probability of (Subscript[ρ, 13])(Subscript[δ, a])(Subscript[δ, b']) since all of these events and actions are assumed to be independent of each other.

Finally marginalize this mother-of-all-tables over the four r.v.’s (A,B,(Subscript[D, A]),(Subscript[D, B])) to get

Table 11. The Bell real model in terms of the random vector ((A,B,(Subscript[D, A]),(Subscript[D, B]))obtained by marginalizing Table 10 over (A,B,(Subscript[D, A]),(Subscript[D, B]))

pmfBellCombinedReducedTable

For completeness, do a sanity check, and marginalize over the first 4 r.v.’s, and note that the original Bell realism model is recovered, as expected.

Table 12. Recovery of the standard Bell realism model by marginalizing Table 10 over ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2]))

pmfBellMargin1234=Assuming[Total[ρ16]==1&&(Subscript[δ, a])+(Subscript[δ, a'])==1&&(Subscript[δ, b])+(Subscript[δ, b'])==1,Simplify[marginalize[pmfBellCombined,{1,2,3,4}]]];
Grid[Prepend[Table[Join[pmfBellMargin1234[[k,1]],{pmfBellMargin1234[[k,2]]}],{k,1,Length[pmfBellMargin1234]}],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[B, 1])","(Subscript[B, 2])","Bell real pmf"}],Frame->All]

Note that the relation between the correlations for the Bell real model as represented in Table 12 compared to Table 9 is as follows: 
⟨(Subscript[A, 1])(Subscript[B, 1])⟩=E[A B|(Subscript[D, A])=-1,(Subscript[D, B])=-1],
⟨(Subscript[A, 1])(Subscript[B, 2])⟩=E[A B|(Subscript[D, A])=-1,(Subscript[D, B])=1], 
 ⟨(Subscript[A, 2])(Subscript[B, 1])⟩=E[A B|(Subscript[D, A])=1,(Subscript[D, B])=-1], 
 ⟨(Subscript[A, 2])(Subscript[B, 2])⟩=E[A B|(Subscript[D, A])=1,(Subscript[D, B])=1].

21. Source of the Bell local model

An excellent discussion of the concept of (Bell) locality can be found in [Shimony 2009] (the “S” in CHSH). Indeed this is an excellent discussion of most, if not all, things to do with Bell’s Theorem. Here is an excerpt:

Excerpt from Shimony article

“The following probabilities are assumed to be well defined:

(5)  p1m (s |a, b, t ) = the probability that the outcome of the measurement performed on 1 is s when m is the complete state, the measurements performed on 1 and 2 respectively are a and b, and the result of the experiment on 2 is t;

(6)  p2m (t |a, b, s ) = the probability that the outcome of the measurement performed on 2 is t when the complete state is m, the measurements performed on 1 and 2 are respectively a and b, and the result of the measurement a is s.

(7)  pm (s, t |a, b ) = the probability that the results of the joint measurements a and b, when the complete state is m, are respectively s and t.

The probability function p will be assumed to be non-negative and to sum to unity when the summation is taken over all allowed values of s and t. (Note that the hidden variables theories considered in Section 1 can be subsumed under this conceptual framework by restricting the values of the probability function p to 1 and 0, the former being identified with the truth value “true” and the latter to the truth value “false”. )

A further feature of the conceptual framework is locality, which is understood as the conjunction of the following Independence Conditions:

Remote Outcome Independence (this name is a neologism, but an appropriate one, for what is commonly called outcome independence)

(8a)  p1m (s |a, b, t ) ≡ p1m (s |a, b ) is independent of t,

(8b)  p2m (t |a, b, s ) ≡ p2m (t |a, b ) is independent of s;

(Note that Eqs. (8a) and (8b) do not preclude correlations of the results of the experiment a on 1 and the experiment b on 2; they say rather that if the complete state m is given, the outcome s of the experiment on 1 provides no additional information regarding the outcome of the experiment on 2, and conversely.)

Remote Context Independence (this also is a neologism, but an appropriate one, for what is commonly called parameter independence):

(9a)  p1m (s |a, b ) ≡ p1m (s | a ) is independent of b,

(9b)  p 2m (t |a, b ) ≡ p2m (t | b ) is independent of a.

Jarrett (1984) and Bell (1990) demonstrated the equivalence of the conjunction of (8a,b) and (9a,b) to the factorization condition:

(10)  pm (s, t |a, b ) = p1m(s |a )p2m(t |b ),
and likewise for (a, b′), (a′, b ) , and (a′,b′) substituted for (a, b ).

The factorizability condition Eq. (10) is also often referred to as Bell locality. It should be emphasized that at the present stage of exposition, however, Bell locality is merely a mathematical condition within a conceptual framework, to which no physical significance has been attached — in particular no connection to the locality of Special Relativity Theory, although such a connection will be made later when experimental applications of Bell’s theorem will be discussed.”

End of excerpt from Shimony article

Translating this to the terminology and notation of this presentation, and using the factorization condition (10) above:

Definition ( Bell locality): A probability model in the (A,B,(Subscript[D, A]),(Subscript[D, B])) format will be called Bell local if P(A=s,B=t|(Subscript[D, A])=(Subscript[ϵ, A]),(Subscript[D, B])=(Subscript[ϵ, B]))=P(A=s|(Subscript[D, A])=(Subscript[ϵ, A])) *P(B=t|(Subscript[D, B])=(Subscript[ϵ, B])) for all (s,t,(Subscript[ϵ, A]),(Subscript[ϵ, B]))∈(({-1,1})^4). That is, A and B are  are conditionally independent given the values of (Subscript[D, A]) and (Subscript[D, B]), respectively.

Derivation of the form of the Bell local model

Assume there are non-negative real numbers (Subscript[α, 1]),(Subscript[α, 2]),(Subscript[α, 3]),(Subscript[α, 4]) and (Subscript[β, 1]),(Subscript[β, 2]),(Subscript[β, 3]),(Subscript[β, 4]) such that the conditional probabilities for (A|(Subscript[D, A])) and (B,(Subscript[D, B])) are given respectively as in the first two tables below.

Assuming[(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1,Refine[{displayPMF[conditionalize[pmfBellLocal,{1},{3}],{"A","(Subscript[D, A])","Probabilities for (A|(Subscript[D, A]))"}],displayPMF[conditionalize[pmfBellLocal,{2},{4}],{"B","(Subscript[D, B])","Probabilities for (B|(Subscript[D, B]))"}],displayPMF[conditionalize[pmfBellLocal,{1,2},{3,4}],{"A","B","(Subscript[D, A])","(Subscript[D, B])","Probabilities for ((A,B)|((Subscript[D, A]),(Subscript[D, B])))"}]}]]/.{1-(Subscript[α, 3])->(Subscript[α, 1]),1-(Subscript[α, 4])->(Subscript[α, 2]),1-(Subscript[β, 3])->(Subscript[β, 1]),1-(Subscript[β, 4])->(Subscript[β, 2])}

Note that the third column in each table does not represent a pmf. These are just conditional probabilities. However, note that (Subscript[α, 1])+(Subscript[α, 3])=P(A=-1|(Subscript[D, A])=-1)+P(A=1|(Subscript[D, A ])=-1)=1 and (Subscript[α, 2])+(Subscript[α, 4])=P(A=-1|(Subscript[D, A])=1)+P(A=1|(Subscript[D, A ])=1)=1 and similarly for the β’s.
The products (Subscript[α, j])(Subscript[β, k]) in the “Probabilities” column of the last table were chosen specifically to impose Bell locality on the model, according to the factorization condition as described by Shimony above, that is,
P(A=s,B=t|(Subscript[D, A])=(Subscript[ϵ, A]),(Subscript[D, B])=(Subscript[ϵ, B]))=P(A=s|(Subscript[D, A])=(Subscript[ϵ, A])) *P(B=t|(Subscript[D, B])=(Subscript[ϵ, B])) for all (s,t,(Subscript[ϵ, A]),(Subscript[ϵ, B]))∈(({-1,1})^4).

Finally, since the probabilities in the last table above represent conditional probabilities given the choices for the detector angle settings, the full joint probability model can be synthesized by multiplying each of these conditional probabilities with the associated probabilities for choosing the detector angle settings. This results in the form of the Bell local model as defined in Table 2 at the very beginning, namely,

pmfBellLocalTable

22. Proof that any Bell local model is also Bell real

Consider the two incarnations of the Bell real model as represented in Table 11 and Table 12 above. They both have the same set of correlations, namely...

correlationsBellReal//MatrixForm

...so any assignment of (Subscript[ρ, k])’s for one model produces the same correlations for the other model (and by extension, the same s-functions).

By this equivalence of the two models, and by applying the Consistency Theorem to the “original” model of Table 12 (in the statement of the Consistency Theorem, set
(Subscript[X, 1])=(Subscript[A, 1]),(Subscript[X, 2])=(Subscript[A, 2]),(Subscript[X, 3])=(Subscript[B, 1]), and (Subscript[X, 4])=(Subscript[B, 2])), it is sufficient to show that the Bell local correlations satisfy the CHSH inequalities. Proceed to do this.

Using Mathematica’s “Maximize” and “Minimize”, the max and min of the s-functions are calculated to be +2 and -2 for (Subscript[s, 1]),(Subscript[s, 2]),(Subscript[s, 3]),(Subscript[s, 4]) applied to the Bell local correlations, as shown below.
First recall the Bell local s-functions, (Subscript[s, 1]),(Subscript[s, 2]),(Subscript[s, 3]),(Subscript[s, 4]) from top to bottom.

sBellLocal//MatrixForm

Now compute max and min for (Subscript[s, 1]),(Subscript[s, 2]),(Subscript[s, 3]),(Subscript[s, 4]).

{solMax1=Maximize[{sBellLocal[[1]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]],
solMin1=Minimize[{sBellLocal[[1]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]],

solMax2=Maximize[{sBellLocal[[2]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]],
solMin2=Minimize[{sBellLocal[[2]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]],

solMax3=Maximize[{sBellLocal[[3]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]],
solMin3=Minimize[{sBellLocal[[3]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]],

solMax4=Maximize[{sBellLocal[[4]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]],
solMin4=Minimize[{sBellLocal[[4]],(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0},Join[aliceProbs,bobProbs]]}//MatrixForm

Now do a sanity check that these solutions produce s-function values between -2 and 2. Check! All output values are between -2 and 2. The ones marked in Red denote the s-functions that were maximized or minimized. The ones in Black are calculated relative to a Max or Min for a different (Subscript[s, k]). They  may or may not be a Max or Min, but they are always between -2 and +2.

Table 13. s-function max and mins for the Bell local model

Grid[Prepend[Prepend[Transpose[Join[{{"(Subscript[s, 1])","(Subscript[s, 2])","(Subscript[s, 3])","(Subscript[s, 4])"}},
{sBellLocal/.solMax1[[2]],
sBellLocal/.solMin1[[2]],

sBellLocal/.solMax2[[2]],
sBellLocal/.solMin2[[2]],

sBellLocal/.solMax3[[2]],
sBellLocal/.solMin3[[2]],

sBellLocal/.solMax4[[2]],
sBellLocal/.solMin4[[2]]}]],
{"s-function \nmaximized/minimized","Max \nargument","Min  \nargument","Max  \nargument","Min  \nargument","Max  \nargument","Min  \nargument","Max  \nargument","Min  \nargument"}],
{"","(Subscript[s, 1]) evaluated at",SpanFromLeft,"(Subscript[s, 2]) evaluated at",SpanFromLeft,"(Subscript[s, 3]) evaluated at",SpanFromLeft,"(Subscript[s, 4]) evaluated at",SpanFromLeft}],ItemStyle->{Automatic,Automatic,{{3,2}->Red,{3,3}->Red,{4,4}->Red,{4,5}->Red,{5,6}->Red,{5,7}->Red,{6,8}->Red,{6,9}->Red}},Frame->All]

Hence any Bell local model is always also Bell real.

However, not every Bell real model is Bell local

For pictorial confirmation, see row 3 of Table 5. It is clear from this picture that an uncountably infinite number of Bell real correlations are off the Bell local surface. But for one specific counterexample, consider (w,x,y,z)=(0,1,-1,0).
First solve for the Bell real parameters that give rise to this 4-tuple of correlations.

LinearProgramming[ConstantArray[1,16],matSigns,{0,1,-1,0}]

Now check that (0,1,-1,0) does indeed arise from this parameter assignment. Check!

subs={(Subscript[ρ, 3])->1/2,(Subscript[ρ, 5])->1/2};
correlationsBellReal/.subs/.ρ0

Now show that (0,1,-1,0) cannot be achieved by any assignment of parameters to a Bell local model. The solution set is empty, as expected. QED

Solve[correlationsBellLocal=={0,1,-1,0}&&(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0,Join[aliceProbs,bobProbs]]

Here is the picture of what has just been proved. Namely, the point (x,y,z)=(1,-1,0)  (purple dot) is in the Bell real region (blue), but it is way off the Bell local contour (green triangles PLUS the vertical line from (0,0,-1) to (0,0,1)). Note that this is the 3D-slice corresponding to w=0, so the 4-tuple of correlations in question is the one specified at the top, namely (w,x,y,z)=(0,1,-1,0).

Show[cvhSeeThruPlots[[3]],bellLocalPlots[[3]],Graphics3D[{Purple,PointSize[Large],Point[{1,-1,0}]}]]

24. QM and Bell real combinations

Show examples of specific parameter assignments for the cases:
1. QM and Bell real
2. QM and not Bell real
3. Not QM and Bell real
4. Not QM and not Bell real

QM and Bell real

Trivial “uniform pmf“ example that produces the Minimal model -- all 0 correlations.

correlationsQM/.{(Subscript[θ, 1])->135 Degree,(Subscript[θ, 2])->45 Degree,(Subscript[θ, 3])->45 Degree,(Subscript[θ, 4])->45 Degree}
correlationsBellReal/.ρ16SubsUniform/.subsδUniform

{ρ16SubsUniform,subsδUniform}

More substantial examples would be all of the Bell local models (which are also Bell real!) that are also QM as shown in the Fig. below -- on the red lines.

Show[bellLocalPlots[[3]],qmSeeThruPlots2[[3]],plotQMBellLocalLines,Graphics3D[{Purple,PointSize[0.02],Point[{0,-.65,.65}]}],AxesLabel->{"x","y","z"}]

QM and not Bell real

Pick the notorious maximum deviation set of angles for the QM model that the Bell real model cannot reach because it must satisfy the Bell CHSH inequalities.

correlationsQM/.θSubsQM
sBell.%//MatrixForm//Chop

Thus the first s-function (Subscript[s, 1])=2(Sqrt[2]) for the QM model, but any Bell real s-function must be between -2 and 2.

Not QM and Bell real

Pick for example the point (0,0,1) in the middle of the square on the top face, w=0 case, i.e, the set of correlations (0,0,0,1). See the purple dot.

Show[qmSeeThruPlots2[[3]],bellRealPlots[[3]],bellRealVertices[{-1,-0.5,0,0.5,1}[[3]]],Graphics3D[{Purple,PointSize[Large],Point[{0,0,1}]}],Graphics3D[{Black,PointSize[Large],Point[{1,-1,-1}]}],AxesLabel->{"x","y","z"},ImageSize->500]

Pick a set of four (Subscript[ρ, k])’s equal to 1/4 that makes the first 3 rows = 0 and the last row = 1. The choice {(Subscript[ρ, 1 ])->1/4,(Subscript[ρ, 3])->1/4,(Subscript[ρ, 6 ])->1/4,(Subscript[ρ, 8])->1/4} is one that works.

correlationsBellReal//MatrixForm

ρ16/.{(Subscript[ρ, 1 ])->1/4,(Subscript[ρ, 3])->1/4,(Subscript[ρ, 6 ])->1/4,(Subscript[ρ, 8])->1/4}/.ρ0
correlationsBellReal/.{(Subscript[ρ, 1 ])->1/4,(Subscript[ρ, 3])->1/4,(Subscript[ρ, 6 ])->1/4,(Subscript[ρ, 8])->1/4}/.ρ0

Now this cannot be achieved by the QM correlations because (Subscript[θ, 4]) has to be some integer multiple of 2π and the other three angles of the form (2k+1)(π/2). Then (Subscript[θ, 1])=(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4]) cannot be satisfied, else (2(Subscript[k, 1])+1)(π/2)=(2(Subscript[k, 2])+1)(π/2)+(2(Subscript[k, 3])+1)(π/2)+2k π which yields 4k =2((Subscript[k, 1])-(Subscript[k, 2])-(Subscript[k, 3]))-1, that is a contradiction equating an even integer to an odd integer.

correlationsQM//MatrixForm

Not QM and not Bell real

This is easy in the situation pictured in the above figure for w=0. Just pick a point far enough into any of the four “corners” to miss both the QM contour and the Bell real region. For example, the point (1,-1,-1) corresponding to the correlations (0,1,-1,-1). See the black dot at the bottom front. Obviously not in the Bell region because not all of the CHSH inequalities are satisfied -- the second s-function has value -3 < -2.

sBell.{0,1,-1,-1}

Also cannot be achieved by the QM model since (Subscript[θ, 1]) would have to be of the form (2(Subscript[k, 1])+1)(π/2), (Subscript[θ, 2]) of the form 2(Subscript[k, 2])π, (Subscript[θ, 3]) of the form (2(Subscript[k, 3])+1)π, and (Subscript[θ, 4])of the form (2(Subscript[k, 4])+1)π, meaning it must be the case that (2(Subscript[k, 1])+1)(π/2)=2(Subscript[k, 2])π+(2(Subscript[k, 3])+1)π+(2(Subscript[k, 4])+1)π,  or (2(Subscript[k, 1])+1)π=4(Subscript[k, 2])π+2(2(Subscript[k, 3])+1)π+2(2(Subscript[k, 4])+1)π or 2(Subscript[k, 1])+1=4(Subscript[k, 2])+2(2(Subscript[k, 3])+1)+2(2(Subscript[k, 4])+1), a contradiction that an odd integer equals an even integer.

Initialization

sumOfRandomReals[] generates “numberOfTuples” “numberOfSummands”-tuples of random real numbers that sum to “sum”, all bounded below by “min”.

sumOfRandomReals[min_,sum_,numberOfSummands_,numberOfTuples_]:=Differences[#]&/@(Append[#,sum]&/@(Prepend[#,min]&/@(Sort[#]&/@RandomReal[{min,sum},{numberOfTuples,numberOfSummands-1}])))

Probability calculus.

expectations[outcomesAndProbs_]:=
Total[Table[outcomesAndProbs[[k,1]]*outcomesAndProbs[[k,2]],{k,1,Length[outcomesAndProbs]}]];
sMatrix={{-1,1,1,1},{1,-1,1,1},{1,1,-1,1},{1,1,1,-1}};
sFunction[outcomesAndProbs_]:={1,-1,1,1}.expectations[outcomesAndProbs];

marginalSelect[outcomesAndProbs_,indices_]:=Module[{},
SortBy[Table[{outcomesAndProbs[[k,1,indices]],outcomesAndProbs[[k,2]]},{k,1,Length[outcomesAndProbs]}],First]
];
marginalize[outcomesAndProbs_,indices_]:=Module[{select,gather},
select=marginalSelect[outcomesAndProbs,indices];
gather=GatherBy[select,First];
Table[{gather[[k,1,1]],Total[gather[[k,All,2]]]},{k,1,Length[gather]}]
];
probability[pmf_,outcomes_]:=Module[{pos},pos=Flatten[Table[Position[pmf[[All,1]],outcomes[[k]]],{k,1,Length[outcomes]}]];
Total[pmf[[pos,2]]]];
conditionalize[outcomesAndProbs_,conditioned_,conditional_]:=Module[{indicesAll,marginAll,marginRight,gather,indicesLeft,indicesRight},
indicesLeft=Sort[conditioned];
indicesRight=Sort[conditional];
indicesAll=Join[indicesLeft,indicesRight];
marginAll=marginalize[outcomesAndProbs,indicesAll];
marginRight=marginalize[outcomesAndProbs,indicesRight];
(*Notice this next statement ASSUMES that the conditioning variables are in the final -Length[indicesRight] positions!!*)
SortBy[Table[{marginAll[[k,1]],marginAll[[k,2]]/(marginRight[[Position[marginRight[[All,1]],marginAll[[k,1,-Length[indicesRight];;-1]]][[1]],2]][[1]])//Simplify},{k,1,Length[marginAll]}
],First]];

Check the independence of two marginal or conditional pmf’s against their combination. pmf1 and pmf2 are the two components of the combination pmf3. The two smaller ones are independent if it can be shown that for every combination of probability inputs from them, their product is the probability of the associated outcome obtained by joining together the outcomes from the two smaller (space,pmf)’s. The ‘perm’ argument is a permutation (indicated for example by Cycles[{{2,3]]]) which re-orders the outcomes of the pmf products to align with the ordering of the combined pmf. For example, if we want to check (A,(Subscript[D, A])) and (B,(Subscript[D, B])) as being independent, then the combined pmf is in the order (A,(Subscript[D, A]),B,(Subscript[D, B])), but we need it to be in the order  (A,B, (Subscript[D, A]),(Subscript[D, B])) --  so we need to re-order  (for example) a Join[{-1,1},{-1,-1}]= {-1,1,-1,-1} by transposing the middle two elements (Cycles[{{2,3}}]) to present {-1,-1,1,-1} to the probability calculator comparison instead, i.e., in the order (A,B, (Subscript[D, A]),(Subscript[D, B])).

independenceCheck[pmf1_,pmf2_,pmf3_,perm_]:=Module[{outcome,outcomes1,outcomes2,outcomes3,prob,probs1,probs2,probs3,pos},
outcomes1=pmf1[[All,1]];
outcomes2=pmf2[[All,1]];
outcomes3=pmf3[[All,1]];
probs1=pmf1[[All,2]];
probs2=pmf2[[All,2]];
probs3=pmf3[[All,2]];

Table[
outcome=Permute[Join[outcomes1[[i]],outcomes2[[j]]],perm];
pos=Position[outcomes3,outcome][[1,1]];
prob=pmf3[[pos,2]];
probability[pmf1,{outcomes1[[i]]}]*probability[pmf2,{outcomes2[[j]]}]==prob,{i,1,Length[outcomes1]},{j,1,Length[outcomes2]}]
];

Display utility for pmf’s that recognizes the ordering of the r.v.’s (e.g. in a call to ‘marginalize’ or ‘conditionalize’) The resulting outcomes have been re-ordered to respect the marginalization or conditionalization specifications.
Second routine produces three conditional displays (A|(Subscript[D, A])), (B|(Subscript[D, B])), and ((A,(Subscript[D, A]))|(B,(Subscript[D, B]))) plus an “independenceCheck” to give some intuition to any claim of Bell locality.

displayPMF[pmf_,header_]:=Module[{},
Grid[Prepend[Table[Join[ If[ListQ[pmf[[k,1]]],pmf[[k,1]],{pmf[[k,1]]}],{  pmf[[k,2]]}],{k,1,Length[ pmf]}],header],Frame->All]
];
conditionalDisplay[pmf_,perm_,sortBy_,condLeft1_,condRight1_,condLeft2_,condRight2_,subs_,header1_,header2_,header3_]:=Module[{(*pmf1,pmf2,pmf3,condLeft,condRight*)},
pmf1=pmf/.subs;
pmf2=conditionalize[pmf1,Sort[Join[condLeft1,condLeft2]],Sort[Join[condRight1,condRight2]]];
pmf3=SortBy[Table[{Permute[pmf2[[k,1]],perm],pmf2[[k,2]]},{k,1,Length[pmf2]}],#[[1,sortBy]]&];
{displayPMF[conditionalize[pmf1,Sort[condLeft1],Sort[condRight1]],header1],displayPMF[conditionalize[pmf1,Sort[condLeft2],Sort[condRight2]],header2],displayPMF[pmf3,header3],
independenceCheck[conditionalize[pmf1,condLeft1,condRight1],conditionalize[pmf1,condLeft2,condRight2],conditionalize[pmf1,Sort[Join[condLeft1,condLeft2]],Sort[Join[condRight1,condRight2]]],perm]//MatrixForm}];

Basic notation for Bell space description

tuples2=Tuples[{-1,1},2];
tuples4=Tuples[{-1,1},4];
tuples4Repeat=Flatten[Table[ConstantArray[tuples4[[k]],4],{k,1,Length[tuples4]}],1];

ρ16=Array[(Subscript[ρ, #])&,16];
γ0={(Subscript[γ, 1])->0,(Subscript[γ, 2])->0,(Subscript[γ, 3])->0,(Subscript[γ, 4])->0,(Subscript[γ, 5])->0,(Subscript[γ, 6])->0,(Subscript[γ, 7])->0,(Subscript[γ, 8])->0,(Subscript[γ, 9])->0,(Subscript[γ, 10])->0,(Subscript[γ, 11])->0,(Subscript[γ, 12])->0,(Subscript[γ, 13])->0,(Subscript[γ, 14])->0,(Subscript[γ, 15])->0,(Subscript[γ, 16])->0};
ρ16Repeat=Flatten[Table[ConstantArray[(Subscript[ρ, k]),4],{k,1,16}]];

γ16=Array[(Subscript[γ, #])&,16];
γ16Repeat=Flatten[Table[ConstantArray[(Subscript[γ, k]),4],{k,1,16}]];

λ16=Array[(Subscript[λ, #])&,16];
λ16Repeat=Flatten[Table[ConstantArray[(Subscript[λ, k]),4],{k,1,16}]];

ρ0={(Subscript[ρ, 1])->0,(Subscript[ρ, 2])->0,(Subscript[ρ, 3])->0,(Subscript[ρ, 4])->0,(Subscript[ρ, 5])->0,(Subscript[ρ, 6])->0,(Subscript[ρ, 7])->0,(Subscript[ρ, 8])->0,(Subscript[ρ, 9])->0,(Subscript[ρ, 10])->0,(Subscript[ρ, 11])->0,(Subscript[ρ, 12])->0,(Subscript[ρ, 13])->0,(Subscript[ρ, 14])->0,(Subscript[ρ, 15])->0,(Subscript[ρ, 16])->0};
ρ16CondsNonNeg=And@@(#≥0&/@ρ16);
ρ16CondsSum1=Total[ρ16]==1;
ρ16SubsSum1={(Subscript[ρ, 1])+(Subscript[ρ, 2])+(Subscript[ρ, 3])+(Subscript[ρ, 4])+(Subscript[ρ, 5])+(Subscript[ρ, 6])+(Subscript[ρ, 7])+(Subscript[ρ, 8])+(Subscript[ρ, 9])+(Subscript[ρ, 10])+(Subscript[ρ, 11])+(Subscript[ρ, 12])+(Subscript[ρ, 13])+(Subscript[ρ, 14])+(Subscript[ρ, 15])+(Subscript[ρ, 16])->1};
ρ16SubsUniform=Table[(Subscript[ρ, k])->1/16,{k,1,16}];

sBell={{-1,1,1,1},{1,-1,1,1},{1,1,-1,1},{1,1,1,-1}};
ρMat={{1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,0},{1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0},{1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0},{1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0},{0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0},{0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0},{0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0},{0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0},{0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0},{0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0},{0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0},{0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0},{0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1},{0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1},{0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1},{0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1}};
%//MatrixForm

Define a function to convert each 4-tuple from ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) to a corresponding list of four 4-tuples from ((Subscript[D, A]),(Subscript[D, B]),A,B).
Also the inverse: from ((Subscript[D, A]),(Subscript[D, B]),A,B) to ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])).

convertBell[lst_]:={{-1,-1,lst[[1]],lst[[3]]},{-1,1,lst[[1]],lst[[4]]},{1,-1,lst[[2]],lst[[3]]},{1,1,lst[[2]],lst[[4]]}};
convertBellInverse[lst_]:={{-1,-1,lst[[1]],lst[[3]]},{-1,1,lst[[1]],lst[[4]]},{1,-1,lst[[2]],lst[[3]]},{1,1,lst[[2]],lst[[4]]}};

Define set of all  outcomes associated with  the ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) and ((Subscript[D, A]),(Subscript[D, B]),A,B) 4-tuple combinations.

convertTuplesBell=Flatten[Table[convertBell[tuples4[[k]]],{k,1,Length[tuples4]}],1];
pmfAliceCombinedOutcomes=Flatten[Table[{Join[tuples4Repeat[[k]],convertTuplesBell[[k]]]},{k,1,Length[tuples4Repeat]}],1];

Now define the probabilities associated with the ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) and ((Subscript[D, A]),(Subscript[D, B]),A,B) 4-tuple combinations.

detectorProductProbs={ (Subscript[δ, a]) (Subscript[δ, b]), (Subscript[δ, a]) (Subscript[δ, b']), (Subscript[δ, a']) (Subscript[δ, b]),(Subscript[δ, a']) (Subscript[δ, b'])};
rTmp=Riffle[detectorProductProbs,detectorProductProbs];
detectorProductProbsRepeated=Riffle[rTmp,rTmp];
pmfDetector=Table[{tuples2[[k]],detectorProductProbs[[k]]},{k,1,Length[tuples2]}];

Assumptions and substitution lists

αβAssumptions=(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0&&(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1;
δAssumptions= (Subscript[δ, a])≥0&&(Subscript[δ, a'])≥0&&(Subscript[δ, b])≥0&&(Subscript[δ, b'])≥0&&(Subscript[δ, a])+(Subscript[δ, a'])==1&&(Subscript[δ, b])+(Subscript[δ, b'])==1;
ρAssumptions=(Subscript[ρ, 1])≥0&&(Subscript[ρ, 2])≥0&&(Subscript[ρ, 3])≥0&&(Subscript[ρ, 4])≥0&&(Subscript[ρ, 5])≥0&&(Subscript[ρ, 6])≥0&&(Subscript[ρ, 7])≥0&&(Subscript[ρ, 8])≥0&&(Subscript[ρ, 9])≥0&&(Subscript[ρ, 10])≥0&&(Subscript[ρ, 11])≥0&&(Subscript[ρ, 12])≥0&&(Subscript[ρ, 13])≥0&&(Subscript[ρ, 14])≥0&&(Subscript[ρ, 15])≥0&&(Subscript[ρ, 16])≥0&&Total[ρ16]==1;
γAssumptions=(Subscript[γ, 1])≥0&&(Subscript[γ, 2])≥0&&(Subscript[γ, 3])≥0&&(Subscript[γ, 4])≥0&&(Subscript[γ, 5])≥0&&(Subscript[γ, 6])≥0&&(Subscript[γ, 7])≥0&&(Subscript[γ, 8])≥0&&(Subscript[γ, 9])≥0&&(Subscript[γ, 10])≥0&&(Subscript[γ, 11])≥0&&(Subscript[γ, 12])≥0&&(Subscript[γ, 13])≥0&&(Subscript[γ, 14])≥0&&(Subscript[γ, 15])≥0&&(Subscript[γ, 16])≥0&&Total[γ16]==1;

θSubsQM={(Subscript[θ, 1])->67.5°,(Subscript[θ, 2])->22.5°,(Subscript[θ, 3])->22.5°,(Subscript[θ, 4])->22.5°,(Subscript[δ, a])->1/2,(Subscript[δ, a'])->1/2,(Subscript[δ, b])->1/2,(Subscript[δ, b'])->1/2};
subsθSum={(Subscript[θ, 1])->(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4])};
subsθMinimal={(Subscript[θ, 2])->45 Degree,(Subscript[θ, 3])->45 Degree, (Subscript[θ, 4])->45 Degree};
subsθInfamous={(Subscript[θ, 2])->22.5 Degree,(Subscript[θ, 3])->22.5 Degree, (Subscript[θ, 4])->22.5 Degree};
subsδMinimal={(Subscript[δ, a])->1/2,(Subscript[δ, b])->1/2};
subsδUniform={(Subscript[δ, a])->1/2,(Subscript[δ, b])->1/2,(Subscript[δ, a'])->1/2,(Subscript[δ, b])'->1/2};
subsρMinimal={(Subscript[ρ, 1])->1/16,(Subscript[ρ, 2])->1/16,(Subscript[ρ, 3])->1/16,(Subscript[ρ, 4])->1/16,(Subscript[ρ, 5])->1/16,(Subscript[ρ, 6])->1/16,(Subscript[ρ, 7])->1/16,(Subscript[ρ, 8])->1/16,(Subscript[ρ, 9])->1/16,(Subscript[ρ, 10])->1/16,(Subscript[ρ, 11])->1/16,(Subscript[ρ, 12])->1/16,(Subscript[ρ, 13])->1/16,(Subscript[ρ, 14])->1/16,(Subscript[ρ, 15])->1/16,(Subscript[ρ, 16])->1/16};
subsγMinimal={(Subscript[γ, 1])->(1/16),(Subscript[γ, 2])->(1/16),(Subscript[γ, 3])->(1/16),(Subscript[γ, 4])->(1/16),(Subscript[γ, 5])->(1/16),(Subscript[γ, 6])->(1/16),(Subscript[γ, 7])->(1/16),(Subscript[γ, 8])->(1/16),(Subscript[γ, 9])->(1/16),(Subscript[γ, 10])->(1/16),(Subscript[γ, 11])->(1/16),(Subscript[γ, 12])->(1/16),(Subscript[γ, 13])->(1/16),(Subscript[γ, 14])->(1/16),(Subscript[γ, 15])->(1/16),(Subscript[γ, 16])->(1/16)};
subsγ4={(Subscript[γ, 1])->(1/4),(Subscript[γ, 2])->(1/4),(Subscript[γ, 3])->(1/4),(Subscript[γ, 4])->(1/4),(Subscript[γ, 5])->0,(Subscript[γ, 6])->0,(Subscript[γ, 7])->0,(Subscript[γ, 8])->0,(Subscript[γ, 9])->0,(Subscript[γ, 10])->0,(Subscript[γ, 11])->0,(Subscript[γ, 12])->0,(Subscript[γ, 13])->0,(Subscript[γ, 14])->0,(Subscript[γ, 15])->0,(Subscript[γ, 16])->0};
subsρ4={(Subscript[ρ, 1])->(1/4),(Subscript[ρ, 2])->(1/4),(Subscript[ρ, 3])->(1/4),(Subscript[ρ, 4])->(1/4),(Subscript[ρ, 5])->0,(Subscript[ρ, 6])->0,(Subscript[ρ, 7])->0,(Subscript[ρ, 8])->0,(Subscript[ρ, 9])->0,(Subscript[ρ, 10])->0,(Subscript[ρ, 11])->0,(Subscript[ρ, 12])->0,(Subscript[ρ, 13])->0,(Subscript[ρ, 14])->0,(Subscript[ρ, 15])->0,(Subscript[ρ, 16])->0};
subsδUniform={(Subscript[δ, a])->1/2,(Subscript[δ, a'])->1/2,(Subscript[δ, b])->1/2,(Subscript[δ, b'])->1/2};
subsαβUniform={(Subscript[α, 1])->1/2,(Subscript[α, 3])->1/2,(Subscript[α, 2])->1/2,(Subscript[α, 4])->1/2,(Subscript[β, 1])->1/2,(Subscript[β, 3])->1/2,(Subscript[β, 2])->1/2,(Subscript[β, 4])->1/2};
subsBL1={(Subscript[α, 1])->.1,(Subscript[α, 2])->.6,(Subscript[α, 3])->.9,(Subscript[α, 4])->.4,(Subscript[β, 1])->.3,(Subscript[β, 2])->.5,(Subscript[β, 3])->.7,(Subscript[β, 4])->.5};
subsBL2={(Subscript[α, 1])->1,(Subscript[α, 2])->1,(Subscript[α, 3])->0,(Subscript[α, 4])->0,(Subscript[β, 1])->1,(Subscript[β, 2])->0,(Subscript[β, 3])->0,(Subscript[β, 4])->1};

The models

Maximal model

pmfMaximal=Table[{tuples4[[k]],γ16[[k]]},{k,1,Length[tuples4]}];
pmfMaximalTable=Grid[Prepend[Table[Join[ pmfMaximal[[k,1]],{ pmfMaximal[[k,2]]}],{k,1,Length[ pmfMaximal]}],{"A","B","(Subscript[D, A])","(Subscript[D, B])","Maximal pmf"}],Frame->All]

Maximal correlations and s-functions

pmfCondMaximal=conditionalize[pmfMaximal,{1,2},{3,4}];
pmfMaximalProducts=Table[{pmfCondMaximal[[k,1,1]]*pmfCondMaximal[[k,1,2]],pmfCondMaximal[[k,2]]},{k,1,Length[pmfCondMaximal]}];
{pmfCondMaximal//MatrixForm,pmfMaximalProducts//MatrixForm}
correlationsMaximal=expectations[#]&/@{pmfMaximalProducts[[{1,5,9,13}]],pmfMaximalProducts[[{2,6,10,14}]],pmfMaximalProducts[[{3,7,11,15}]],pmfMaximalProducts[[{4,8,12,16}]]}//Simplify
sMaximal=sMatrix.correlationsMaximal//Simplify

Bell real model

Define set of all  outcomes associated with  the ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) and (A,B,(Subscript[D, A]),(Subscript[D, B])) 4-tuple combinations. We perform the permutation here to account for our new standard of (A,B,(Subscript[D, A]),(Subscript[D, B])) instead of ((Subscript[D, A]),(Subscript[D, B]),A,B).

tuples4Repeat=Flatten[Table[ConstantArray[tuples4[[k]],4],{k,1,Length[tuples4]}],1];
convertTuplesBell=Permute[#,Cycles[{{1,3},{2,4}}]]&/@Flatten[Table[convertBell[tuples4[[k]]],{k,1,Length[tuples4]}],1];
pmfCombinedOutcomes=Flatten[Table[{Join[tuples4Repeat[[k]],convertTuplesBell[[k]]]},{k,1,Length[tuples4Repeat]}],1];

Now define the probabilities associated with the ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) and (A,B,(Subscript[D, A]),(Subscript[D, B])) 4-tuple combinations. Note this does NOT involve the QM conditional probabilities since the outcomes are pre-determined by the Bell “realism” assumptions.

detectorProductProbs={ (Subscript[δ, a]) (Subscript[δ, b]), (Subscript[δ, a]) (Subscript[δ, b']), (Subscript[δ, a']) (Subscript[δ, b]),(Subscript[δ, a']) (Subscript[δ, b'])};
rTmp=Riffle[detectorProductProbs,detectorProductProbs];
detectorProductProbsRepeated=Riffle[rTmp,rTmp];

partialBellProbs=detectorProductProbs[[Flatten[Position[pmfDetector[[All,1]],#]&/@convertTuplesBell[[All,3;;4]]]]]
pmfCombinedBellProbabilities=Table[ρ16Repeat[[k]]*partialBellProbs[[k]],{k,1,Length[partialBellProbs]}]

pmfBellRealism=Table[{tuples4[[k]],ρ16[[k]]},{k,1,Length[tuples4]}];
pmfBellCombined=Table[{pmfCombinedOutcomes[[k]],pmfCombinedBellProbabilities[[k]]},{k,1,Length[pmfCombinedOutcomes]}];
pmfBellCombinedReduced=Collect[SortBy[marginalize[  pmfBellCombined,{5,6,7,8}],First]//Simplify,{(Subscript[δ, a']),(Subscript[δ, b']),(Subscript[δ, a]),(Subscript[δ, b])}];
pmfBellRealismTable=Grid[Prepend[Table[Join[ pmfBellRealism[[k,1]],{ pmfBellRealism[[k,2]]}],{k,1,Length[ pmfBellRealism]}],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[B, 1])","(Subscript[B, 2])","Bell real pmf"}],Frame->All];
pmfConsistencyTable=Grid[Prepend[Table[Join[ pmfBellRealism[[k,1]],{ pmfBellRealism[[k,2]]}],{k,1,Length[ pmfBellRealism]}],{"(Subscript[X, 1])","(Subscript[X, 2])","(Subscript[X, 3])","(Subscript[X, 4])","pmf"}],Frame->All];
subsγToρ=Table[(Subscript[γ, k])->pmfBellCombinedReduced[[k,2]],{k,1,Length[γ16]}];

Mother of all tables

Notice that in the final table we have opted for a non-standard ordering to make the the ordering of the probabilities “pretty”.

pmfBellCombinedTable=Grid[Prepend[Table[Join[pmfBellCombined[[k,1]],{pmfBellCombined[[k,2]]}],{k,1,Length[pmfBellCombined]}],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[B, 1])","(Subscript[B, 2])","A","B","(Subscript[D, A])","(Subscript[D, B])","Combined Bell real pmf"}],Frame->All];
pmfBellRealismTable=Grid[Prepend[Table[Join[pmfBellRealism[[k,1]],{pmfBellRealism[[k,2]]}],{k,1,Length[pmfBellRealism]}],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[B, 1])","(Subscript[B, 2])","Bell real pmf"}],Frame->All];
pmfBellCombinedReducedTable=Grid[Prepend[Table[Join[pmfBellCombinedReduced[[k,1]],{pmfBellCombinedReduced[[k,2]]}],{k,1,Length[pmfBellCombinedReduced]}],{"A","B","(Subscript[D, A])","(Subscript[D, B])","Bell real pmf"}],Frame->All];
pmfBellCombinedNumberedTable=Grid[Transpose[Prepend[Transpose[Prepend[Table[Join[pmfBellCombined[[k,1]],{pmfBellCombined[[k,2]]}],{k,1,Length[pmfBellCombined]}],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[B, 1])","(Subscript[B, 2])","A","B","(Subscript[D, A])","(Subscript[D, B])","Bell real combined pmf"}]],Range[0,64]]],ItemStyle->{Automatic,Automatic,{{51,1}->Red,{51,2}->Red,{51,3}->Red,{51,4}->Red,{51,5}->Red,{51,6}->Red,{51,7}->Red,{51,8}->Red,{51,9}->Red,{51,10}->Red}},Frame->All];

Bell real correlations and s-functions

pmfCondBellCombinedReduced=conditionalize[pmfBellCombinedReduced,{1,2},{3,4}]/.{(Subscript[ρ, 1])+(Subscript[ρ, 2])+(Subscript[ρ, 3])+(Subscript[ρ, 4])+(Subscript[ρ, 5])+(Subscript[ρ, 6])+(Subscript[ρ, 7])+(Subscript[ρ, 8])+(Subscript[ρ, 9])+(Subscript[ρ, 10])+(Subscript[ρ, 11])+(Subscript[ρ, 12])+(Subscript[ρ, 13])+(Subscript[ρ, 14])+(Subscript[ρ, 15])+(Subscript[ρ, 16])->1};
pmfBellCombinedReducedProducts=Table[{pmfCondBellCombinedReduced[[k,1,1]]*pmfCondBellCombinedReduced[[k,1,2]],pmfCondBellCombinedReduced[[k,2]]},{k,1,Length[pmfCondBellCombinedReduced]}];
{pmfCondBellCombinedReduced//MatrixForm,pmfBellCombinedReducedProducts//MatrixForm};
correlationsBellReal=expectations[#]&/@{pmfBellCombinedReducedProducts[[{1,5,9,13}]],pmfBellCombinedReducedProducts[[{2,6,10,14}]],pmfBellCombinedReducedProducts[[{3,7,11,15}]],pmfBellCombinedReducedProducts[[{4,8,12,16}]]}//Simplify;
sBellCombinedReduced=sMatrix.correlationsBellReal//Simplify;

Check that the ((Subscript[A, 1]),(Subscript[A, 2]),(Subscript[B, 1]),(Subscript[B, 2])) and (A,B,(Subscript[D, A]),(Subscript[D, B])) forms of the Bell real models produce identical correlations. Check.

{matSigns.ρ16//MatrixForm,correlationsBellReal//MatrixForm}
matSigns.ρ16==correlationsBellReal

QM Model

Note: Be careful! The definition of the “qmConditionalProbs” assumes the ordering of tuples4 according to ((Subscript[D, A]),(Subscript[D, B]),A,B) ! That is ((Subscript[D, A]),(Subscript[D, B])) vary the slowest --(-1,-1),(-1,-1),(-1,-1),(-1,-1),(-1,1),... “qmProbs” and “pmfQM[[All,2}}” are NOT in the same order!

qmConditionalProbs={(1/2) ((Cos[(Subscript[θ, 1])])^2) ,(1/2) ((Sin[(Subscript[θ, 1])])^2) ,(1/2) ((Sin[(Subscript[θ, 1])])^2) ,(1/2) ((Cos[(Subscript[θ, 1])])^2) ,(1/2) ((Cos[(Subscript[θ, 2])])^2) ,(1/2) ((Sin[(Subscript[θ, 2])])^2) ,(1/2) ((Sin[(Subscript[θ, 2])])^2) ,(1/2) ((Cos[(Subscript[θ, 2])])^2) ,(1/2) ((Cos[(Subscript[θ, 3])])^2) ,(1/2) ((Sin[(Subscript[θ, 3])])^2) ,(1/2) ((Sin[(Subscript[θ, 3])])^2) ,(1/2) ((Cos[(Subscript[θ, 3])])^2) ,(1/2) ((Cos[(Subscript[θ, 4])])^2) ,(1/2) ((Sin[(Subscript[θ, 4])])^2) ,(1/2) ((Sin[(Subscript[θ, 4])])^2) ,(1/2) ((Cos[(Subscript[θ, 4])])^2) };
qmProbs=qmConditionalProbs*detectorProductProbsRepeated;
qmProbsRepeated=Flatten[Table[Table[qmProbs[[k]],{j,1,4}],{k,1,Length[qmProbs]}]];

pmfQM=SortBy[Table[{Permute[tuples4[[k]],Cycles[{{1,3},{2,4}}]],qmProbs[[k]]},{k,1,Length[tuples4]}],First];
pmfQMTable=Grid[Prepend[Table[Join[pmfQM[[k,1]],{pmfQM[[k,2]]}],{k,1,Length[pmfQM]}],{"A","B","(Subscript[D, A])","(Subscript[D, B])","QM pmf"}],Frame->All];
pmfQMBellTable=Grid[Prepend[SortBy[Table[Join[pmfQM[[k,1]],{pmfQM[[k,2]]},{pmfBellCombinedReduced[[k,2]]}],{k,1,Length[pmfQM]}],#[[{3,4}]]&],{"A","B","(Subscript[D, A])","(Subscript[D, B])","QM pmf","Bell real pmf"}],Frame->All];

QM correlations and s-functions

pmfCondQM=conditionalize[pmfQM,{1,2},{3,4}];
pmfQMProducts=Table[{pmfCondQM[[k,1,1]]*pmfCondQM[[k,1,2]],pmfCondQM[[k,2]]},{k,1,Length[pmfCondQM]}];
{pmfCondQM//MatrixForm,pmfQMProducts//MatrixForm}
correlationsQM=expectations[#]&/@{pmfQMProducts[[{1,5,9,13}]],pmfQMProducts[[{2,6,10,14}]],pmfQMProducts[[{3,7,11,15}]],pmfQMProducts[[{4,8,12,16}]]}//Simplify;
sMatrix.correlationsQM//Simplify

Note: Without a restriction such as (Subscript[θ, 1])=(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4]), that is, if these angles were independent free parameters, the max/min s-function values would be ±4 instead of ±2(Sqrt[2]), the so-called Tsirelson bound. Thus the Tsirelson bound is a consequence of the geometry of Fig. in Table 4, namely that in the set of four angle differences, one of them must be the sum of the other three.

Bell local model

aliceProbs ={(Subscript[α, 1]),(Subscript[α, 2]),(Subscript[α, 3]),(Subscript[α, 4])};
bobProbs={(Subscript[β, 1]),(Subscript[β, 2]),(Subscript[β, 3]),(Subscript[β, 4])};
term1=Table[Switch[tuples4[[k,{1,3}]],{-1,-1},(Subscript[α, 1]),{-1,1},(Subscript[α, 2]),{1,-1},(Subscript[α, 3]),{1,1},(Subscript[α, 4])],{k,1,Length[tuples4]}];
term2=Table[Switch[tuples4[[k,{2,4}]],{-1,-1},(Subscript[β, 1]),{-1,1},(Subscript[β, 2]),{1,-1},(Subscript[β, 3]),{1,1},(Subscript[β, 4])],{k,1,Length[tuples4]}];
term3=Table[Switch[tuples4[[k,{3,4}]],{-1,-1},(Subscript[δ, a])(Subscript[δ, b]),{-1,1},(Subscript[δ, a])(Subscript[δ, b']),{1,-1},(Subscript[δ, a'])(Subscript[δ, b]),{1,1},(Subscript[δ, a'])(Subscript[δ, b'])],{k,1,Length[tuples4]}];
aliceBobProbs=Table[term1[[k]]*term2[[k]]*term3[[k]],{k,1,Length[tuples4]}];

pmfBellLocal=Table[{tuples4[[k]],aliceBobProbs[[k]]},{k,1,Length[tuples4]}];
bellLocalProbs=pmfBellLocal[[All,2]];
pmfAlice=marginalize[pmfBellLocal,{1,3}];
pmfBob=marginalize[pmfBellLocal,{2,4}];
pmfBellLocalTable=Grid[Prepend[Table[Join[pmfBellLocal[[k,1]],{pmfBellLocal[[k,2]]}],{k,1,Length[pmfBellLocal]}],{"A","B","(Subscript[D, A])","(Subscript[D, B])","Bell local pmf"}],Frame->All];
pmfAliceTable=Grid[(Prepend[Table[Assuming[(Subscript[δ, b])+(Subscript[δ, b'])==1&&(Subscript[δ, a])+(Subscript[δ, a'])==1&&(Subscript[α, 1])+(Subscript[α, 3])==1&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0,Simplify[Join[pmfAlice[[k,1]],{pmfAlice[[k,2]]}]]]/.{(-1+(Subscript[α, 3]))->-(Subscript[α, 1]),(-1+(Subscript[δ, a']))->-(Subscript[δ, a]),(-1+(Subscript[α, 4]))->-(Subscript[α, 2])},{k,1,Length[pmfAlice]}],{"A","(Subscript[D, A])","(A,(Subscript[D, A])) marginal pmf"}]),Frame->All];
pmfBobTable=Grid[Prepend[Table[Assuming[(Subscript[δ, b])+(Subscript[δ, b'])==1&&(Subscript[δ, a])+(Subscript[δ, a'])==1&&(Subscript[α, 1])+(Subscript[α, 3])==1&&(-1+(Subscript[α, 4]))==-(Subscript[α, 2])&&(Subscript[α, 2])+(Subscript[α, 4])==1&&(Subscript[β, 1])+(Subscript[β, 3])==1&&(Subscript[β, 2])+(Subscript[β, 4])==1&&(Subscript[α, 1])≥0&&(Subscript[α, 2])≥0&&(Subscript[α, 3])≥0&&(Subscript[α, 4])≥0&&(Subscript[β, 1])≥0&&(Subscript[β, 2])≥0&&(Subscript[β, 3])≥0&&(Subscript[β, 4])≥0,Simplify[Join[pmfBob[[k,1]],{pmfBob[[k,2]]}]]]/.{(-1+(Subscript[β, 3]))->-(Subscript[β, 1]),(-1+(Subscript[δ, b']))->-(Subscript[δ, b]),(-1+(Subscript[β, 4]))->-(Subscript[β, 2])},{k,1,Length[pmfBob]}],{"B","(Subscript[D, B])","(B,(Subscript[D, B])) marginal pmf"}],Frame->All];
pmfQMBellLocalTable=Grid[Prepend[Table[Join[tuples4[[k]],{pmfQM[[k,2]]},{pmfBellCombinedReduced[[k,2]]},{pmfBellLocal[[k,2]]}],{k,1,Length[pmfQM]}],Text@Style[#,Bold,Blue]&/@{"A","B","(Subscript[D, A])","(Subscript[D, B])","QM pmf","Bell real pmf","Bell local pmf"}],Frame->All];

Bell local correlations and s-functions

Now compute the correlations and the s-functions. Check!

pmfCondBellLocal=(conditionalize[pmfBellLocal,{1,2},{3,4}]//Simplify)/.{(Subscript[α, 1])+(Subscript[α, 3])->1,(Subscript[α, 2])+(Subscript[α, 4])->1,(Subscript[β, 1])+(Subscript[β, 3])->1,(Subscript[β, 2])+(Subscript[β, 4])->1};
pmfBellLocalProducts=Table[{pmfCondBellLocal[[k,1,1]]*pmfCondBellLocal[[k,1,2]],pmfCondBellLocal[[k,2]]},{k,1,Length[pmfCondBellLocal]}];
{pmfCondBellLocal//MatrixForm,pmfBellLocalProducts//MatrixForm}
correlationsBellLocal=expectations[#]&/@{pmfBellLocalProducts[[{1,5,9,13}]],pmfBellLocalProducts[[{2,6,10,14}]],pmfBellLocalProducts[[{3,7,11,15}]],pmfBellLocalProducts[[{4,8,12,16}]]}//Simplify;
sBellLocal=sMatrix.correlationsBellLocal//Simplify;
%//Column

Minimal model

minimalConditionalProbs=ConstantArray[(1/4),16];
minimalProbs=minimalConditionalProbs*detectorProductProbsRepeated;
minimalProbsRepeated=Flatten[Table[Table[minimalProbs[[k]],{j,1,4}],{k,1,Length[minimalProbs]}]];

pmfMinimal=SortBy[Table[{Permute[tuples4[[k]],Cycles[{{1,3},{2,4}}]],minimalProbs[[k]]},{k,1,Length[tuples4]}],First];
pmfMinimalTable=Grid[Prepend[Table[Join[pmfMinimal[[k,1]],{pmfMinimal[[k,2]]}],{k,1,Length[pmfMinimal]}],{"A","B","(Subscript[D, A])","(Subscript[D, B])","Minimal pmf"}],Frame->All]

Minimal model correlations and s-functions

pmfCondMinimal=conditionalize[pmfMinimal,{1,2},{3,4}];
pmfMinimalProducts=Table[{pmfCondMinimal[[k,1,1]]*pmfCondMinimal[[k,1,2]],pmfCondMinimal[[k,2]]},{k,1,Length[pmfCondMinimal]}];
{pmfCondMinimal//MatrixForm,pmfMinimalProducts//MatrixForm}
correlationsMinimal=expectations[#]&/@{pmfMinimalProducts[[{1,5,9,13}]],pmfMinimalProducts[[{2,6,10,14}]],pmfMinimalProducts[[{3,7,11,15}]],pmfMinimalProducts[[{4,8,12,16}]]}//Simplify
sMatrix.correlationsMinimal//Simplify

Put it all together in a table

modelsTable=Grid[Append[Prepend[Prepend[Table[Join[tuples4[[k]],{pmfMaximal[[k,2]]},{pmfQM[[k,2]]},{pmfBellCombinedReduced[[k,2]]},{pmfBellLocal[[k,2]]},{pmfMinimal[[k,2]]}],{k,1,Length[tuples4]}],Text@Style[#,Bold,Blue]&/@{"A","B","(Subscript[D, A])","(Subscript[D, B])","Maximal","QM","Bell real","Bell local","Minimal"}],{Text@Style["Outcomes",Bold,Red],SpanFromLeft,SpanFromLeft,SpanFromLeft,Text@Style["The five model pmfs",Bold,Red],SpanFromLeft}],{Text@Style["Parameter \ncontsraints",Bold,Blue],SpanFromLeft,SpanFromLeft,SpanFromLeft,"(Subscript[γ, k])≥0 & ∑(Subscript[γ, k])=1","(Subscript[θ, 1])=(Subscript[θ, 2])+(Subscript[θ, 3])+(Subscript[θ, 4]) \n(Subscript[δ, a])+(Subscript[δ, a'])=1, (Subscript[δ, b])+(Subscript[δ, b'])=1","(Subscript[ρ, k])≥0 & ∑(Subscript[ρ, k])=1 \n(Subscript[δ, a])+(Subscript[δ, a'])=1, (Subscript[δ, b])+(Subscript[δ, b'])=1","(Subscript[α, k])≥0, (Subscript[β, k])≥0 \n(Subscript[α, 1])+(Subscript[α, 3])=(Subscript[β, 1])+(Subscript[β, 3])=1 \n(Subscript[α, 2])+(Subscript[α, 4])=(Subscript[β, 2])+(Subscript[β, 4])=1 \n(Subscript[δ, a])+(Subscript[δ, a'])=1, (Subscript[δ, b])+(Subscript[δ, b'])=1","(Subscript[δ, a])+(Subscript[δ, a'])=1, (Subscript[δ, b])+(Subscript[δ, b'])=1"}],Frame->All]

A useful matrix for solving for α’s and β’s in terms of the ρ’s

αMat={{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},{0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1},{0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1}};
%//MatrixForm
βMat={{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0},{0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1},{0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1}};
%//MatrixForm
αβMat=Join[βMat,αMat,{{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}}];
%//MatrixForm

subsTest={(Subscript[ρ, 1])->.1,(Subscript[ρ, 2])->.2,(Subscript[ρ, 5])->.3,(Subscript[ρ, 6])->.2,(Subscript[ρ, 3])->0,(Subscript[ρ, 4])->0,(Subscript[ρ, 7])->0,(Subscript[ρ, 8])->0,(Subscript[ρ, 9])->.05, (Subscript[ρ, 10])->.05,(Subscript[ρ, 11])->0,(Subscript[ρ, 12])->0,(Subscript[ρ, 13])->.05,(Subscript[ρ, 14])->.05,(Subscript[ρ, 15])->0,(Subscript[ρ, 16])->0};
sBell.(correlationsBellReal/.subsTest)//Chop

Bell real case.

Matrix to reconstruct the Bell real versions ((Subscript[ρ, k])’s) corresponding to QM probabilities (more particular and specific than just the correlations).

tmpB=(conditionalize[pmfBellCombinedReduced,{1,2},{3,4}]/.{Total[ρ16]->1})[[All,2]];
ind=Partition[Flatten[Table[Position[ρ16,#]&/@Table[tmpB[[k,j]],{j,1,4}],{k,1,Length[tmpB]}]],4];
matR=Table[Normal[SparseArray[ind[[k]]->{1,1,1,1},16]],{k,1,Length[ind]}];
matRT=Transpose[matR];
{matR//MatrixForm,MatrixRank[matR]}
{matRT//MatrixForm,MatrixRank[matRT]}

Here is the reconstruction of the probs in the pmfBellCombinedReduced pmf -- conditional since we have divided out the detector setting probs.

matR.ρ16//MatrixForm

Now compute the correlations using a modified version of this matrix. Note also that a row of 1’s has been added to the bottom.

matRPlusMinus={matR[[1]]-matR[[4]],matR[[2]]-matR[[3]],matR[[5]]-matR[[8]],matR[[6]]-matR[[7]],matR[[9]]-matR[[12]],matR[[10]]-matR[[11]],matR[[13]]-matR[[16]],matR[[14]]-matR[[15]],ConstantArray[1,16]};
%//MatrixForm
%%.ρ16//MatrixForm
MatrixRank[matRPlusMinus]

Compare LinearSolve “solution” to well--known minimal solution.

Here is a useful matrix that basically captures the signs in the correlation matrix. Easy way to get the Bell correlations --

matSigns={{1,1,-1,-1,1,1,-1,-1,-1,-1,1,1,-1,-1,1,1},{1,-1,1,-1,1,-1,1,-1,-1,1,-1,1,-1,1,-1,1},{1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1},{1,-1,1,-1,-1,1,-1,1,1,-1,1,-1,-1,1,-1,1}};
matSigns//MatrixForm
matSigns.ρ16//MatrixForm
matSigns.ρ16==correlationsBellReal

From Bellevue talk “Bell Theorem Update 4”

ptsZPlus1={{w,1,1},{-1,-w,1},{1,w,1},{-w,-1,1}};
ptsZMinus1={{w,-1,-1},{1,-w,-1},{-1,w,-1},{-w,1,-1}};
ptsXPlus1={{1,w,1},{1,-1,-w},{1,1,w},{1,-w,-1}};
ptsXMinus1={{-1,w,-1},{-1,1,-w},{-1,-1,w},{-1,-w,1}};
ptsYPlus1={{w,1,1},{-1,1,-w},{1,1,w},{-w,1,-1}};
ptsYMinus1={{w,-1,-1},{1,-1,-w},{-1,-1,w},{-w,-1,1}};

Define some points in 3D and 4D space corresponding to fixed values of the first coordinate w.

ptsw3=DeleteDuplicates[Join[ptsZPlus1,ptsZMinus1,ptsXPlus1,ptsXMinus1,ptsYPlus1,ptsYMinus1]];
%//MatrixForm
ptsw4=Prepend[#,w]&/@ptsw3;
%//MatrixForm
ptsw3Plus1=DeleteDuplicates[ptsw3/.{w->1}];
%//MatrixForm
ptsw3Minus1=DeleteDuplicates[ptsw3/.{w->-1}];
%//MatrixForm
ptsw4Plus1=DeleteDuplicates[ptsw4/.{w->1}];
%//MatrixForm
ptsw4Minus1=DeleteDuplicates[ptsw4/.{w->-1}];
%//MatrixForm

For sufficiency part of Fine Theorem proof: Determination of (Subscript[ρ, k])’s that reproduce (Subscript[V, 4]).

lst={};
Table[rSubsTmp=ReplacePart[ρ0,{{i,2}->1/2(1+w),{j,2}->1/2(1-w)}];
corr=correlationsBellReal/.rSubsTmp//Simplify;
pos=Position[ptsw4,corr];
If[pos≠{},AppendTo[lst,{pos[[1,1]],{i,j},corr}]],{i,1,Length[ρ0]-1},{j,i,Length[ρ0]}];
lstTmp=Sort[lst,#1[[1]]<#2[[1]]&];
lst1=Table[lstTmp[[k]],{k,1,Length[lstTmp]-1,2}];
ρSubs1=Table[ReplacePart[ρ0,{{lst1[[k,2,1]],2}->1/2(1+w),{lst1[[k,2,2]],2}->1/2(1-w)}],{k,1,Length[lst1]}];

lst={};
Table[rSubsTmp=ReplacePart[ρ0,{{i,2}->1/2(1-w),{j,2}->1/2(1+w)}];
corr=correlationsBellReal/.rSubsTmp//Simplify;
pos=Position[ptsw4,corr];
If[pos≠{},AppendTo[lst,{pos[[1,1]],{i,j},corr}]],{i,1,Length[ρ0]-1},{j,i,Length[ρ0]}];
lstTmp=Sort[lst,#1[[1]]<#2[[1]]&];
lst2=Table[lstTmp[[k]],{k,1,Length[lstTmp]-1,2}];
ρSubs2=Table[ReplacePart[ρ0,{{lst2[[k,2,1]],2}->1/2(1-w),{lst2[[k,2,2]],2}->1/2(1+w)}],{k,1,Length[lst2]}];

Case w=-1:

ρ16WMinus1Subs1=DeleteDuplicates[(ρ16/.#&)/@ρSubs1/.{w->-1}];
ρ16WMinus1Subs1//MatrixForm
correlationsWMinus1Subs1=DeleteDuplicates[(correlationsBellReal/.#&)/@ρSubs1/.{w->-1}];
extPtsWMinus1Subs1=DeleteDuplicates[ptsw4/.{w->-1}];
{correlationsWMinus1Subs1//MatrixForm,
extPtsWMinus1Subs1//MatrixForm}

Case -1≤w≤1:

ρ16Subs1=(ρ16/.#&)/@ρSubs1;
ρ16Subs1//MatrixForm
correlationsSubs1=(correlationsBellReal/.#&)/@ρSubs1//Simplify;
{correlationsSubs1//MatrixForm,ptsw4//MatrixForm}

r matrix and correlations using ρSubs2. Compare to similar structures above using ρSubs1. ρ16Subs1 has all zeros in the last two columns whereas ρ16Subs2 has all 0’s in the first two columns.

ρ16Subs2=(ρ16/.#&)/@ρSubs2;
ρ16Subs2//MatrixForm
correlationsSubs2=(correlationsBellReal/.#&)/@ρSubs2//Simplify;
{correlationsSubs2//MatrixForm,ptsw4//MatrixForm}

Case w=1: (note we use ρSubs2 here as opposed to ρSubs1 -- ρSubs1 requires 6 (Subscript[r, k])  pmf’s instead of 4)

ρ16WPlus1Subs2=DeleteDuplicates[(ρ16/.#&)/@ρSubs2/.{w->1}];
ρ16WPlus1Subs2//MatrixForm
correlationsWPlus1Subs2=DeleteDuplicates[(correlationsBellReal/.#&)/@ρSubs2/.{w->1}];
extPtsWPlus1Subs2=DeleteDuplicates[ptsw4/.{w->1}];
{correlationsWPlus1Subs2//MatrixForm,
extPtsWPlus1Subs2//MatrixForm}

Some solutions and substitution lists

Let  p=(({{w}, {x}, {y}, {z}}))=(({{Cos[2 (Subscript[θ, 1])]}, {Cos[2 (Subscript[θ, 2])]}, {Cos[2 (Subscript[θ, 3])]}, {Cos[2 (Subscript[θ, 4])]}}))  where (Subscript[θ, 1])=20°,(Subscript[θ, 2])=30°,(Subscript[θ, 3])=-50°,(Subscript[θ, 4])=40°. Then p=(({{0.766044443118978`}, {0.5`}, {-0.17364817766693033`}, {0.17364817766693033`}})) and the QM S-test functions result in (({{-0.266044443118978`}, {0.266044443118978`}, {1.6133407984528385`}, {0.9187480877851173`}})), with all values between -2 and +2, so there is a solution.

Solve for the α coefficients that make p=α V.

(ptsw4/.{w->Cos[40Degree]});
%//N
correlationsQM;
correlationsQM/.subsθSum/.{(Subscript[θ, 2])->30 Degree,(Subscript[θ, 3])->-50 Degree,(Subscript[θ, 4])->40 Degree};
%//N//MatrixForm
sBell.%;
%//N//MatrixForm

(ptsw4/.{w->Cos[40 Degree]})
(correlationsQM/.subsθSum/.{(Subscript[θ, 2])->30 Degree,(Subscript[θ, 3])->-50 Degree,(Subscript[θ, 4])->40 Degree})
α12.(ptsw4/.{w->Cos[40 Degree]})==(correlationsQM/.subsθSum/.{(Subscript[θ, 2])->30 Degree,(Subscript[θ, 3])->50 Degree,(Subscript[θ, 4])->40 Degree})

α12={(Subscript[α, 1]),(Subscript[α, 2]),(Subscript[α, 3]),(Subscript[α, 4]),(Subscript[α, 5]),(Subscript[α, 6]),(Subscript[α, 7]),(Subscript[α, 8]),(Subscript[α, 9]),(Subscript[α, 10]),(Subscript[α, 11]),(Subscript[α, 12])};
solα=Assuming[Total[α12]==1,Solve[(α12.(ptsw4/.{w->Cos[40 Degree]}))==(correlationsQM/.subsθSum/.{(Subscript[θ, 2])->30 Degree,(Subscript[θ, 3])->-50 Degree,(Subscript[θ, 4])->40 Degree}),α12]]//Simplify

Simplify this solution by making all (Subscript[α, k])’s = 0 for k≤8. Note that the four remaining coefficients are non-negative and sum to 1.

solαSimple=(solα/.Table[(Subscript[α, k])->0,{k,1,8}])[[1]];
%//N
Total[α12/.solα[[1]]/.Table[(Subscript[α, k])->0,{k,1,8}]]//Simplify

Create the substitution list for α. Note the resulting coefficients are non-negative and sum to 1.

subsαSimple=Join[solαSimple,Table[(Subscript[α, k])->0,{k,1,8}]]
α12/.subsαSimple
%//N
Total[%%]//Simplify

Now solve for the (Subscript[r, k])’s that will reproduce the given QM  correlations. Note they are non-negative and sum to 1.

ρSol1=((α12/.solα[[1]]/.Table[(Subscript[α, k])->0,{k,1,8}]).ρ16Subs1)/.{w->Cos[40 Degree]}
%//N
Total[%%]//Simplify

Create the corresponding substitution list for the (Subscript[r, k])’s.

subsQSol=Table[(Subscript[ρ, k])->ρSol1[[k]],{k,1,Length[ρ16]}];

Finally check that the QM correlations have been reproduced within the Bell real model. Also check that the S-test functions are between -2 and +2. Check!!

correlationsBellReal/.subsQSol//Simplify
correlationsQM/.subsθSum/.{(Subscript[θ, 2])->30 Degree,(Subscript[θ, 3])->-50 Degree,(Subscript[θ, 4])->40 Degree}
%==%%
sBell.%%%//N

Plotting

bellRealVertices[ww_]:=Graphics3D[{Red,PointSize[Medium],Point[Rest[#]&/@ptsw4/.{w->ww}]}];
bellLocalVertices[ww_]:=Graphics3D[{Red,PointSize[Medium],Point[Select[Rest[#]&/@ptsw4,#[[3]]==1||#[[3]]==-1&]/.{w->ww}]}];

threeDRegions=Table[DeleteDuplicates[Rest[#]&/@(ptsw4/.{w->k})],{k,{-1,-1/2,0,1/2,1}}];
(#//MatrixForm)&/@threeDRegions
bellRealPlots=ConvexHullMesh[#,(*MeshCellStyle->{{0,All}->{PointSize[Medium],Red}},*)Ticks->{{-1,0,1},{-1,0,1},{-1,0,-1}},FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}}(*,  PlotTheme->"Detailed"*)]&/@threeDRegions;
cvhSeeThruPlots=ConvexHullMesh[#,MeshCellStyle->{{2,All}->Opacity[0.5,LightBlue](*,{0,All}->{PointSize[Medium],Red}*)},Ticks->{{-1,0,1},{-1,0,1},{-1,0,-1}},FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}}]&/@threeDRegions;

qmExtremePt={Cos[2(Subscript[θ, 1])],Cos[2(Subscript[θ, 2])],Cos[2(Subscript[θ, 3])] ,Cos[2(Subscript[θ, 4])]}/.{(Subscript[θ, 1])->22.5 Degree,(Subscript[θ, 2])->67.5 Degree,(Subscript[θ, 3])->22.5 Degree,(Subscript[θ, 4])->22.5 Degree};
qmExtremePtPlot=Graphics3D[{Purple,PointSize[Large],Point[Rest[qmExtremePt]]}];
cvhPlotExt=ConvexHullMesh[DeleteDuplicates[Rest[#]&/@(ptsw4/.{w->1/2Sqrt[2]})],(*MeshCellStyle->{{0,All}->{PointSize[Large],Red}},*)  PlotTheme->"Detailed"];
cvhPlotNotExt=ConvexHullMesh[DeleteDuplicates[Rest[#]&/@(ptsw4/.{w->-1})],(*MeshCellStyle->{{0,All}->{PointSize[Large],Red}},*)  PlotTheme->"Detailed"];
comparoPlot1=Show[ParametricPlot3D[{ Cos[2 (α-γ-δ)], Cos[2 γ], Cos[2 δ]}/.{α->22.5 Degree},{γ,-π/2,π/2},{δ,-π/2,π/2},PlotStyle->{Yellow,Opacity[0.4]},Mesh->None],cvhPlotExt,qmExtremePtPlot,PlotLabel->"Bell region and QM contour with w=cos(45°)=1/2(Sqrt[2]) \nshowing extreme QM point (purple dot)"];
comparoPlot2=Show[ParametricPlot3D[{ Cos[2 (α-γ-δ)], Cos[2 γ], Cos[2 δ]}/.{α->22.5 Degree},{γ,-π/2,π/2},{δ,-π/2,π/2},PlotStyle->{Yellow,Opacity[0.4]},Mesh->None],cvhPlotNotExt,qmExtremePtPlot,PlotLabel->"Bell region with w=-1 and QM contour with w=cos(45°)=1/2\!(Sqrt[2]) \nNote extreme QM point (purple dot) is inside the tetrahedron"];

angleSubs=1/2ArcCos[#]&/@{-1,-1/2(Sqrt[2]),-1/2,0,1/2,1/2(Sqrt[2]),1};

qmParamPlots=Table[ParametricPlot3D[{ Cos[2 (α-γ-δ)], Cos[2 γ], Cos[2 δ]}/.{α->angle},{γ,-π/2,π/2},{δ,-π/2,π/2}],{angle,{-π/2,-π/4,-π/8,0,π/8,π/4,π/2}}];

qmSeeThruPlots=Table[ParametricPlot3D[{ Cos[2 (α-γ-δ)], Cos[2 γ], Cos[2 δ]}/.{α->angle},{γ,-π/2,π/2},{δ,-π/2,π/2},AxesLabel->{"<(Subscript[X, 2])>","<(Subscript[X, 3])>","<(Subscript[X, 4])>"},PlotStyle->{Yellow,Opacity[0.4]},Mesh->None,(*PlotLabel->"QM expectations \n3D-slice at <(Subscript[X, 1])> =  cos("<>ToString[2angle/N[Degree]]<> "°) = "<>ToString[N[Cos[2angle ]]//Chop],*)PlotPoints->50,ImageSize->300],{angle,{-π/2,-π/4,-π/8,0,π/8,π/4,π/2}}];

qmPlots=Table[ParametricPlot3D[{ Cos[2 (α-γ-δ)], Cos[2 γ], Cos[2 δ]}/.{α->angle},{γ,-π/2,π/2},{δ,-π/2,π/2},Ticks->{{-1,0,1},{-1,0,1},{-1,0,-1}},ImageSize->200,FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}}],{angle,{π/2,π/3,π/4,π/6,0}}];
qmSeeThruPlots2=Table[ParametricPlot3D[{ Cos[2 (α-γ-δ)], Cos[2 γ], Cos[2 δ]}/.{α->angle},{γ,-π/2,π/2},{δ,-π/2,π/2},(*AxesLabel->{"x","y>","z"},*)Ticks->{{-1,0,1},{-1,0,1},{-1,0,-1}},FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}},PlotStyle->{Yellow,Opacity[0.4]},Mesh->None,(*PlotLabel->"QM and Bell correlations \n3D-slice at w = cos("<>ToString[2angle/N[Degree]]<> "°) = "<>ToString[N[Cos[2angle ]]//Chop],*)PlotPoints->50,ImageSize->200],{angle,{π/2,π/3,π/4,π/6,0}}];

ptMaxQM=Graphics3D[{Purple,PointSize[Large],Point[{22.5 ,22.5 ,2(Sqrt[2])}]}];
planeOneHalfAchievable=Show[ParametricPlot3D[{x,y,2},{x,-90,0},{y,-90-x,x+90},PlotStyle->Green],ParametricPlot3D[{x,y,2},{x,0,90},{y,x-90,90-x},PlotStyle->Green]];

expQM[θ2_,θ3_,θ4_]:={Cos[2(θ2+θ3+θ4)],Cos[2θ2],Cos[2θ3],Cos[2θ4]};
sQM[θ2_,θ3_,θ4_]:={-1,1,1,1}.expQM[θ2,θ3,θ4];
sQMDeg[θ2_,θ3_,θ4_]:=sQM[θ2 Degree,θ3 Degree,θ4 Degree];

comparoPlot3=Show[ParametricPlot3D[{γ,δ,sQMDeg[22.5 ,γ,δ]},{γ,-90,90},{δ,-90,90},PlotPoints->35],
ContourPlot3D[sQMDeg[22.5 ,γ,δ]==2,{γ,-90,90},{δ,-90,90},{z,2-.01,2+.01}],ptMaxQM,planeOneHalfAchievable,AxesLabel->{"(Subscript[θ, 3])°","(Subscript[θ, 4])°","(Subscript[S, 1])"},PlotRange->All,BoxRatios->{1,1,0.5},FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}},PlotLabel->"Green rectangle: Max Bell (Subscript[S, 1])=2 \nPurple dot: Max QM (Subscript[S, 1]) = 2(Sqrt[2])  \nYellow surface: QM (Subscript[S, 1]) values for (Subscript[Q, 11])=-(1/(Sqrt[2]))  \n Black curve (intersection): Simultaneous Bell & QM achievable value 2 ",ImageSize->500];

subsαβ={{{(Subscript[α, 1])->1,(Subscript[β, 1])->0},{(Subscript[α, 1])->0,(Subscript[β, 1])->1}},{{(Subscript[α, 1])->1,(Subscript[β, 1])->1/4},{(Subscript[α, 1])->1/4,(Subscript[β, 1])->1}},{{(Subscript[α, 1])->(1/2),(Subscript[β, 1])->1},{(Subscript[α, 1])->1,(Subscript[β, 1])->1/2},{(Subscript[α, 1])->(1/2),(Subscript[β, 1])->1/2}},{{(Subscript[α, 1])->1,(Subscript[β, 1])->(3/4)},{(Subscript[α, 1])->3/4,(Subscript[β, 1])->1}},{{(Subscript[α, 1])->1,(Subscript[β, 1])->1},{(Subscript[α, 1])->0,(Subscript[β, 1])->0}}};
subsαβReduce={(Subscript[α, 3])->1-(Subscript[α, 1]),(Subscript[α, 4])->1-(Subscript[α, 2]),(Subscript[β, 3])->1-(Subscript[β, 1]),(Subscript[β, 4])->1-(Subscript[β, 2])};
corrBellReduce=correlationsBellLocal/.subsαβReduce;
corrBellTable=(Table[corrBellReduce/.subsαβ[[k]],{k,1,Length[subsαβ]}])//Simplify;
(*Keep the Cosine version around in case it's needed later. However, it needs to be updated to include the multiple 'sheets' that the revised version revealed.*)
(*bellLocalPlots=ParametricPlot3D[{{Cos[2 θ2 Degree],Cos[2θ3 Degree],-Cos[2θ2 Degree] Cos[2θ3 Degree]},{Cos[2 θ2 Degree],-Cos[2θ2 Degree] Cos[2θ3 Degree],Cos[2 θ3 Degree]},{-Cos[2θ2 Degree] Cos[2θ3 Degree],Cos[2 θ2 Degree],Cos[2θ3 Degree]}},{θ2,0,90},{θ3,0,90},AxesLabel->{"x","y","z"},FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}}];*)
bellLocalPlots=Table[Show[ParametricPlot3D[Table[Rest[corrBellTable[[k,j]]],{j,1,Length[corrBellTable[[k]]]}],{(Subscript[α, 2]),0,1},{(Subscript[β, 2]),0,1},Ticks->{{-1,0,1},{-1,0,1},{-1,0,-1}},PlotStyle->Green,PlotRange->{{-1,1},{-1,1},{-1,1}},FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}}](*,bellLocalVertices[{-1,-0.5,0,0.5,1}[[k]]]*)],{k,1,Length[subsαβ]}];

maximalSeeThruPlot=RegionPlot3D[-1≤x≤1&&-1≤y≤1&&-1≤z≤1,{x,-1,1},{y,-1,1},{z,-1,1},Ticks->{{-1,0,1},{-1,0,1},{-1,0,-1}},(*AxesLabel->{"x","y","z"},*)PlotStyle->{LightGreen,Opacity[0.2]},Mesh->None,PlotRange->{{-1,1},{-1,1},{-1,1}},FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}}];
minimalPlot=Graphics3D[{Red,PointSize[Large],Point[{0,0,0}]},PlotRange->{{-1,1},{-1,1},{-1,1}},Axes->True,Ticks->{{-1,0,1},{-1,0,1},{-1,0,-1}},(*AxesLabel->{"x","y","z"},*)FaceGrids->{{0,1,0},{-1,0,0},{0,0,-1}}];

Create the graphics for the lines that these correlations have in common when w=0.

ptsQMBellLocal={{0,0,0,0},{0,0,-1,-1},{0,0,-1,1},{0,0,1,-1},{0,0,1,1},{0,-1,0,-1},{0,-1,0,1},{0,1,0,-1},{0,1,0,1}};
linesQMBellLocal=Table[{{0,0,0},Rest[ptsQMBellLocal[[k]]]},{k,2,Length[ptsQMBellLocal]}];
plotQMBellLocalLines=Graphics3D[{Red,Thickness[.01],Line[#]}]&/@linesQMBellLocal;

QM s-function max/min calculations

dsQM=D[sQM[θ2,θ3,θ4],{{θ2,θ3,θ4}}];
ds2QM=D[sQM[θ2,θ3,θ4],{{θ2,θ3,θ4},2}];
dsQMSubsGeneric=Solve[dsQM=={0,0,0},{θ2,θ3,θ4}]//Simplify;
dsQMSubs=dsQMSubsGeneric/.{C[1]->0,C[2]->0,C[3]->0}
dsQMSubsGenericα0=Solve[(dsQM/.{θ2+θ3+θ4->0})=={0,0,0},{θ2,θ3,θ4}];
dsQMSubsα0=dsQMSubsGenericα0/.{C[1]->0,C[2]->0,C[3]->0}

Checks

dsQM/.#&/@dsQMSubs//Simplify
ds2QM/.#&/@dsQMSubs//Simplify

Find the critical points and the extreme points.

qmCritPts=Table[{θ2,θ3,θ4}/.dsQMSubs[[k]],{k,1,Length[dsQMSubs]}]
qmExtremePts=Table[Join[Rest[qmCritPts[[k]]],{sQM@@qmCritPts[[k]]}],{k,1,Length[qmCritPts]}]

Evaluate the QM expectations at the critical points. Same for (Subscript[s, QM]).

qmExpCrit=correlationsQM/.{(Subscript[θ, 1])->θ2+θ3+θ4,(Subscript[θ, 2])->θ2,(Subscript[θ, 3])->θ3,(Subscript[θ, 4])->θ4}/.#&/@dsQMSubs
qmSCrit=sQM[θ2,θ3,θ4]/.#&/@dsQMSubs

Eight are local extrema (±1/2) and 4 are global extrema (±1/2(Sqrt[2])). 
Note that all global extrema are at points where (Subscript[θ, 2])=(Subscript[θ, 3])=(Subscript[θ, 4]), and are odd multiples of π/8:

dsQMSubs[[5;;8]]
Table[Cos[2(θ2+θ3+θ4)]/.%[[k]],{k,1,Length[%]}]
%//N

Check to see whether the Hessians are positive (negative) semi-definite or even positive (negative) definite. Thus the 5th and 8th (positive definite) indicate minima and the 6th and 7th (negative definite) indicate maxima.

PositiveSemidefiniteMatrixQ[#]&/@(ds2QM/.#&/@dsQMSubs)
PositiveDefiniteMatrixQ[#]&/@(ds2QM/.#&/@dsQMSubs)
NegativeSemidefiniteMatrixQ[#]&/@(ds2QM/.#&/@dsQMSubs)
NegativeDefiniteMatrixQ[#]&/@(ds2QM/.#&/@dsQMSubs)

The positive definite ones are the 5th and 8th solutions.  Both are at the global minima -2(Sqrt[2]).

dsQMSubs[[{5,8}]]
{-1,1,1,1}.correlationsQM/.{(Subscript[θ, 1])->θ2+θ3+θ4,(Subscript[θ, 2])->θ2,(Subscript[θ, 3])->θ3,(Subscript[θ, 4])->θ4}/.#&/@%
(ds2QM/.#&/@dsQMSubs)[[6]]//MatrixForm
(ds2QM/.#&/@dsQMSubs)[[7]]//MatrixForm

The negative definite ones are the 6th and 7th solutions.  Both are at the global maxima 2(Sqrt[2]).

dsQMSubs[[{6,7}]]
{-1,1,1,1}.correlationsQM/.{(Subscript[θ, 1])->θ2+θ3+θ4,(Subscript[θ, 2])->θ2,(Subscript[θ, 3])->θ3,(Subscript[θ, 4])->θ4}/.#&/@%
(ds2QM/.#&/@dsQMSubs)[[5]]//MatrixForm
(ds2QM/.#&/@dsQMSubs)[[8]]//MatrixForm

Where are these points among the QM correlations? 
First points (they are repeated) are min points.
Second points (they are repeated) are max points.

correlationsQM/.{(Subscript[θ, 1])->θ2+θ3+θ4,(Subscript[θ, 2])->θ2,(Subscript[θ, 3])->θ3,(Subscript[θ, 4])->θ4}/.#&/@dsQMSubs[[{5,8}]]
correlationsQM/.{(Subscript[θ, 1])->θ2+θ3+θ4,(Subscript[θ, 2])->θ2,(Subscript[θ, 3])->θ3,(Subscript[θ, 4])->θ4}/.#&/@dsQMSubs[[{6,7}]]

Define list of extreme points -- full 4D version:

qmExtremePts4D=Table[{θ2+θ3+θ4,θ2,θ3,θ4}/.dsQMSubs[[k]],{k,1,Length[dsQMSubs]}]

Outcome Pads

realmOutcomes=Riffle[Permutations[{1,0,0,0}],Permutations[{-1,0,0,0}]];
realmOutcomes//MatrixForm

pad1Plus={"1","","",""};
pad1Minus={"-1","","",""};
pad2Plus={"","1","",""};
pad2Minus={"","-1","",""};
pad3Plus={"","","1",""};
pad3Minus={"","","-1",""};
pad4Plus={"","","","1"};
pad4Minus={"","","","-1"};
outcomes={pad1Plus,pad1Minus,pad2Plus,pad2Minus,pad3Plus,pad3Minus,pad4Plus,pad4Minus};randomOutcomes=Table[Prepend[RandomChoice[outcomes],Text@Style[ToString[k],Bold,Blue]],{k,1,10}];

pad0=Join[{Text@Style["Trial",Bold,Blue]},Map[Text@Style[#,Bold,Red]&,{"1","2","3","4"}]];
pad02=Join[{Text@Style["Iteration",Bold,Blue]},Map[Text@Style[#,Bold,Red]&,{"1","2","3","4"}]];
pad00={"",Text@Style["Door",Bold,Red],SpanFromLeft};
pad00DetectorSetting={"",Text@Style["Measurement setting",Bold,Red],SpanFromLeft};
pad000={Text@Style["Outcomes",Bold,Purple],SpanFromLeft};
padDots={"...","...","...","...","..."};

padN=Prepend[RandomChoice[outcomes],Text@Style["N",Bold,Blue]];
padSums=Map[Text@Style[#,Bold,Black]&,{"Sums","((Subscript[N, 1]^(+)),(Subscript[N, 1]^(-)))","((Subscript[N, 2]^(+)),(Subscript[N, 2]^(-)))","((Subscript[N, 3]^(+)),(Subscript[N, 3]^(-)))","((Subscript[N, 4]^(+)),(Subscript[N, 4]^(-)))"}];
randomOutcomes=Table[Prepend[RandomChoice[outcomes],Text@Style[ToString[k],Bold,Blue]],{k,1,10}];
sumsObserverPad=Grid[Prepend[Append[Append[Append[Prepend[Prepend[randomOutcomes,pad0],pad00],padDots],padN],padSums],pad000],Frame->All];

padExpectations=Map[Text@Style[#,Bold,Black]&,{"Correlation estimates","(((Subscript[N, 1]^(+)) - (Subscript[N, 1]^(-)))/((Subscript[N, 1]^(+)) + (Subscript[N, 1]^(-))))","(((Subscript[N, 2]^(+)) - (Subscript[N, 2]^(-)))/((Subscript[N, 2]^(+)) + (Subscript[N, 2]^(-))))","(((Subscript[N, 3]^(+)) - (Subscript[N, 3]^(-)))/((Subscript[N, 3]^(+)) + (Subscript[N, 3]^(-))))","(((Subscript[N, 4]^(+)) - (Subscript[N, 4]^(-)))/((Subscript[N, 4]^(+)) + (Subscript[N, 4]^(-))))"}];
padExpectations2=Map[Text@Style[#,Bold,Black]&,{"Correlation \nestimates","(Subscript[(OverHat[μ]), 1])=(((Subscript[N, 1]^(+)) - (Subscript[N, 1]^(-)))/N)","(Subscript[(OverHat[μ]), 2])=(((Subscript[N, 2]^(+)) - (Subscript[N, 2]^(-)))/N)","(Subscript[(OverHat[μ]), 3])=(((Subscript[N, 3]^(+)) - (Subscript[N, 3]^(-)))/N)","(Subscript[(OverHat[μ]), 4])=(((Subscript[N, 4]^(+)) - (Subscript[N, 4]^(-)))/N)"}];
padS2={Text@Style["(OverHat[S]) ",Bold,Black],Text@Style["(Subscript[(OverHat[μ]), 1])-(Subscript[(OverHat[μ]), 2])+(Subscript[(OverHat[μ]), 3])+(Subscript[(OverHat[μ]), 4])",Bold,Black],SpanFromLeft};
padS={Text@Style["s estimate",Bold,Black],Text@Style["(OverHat[s])=(((Subscript[N, 1]^(+)) - (Subscript[N, 1]^(-)))/((Subscript[N, 1]^(+)) + (Subscript[N, 1]^(-))))-(((Subscript[N, 2]^(+)) - (Subscript[N, 2]^(-)))/((Subscript[N, 2]^(+)) + (Subscript[N, 2]^(-))))+(((Subscript[N, 3]^(+)) - (Subscript[N, 3]^(-)))/((Subscript[N, 3]^(+)) + (Subscript[N, 3]^(-))))+(((Subscript[N, 4]^(+)) - (Subscript[N, 4]^(-)))/((Subscript[N, 4]^(+)) + (Subscript[N, 4]^(-))))",Bold,Black],SpanFromLeft};

randomOutcomes=Table[Prepend[RandomChoice[outcomes],Text@Style[ToString[k],Bold,Blue]],{k,1,10}];
randomOutcomes2=Table[Prepend[RandomChoice[realmOutcomes],Text@Style[ToString[k],Bold,Blue]],{k,1,10}];
padN2=Prepend[RandomChoice[realmOutcomes],Text@Style["N",Bold,Blue]];
sPad=Grid[Prepend[Append[Append[Append[Append[Append[Prepend[Prepend[randomOutcomes,pad0],pad00DetectorSetting],padDots],padN],padSums],padExpectations],padS],pad000],Frame->All];
sPadBrief=Grid[Prepend[Append[Append[Append[Prepend[Prepend[randomOutcomes,pad0],pad00],padDots],padN],padSums],pad000],Frame->All];
sPad2=Grid[Prepend[Append[Append[Append[Append[Append[Prepend[Prepend[randomOutcomes2,pad02],pad00DetectorSetting],padDots],padN2],padSums],padExpectations2],padS2],pad000],Frame->All];

Graphics definitions for the QM discussion

Could probably cut this way down since we only use one of the original diagrams (diagramGeneric). But it’s easier just to keep everything for now.

circle=Graphics[{Blue,Dotted,Circle[{0,0},1]}];
arc1=Graphics[Circle[{0,0},.2,{-45 Degree,45 Degree}]];
arc2=Graphics[Circle[{0,0},.2,{45 Degree,90 Degree}]];
arc3=Graphics[Circle[{0,0},.2,{-45 Degree,-90 Degree}]];
arc4=Graphics[Circle[{0,0},.2,{-45 Degree,-135 Degree}]];
arc5=Graphics[Circle[{0,0},.2,{-90 Degree,-135 Degree}]];
spinup=Graphics[{Red,Opacity[0.5],Thickness[0.01],Arrow[{{0,0},{0,1.25}}]}];
spindown=Graphics[{Red,Opacity[0.5],Thickness[0.01],Arrow[{{0,0},{0,-1.25}}]}];
aMeasure=Graphics[{Arrow[{{0,0},{Cos[90 Degree],Sin[90 Degree]}}]}];
bMeasure=Graphics[{Arrow[{{0,0},{Cos[67.5 Degree],Sin[67.5 Degree]}}]}];
aPrimeMeasure=Graphics[{Arrow[{{0,0},{Cos[43 Degree],Sin[43 Degree]}}]}];
bPrimeMeasure=Graphics[{Arrow[{{0,0},{Cos[22.5 Degree],Sin[22.5 Degree]}}]}];

spin45=Graphics[{Red,Opacity[0.5],Thickness[0.01],Arrow[{{0,0},1.25{Cos[45 Degree],Sin[45 Degree]}}]}];
spinm45=Graphics[{Red,Opacity[0.5],Thickness[0.01],Arrow[{{0,0},1.25{Cos[-135 Degree],Sin[-135 Degree]}}]}];
measure10=Graphics[{Arrow[{{0,0},{Cos[80 Degree],Sin[80 Degree]}}]}];
measure25=Graphics[{Arrow[{{0,0},{Cos[65 Degree],Sin[65 Degree]}}]}];
measure145=Graphics[{Arrow[{{0,0},{Cos[-55 Degree],Sin[-55 Degree]}}]}];
measure170=Graphics[{Arrow[{{0,0},{Cos[-80 Degree],Sin[-80 Degree]}}]}];
measurem135=Graphics[{Arrow[{{0,0},{Cos[-135 Degree],Sin[-135 Degree]}}]}];
arc10=Graphics[Circle[{0,0},.2,{90 Degree,80 Degree}]];
arc25=Graphics[Circle[{0,0},.2,{80 Degree,65 Degree}]];
arc145=Graphics[Circle[{0,0},.2,{65 Degree,-55 Degree}]];
arc170=Graphics[Circle[{0,0},.2,{-55 Degree,-80 Degree}]];
arc22p5=Graphics[Circle[{0,0},.6,{22.5 Degree,45 Degree}]];
arc45=Graphics[Circle[{0,0},.6,{45 Degree,67.5 Degree}]];
arc67p5=Graphics[Circle[{0,0},.6,{67.5 Degree,90 Degree}]];
arc67p5Total=Graphics[Circle[{0,0},.2,{22.5 Degree,90 Degree}]];
arc22p5Std=Graphics[Circle[{0,0},1.2,{0 Degree,22.5 Degree}]];
arc45Std=Graphics[Circle[{0,0},1.4,{0 Degree,45 Degree}]];
arc67p5Std=Graphics[Circle[{0,0},1.6,{0 Degree,67.5 Degree}]];
arc90Std=Graphics[Circle[{0,0},1.8,{0 Degree,90 Degree}]];

(*The α's are the drawing angles. They are not necessarily the θ's*)
(*angleVectors=Graphics[{Arrow[{{0,0},{Cos[α1 ],Sin[α1 ]}}],Arrow[{{0,0},{Cos[α2 ],Sin[α2 ]}}],Arrow[{{0,0},{Cos[α3 ],Sin[α3 ]}}],Arrow[{{0,0},{Cos[α4 ],Sin[α4 ]}}]}];*)
angleVectors=Graphics[{Arrow[{{0,0},d4{Cos[α1 ],Sin[α1 ]}}],Arrow[{{0,0},d3{Cos[α2 ],Sin[α2 ]}}],Arrow[{{0,0},d2{Cos[α3 ],Sin[α3 ]}}],Arrow[{{0,0},d1{Cos[α4 ],Sin[α4 ]}}]}];
angles1={α1->90  Degree,α2->67.5  Degree,α3->43 Degree,α4->22.5  Degree};
angles2={α1->90  Degree,α2->22.5 Degree ,α3->-45  Degree,α4->-112.5  Degree };
angles3={α1->77 Degree,α2->55 Degree ,α3->33  Degree,α4->11  Degree };
angles4={α1->0 Degree,α2->22.5 Degree ,α3->45  Degree,α4->67.5  Degree };
distances={d1->1,d2->1,d3->1,d4->1};
distances3={d1->1.8,d2->1.6,d3->1.4,d4->1.2};
distances4={d1->1.2,d2->1.4,d3->1.6,d4->1.8};

insetAngles1={
Inset[Text[Style["a",FontSize->16,Bold]],d1{Cos[α1 ],Sin[α1 ]}],
Inset[Text[Style["b",FontSize->16,Bold]],d2{Cos[α2 ],Sin[α2 ]}],
Inset[Text[Style["a'",FontSize->16,Bold]],d3{Cos[α3 ],Sin[α3 ]}],
Inset[Text[Style["b'",FontSize->16,Bold]],d4{Cos[α4 ],Sin[α4 ]}]
};
insetAngles2={
Inset[Text[Style["π/8",FontSize->16,Bold]],1/3({Cos[α1 ],Sin[α1 ]}+{Cos[α2 ],Sin[α2 ]})],
Inset[Text[Style["π/8",FontSize->16,Bold]],1/3({Cos[α2 ],Sin[α2 ]}+{Cos[α3 ],Sin[α3 ]})],
Inset[Text[Style["π/8",FontSize->16,Bold]],1/3({Cos[α3 ],Sin[α3 ]}+{Cos[α4 ],Sin[α4 ]})]
};

insetAngles3={
Inset[Text[Style["3π/8",FontSize->16,Bold]],1/3({Cos[α1 ],Sin[α1 ]}+{Cos[α2 ],Sin[α2 ]})],
Inset[Text[Style["3π/8",FontSize->16,Bold]],1/3({Cos[α2 ],Sin[α2 ]}+{Cos[α3 ],Sin[α3 ]})],
Inset[Text[Style["3π/8",FontSize->16,Bold]],1/3({Cos[α3 ],Sin[α3 ]}+{Cos[α4 ],Sin[α4 ]})]
};

insetAngles4={
Inset[Text[Style["(Subscript[θ, 2])",FontSize->16,Bold]],1/3({Cos[α1 ],Sin[α1 ]}+{Cos[α2 ],Sin[α2 ]})],
Inset[Text[Style["(Subscript[θ, 3])",FontSize->16,Bold]],1/3({Cos[α2 ],Sin[α2 ]}+{Cos[α3 ],Sin[α3 ]})],
Inset[Text[Style["(Subscript[θ, 4])",FontSize->16,Bold]],1/3({Cos[α3 ],Sin[α3 ]}+{Cos[α4 ],Sin[α4 ]})],
Inset[Text[Style["(Subscript[θ, 1])",FontSize->16,Bold]],1/8({Cos[α2 ],Sin[α2 ]}+{Cos[α3 ],Sin[α3 ]})]
};

insetAngles5={
Inset[Text[Style["22.5°",FontSize->16,Bold]],1/3({Cos[α1 ],Sin[α1 ]}+{Cos[α2 ],Sin[α2 ]})],
Inset[Text[Style["22.5°",FontSize->16,Bold]],1/3({Cos[α2 ],Sin[α2 ]}+{Cos[α3 ],Sin[α3 ]})],
Inset[Text[Style["22.5°",FontSize->16,Bold]],1/3({Cos[α3 ],Sin[α3 ]}+{Cos[α4 ],Sin[α4 ]})],
Inset[Text[Style["67.5°",FontSize->16,Bold]],1/8({Cos[α2 ],Sin[α2 ]}+{Cos[α3 ],Sin[α3 ]})]
};

rightPointer=Graphics[{Blue,Arrowheads[0.1],Thick,Arrow[{{-.5,0},{.5,0}}]}];
leftPointer=Graphics[{Blue,Arrowheads[0.1],Thick,Arrow[{{.8,0},{-.8,.5}}]}];
xAxis=Graphics[{Blue,Arrowheads[0.1],Thin,Dotted, Arrow[{{0,0},{1.8,0}}]}];

diagramP8=Show[circle,angleVectors/.angles1/.distances,Epilog->{insetAngles1,insetAngles2}/.angles1/.distances];
diagramP38=Show[circle,angleVectors/.angles2/.distances,Epilog->{insetAngles1,insetAngles3}/.angles2/.distances];
diagram22P5=Show[circle,angleVectors/.angles4/.distances,Epilog->{insetAngles1,insetAngles3}/.angles4/.distances];
diagram67P5=Show[circle,xAxis,angleVectors/.angles1/.distances4,arc22p5Std,arc45Std,arc67p5Std,arc90Std,arc22p5,arc45,arc67p5,arc67p5Total,Epilog->{insetAngles1/.angles3/.distances3,insetAngles5/.angles1},ImageSize->400];
diagramGeneric=Show[circle,xAxis,angleVectors/.angles1/.distances4,arc22p5Std,arc45Std,arc67p5Std,arc90Std,arc22p5,arc45,arc67p5,arc67p5Total,Epilog->{insetAngles1/.angles3/.distances3,insetAngles4/.angles1},ImageSize->400];

End Initialization