# Quantum Violation of Bell’s Inequality:
a misunderstanding based on
a mathematical error of neglect

Frank Lad, email<EMAIL_ADDRESS>
Department of Mathematics and Statistics, University of Canterbury

(29 March 2020)

###### Abstract

The fabled violation of Bell’s inequality by the probabilistic specifications
of quantum mechanics is shown to derive from a mathematical error. The
inequality, designed to assess consequences of Einstein’s principle of local
realism, pertains to four polarization products on the same pair of photons
arising in a gedankenexperiment. The summands of the CHSH quantity
$s(\lambda)$ inhere four symmetric functional relations which have long been
neglected in analytic considerations. Its expectation $E[s(\lambda)]$ is not
the sum of four “marginal” expectations from a joint distribution, as quantum
theory explicitly avoids such a specification. Rather, $E[s(\lambda)]$ has
four distinct representations as the sum of three expectations of polarization
products plus the expectation of a fourth which is restricted to equal a
function value determined by the other three. Analysis using Bruno de
Finetti’s fundamental theorem of prevision (FTP) yields only a bound for
$E(s)$ within $(1.1213,2]$, surely not $2\sqrt{2}$ at all. The 4-D polytope of
cohering joint $P_{++}$ probabilities at the four stipulated angle settings
are displayed as passing through 3-D space. Aspect’s “estimation” is based on
polarization products from different photon pairs that do not have embedded
within them the inhering functional relations. When you do actively embed the
restrictions into Aspect’s estimation procedure, it yields an estimate of
1.7667, although this is not and cannot be definitive.

Keywords: Bell inequality defiance, CHSH formulation, fundamental theorem of
probability, probability bounds, 4-dimensional cuts

##  1 Introduction

As surprising as this may sound, claims that probabilistic specifications of
quantum mechanics defy the mathematical prescription known as Bell’s
inequality are just plain wrong. This may be difficult for you to accept,
depending on how wedded you are to the outlook that gives rise to them. You
will not be alone. The eminent journal Nature (2015, 526, 649-650)
flamboyantly announced to its readership the “Death by experiment for local
realism” as an introduction to its publication of experimental results
achieved at the Technical University of Delft. These were proclaimed to have
closed simultaneously all seven loopholes that had been suggested as possible
explanations of the touted violations of the inequality. In the eyes of the
professional physics community, the matter is now closed. My claim is that the
touted violation of the inequality derives from a mathematical mistake, an
error of neglect. Its recognition relies only on a basic understanding of
functions of many variables and on standard features of applied linear
algebra. This presentation is designed for any sophisticated reader not put
off by equations per se, who has followed this issue at least at the level of
popular description of scientific activity.

It is clear in his own writings that John Bell himself was puzzled by the
implications of his inequality (1964, 1966, 1971, 1987). He suspected that
something was wrong with the understanding that the probabilities of quantum
mechanics seem to defy its structure, and he expressed undying confidence that
this error would be discovered in due time. I am making a bold claim that I
have found the error he sought. I accept all probabilistic assertions
supported by quantum theory, and I exhibit their implied support of the
inequality bounds.

I do not contest the experimental results of the Delft group, nor any of the
related experimentation which has followed from the pathbreaking initial work
of Alain Aspect. I do contest the inferences they are purported to support. In
this note I will first review the derivation of the inequality in the context
to which it applies, featuring its relation to Einstein’s principle of local
realism. The review will focus on the CHSH form of the inequality to which
Aspect’s optical experimentation is considered to be relevant. Identifying the
neglected functional relations that are involved in a thought experiment on a
single pair of photons, I will show why the claims to defiance of the
inequality are mistaken, and how to derive the actual implications of quantum
theory for the probabilities under consideration. Further will be shown why
Aspect’s computations (and all subsequent extensions) proposed to exhibit
empirical confirmation of the inequality defiance are ill considered, and how
they ought to be adjusted. This demonstration relies on the computational
mechanics of Bruno de Finetti’s fundamental theorem of probability. The
results are displayed both algebraically and geometrically.

##  2 The physical setup of four experiments providing context
for Bell’s inequality in CHSH form: a 16-D problem

We shall review the setup of an optical variant of Bell’s experiment, designed
by Alain Aspect in the 1980’s to take advantage of a formulation of the
problem proposed by Clauser, Horne, Shimony, and Holt (CHSH, 1969). The
original discussions of the inequality violation were couched in terms of
observations of spins of paired electrons. Although specific algebraic details
differ for the two types of experimental situation, the conclusions reached
would be identical.

An experiment is conducted on a pair of photons traveling in opposite
directions along an axis, ${\bf z}$, from a common source. The direction one
of the photons travels toward detector $A$ on the left is directly opposite to
the direction its paired photon travels toward detector $B$ on the right:
$\,{\bf z}_{A}=-{\bf z}_{B}\,$. At the end of their respective journeys, each
of the paired photons engages polarising material that either allows it to
pass through or to be deflected.

Figure 1: Polarizing material is aligned at the detection stations of $A$ and
$B$, each with two possible choices of direction in the $(x,y)$ dimension
relative to the ${\bf z}$ direction of the incoming photon: direction ${\bf
a}$ or ${\bf a^{\prime}}$ at station $A$, and ${\bf b}$ or ${\bf b^{\prime}}$
at station $B$.

The detection of a photon that passes through the polarizer is designated by
denoting the numerical value of $A=+1$, while the detection of a photon as
blocked is designated by the value of $A=-1$. The polarizer addressed by
photon $A$ is directed at a variable angle ${\bf a^{*}}$ in the $(x,y)$ plane
perpendicular to $\,{\bf z}_{A}$. This polarization direction can be set in
either of two specific angles designated as ${\bf a}$ and ${\bf a^{\prime}}$
in the experimental setups we shall consider. Similarly, the direction of the
polarizer met by the photon at station $B$ can be set at either angle ${\bf
b}$ or ${\bf b^{\prime}}$ in its $(x,y)$ plane. Depending on the specific pair
of polarization angles ${\bf a^{*}}$,${\bf b^{*}}$ chosen for any particular
experiment, we shall observe the paired values of either $(A({\bf a}),B({\bf
b})),\ $ or $(A({\bf a}),B({\bf b^{\prime}})),\ $ or $(A({\bf
a^{\prime}}),B({\bf b}))$, or $(A({\bf a^{\prime}}),B({\bf b^{\prime}}))$.
Since the observations of the $A$ and the $B$ photon detections can each equal
either $+1$ or $-1$ whatever the angle pairing might be, the chosen
observation pair $(A({\bf a^{*}}),B({\bf b^{*}}))$ will equal one of the four
possibilities $(+,+),(+,-),(-,+)$, or $(-,-)$, where we are suppressing here
the needless numeric values of $1$ in each designated pairing.

Experimental choices of the two polarization angle directions yield a specific
relative angle between them at $A$ and $B$ in any given experiment. Using
Aspect’s notation that parentheses around a pair of directions denotes the
relative angle between them, the experimental detection angle settings $({\bf
a}^{*},{\bf z}_{A})$ and $({\bf b}^{*},{\bf z}_{B})$ imply the relative angle
between polarizers at stations $A$ and $B$ in the $(x,y)$ dimension as $({\bf
a}^{*},{\bf b}^{*})$. Bell’s inequality is relevant to this context in which
the two photon polarization directions can be paired at any one of four
distinct relative angles, denoted by the parenthetic pairs $({\bf a},{\bf
b})$, $({\bf a},{\bf b^{\prime}})$, $({\bf a^{\prime}},{\bf b})$, or $({\bf
a^{\prime}},{\bf b^{\prime}})$.

In order to view the relative angles we are talking about, mentally we would
need to swing the $(x,y)$ plane as it is viewed by the photon directed to
station $A$ around by $180^{\circ}$ and superimpose it on the $(x,y)$ plane as
it is viewed by the photon directed to station $B$. In this manner we can
understand the size and meaning of the relative angles between the various
values of polarization orientations ${\bf a^{*}}$ and ${\bf b^{*}}$ as seen
here in Figure $2$.

Figure 2: Directional vectors of the polarization angle settings at the
observation stations $A$ and $B$, viewed in a common axis orientation. The
specific relative angle size settings displayed are the extreme violation
values, a feature to be discussed.

The theory of quantum mechanics motivates specification of probabilities for
the four observable outcome possibilities of the polarization experiment as
depending on the relative angle $({\bf a}^{*},{\bf b}^{*})$ between the
direction vectors of the polarizers at stations $A$ and $B$. For any such
relative angle pairing, the probabilities specified by quantum theory for the
four possible experimental observations $\\{++,+-,-+,--\\}$ are

$P[(A({\bf a}^{*})=+1)(B({\bf b}^{*})=+1)]\ =\ P[(A({\bf a}^{*})=-1)(B({\bf
b}^{*})=-1)]\ =\ \frac{1}{2}\;cos^{2}({\bf a}^{*},{\bf b}^{*})\;$, and

$P[(A({\bf a}^{*})=+1)(B({\bf b}^{*})=-1)]\ =\ P[(A({\bf a}^{*})=-1)(B({\bf
b}^{*})=+1)]\ =\ \frac{1}{2}\;sin^{2}({\bf a}^{*},{\bf b}^{*})\;$. $(1)$

For efficiency in what follows, we shall denote the four probabilities
appearing in equations $(1)$ by $P_{++},P_{--},P_{+-}$, and $P_{-+}\,$ when
the pertinent angle setting is evident.

These four probabilities surely sum to equal $1$, because the sum of
$cos^{2}+sin^{2}$ of any angle equals $1$. A few properties of the joint
probability mass function (pmf) they compose should be noticed. Firstly, the
four probabilities can be specified by the value of any one of them. The
equations $(1)$ stipulate that no matter what the relative angle $({\bf
a}^{*},{\bf b}^{*})$ may be, the values of $P_{++}=P_{--}$, and
$P_{-+}=P_{+-}$. Since the four probabilities do sum to $1$ then, the
specification of $P_{++}$ as the value $p$, for example, implies that the pmf
vector $[P_{++},P_{--},P_{+-},P_{-+}]\,$ would be $[p,p,(1-2p)/2,(1-2p)/2]$.
Another implication of this feature is that the probabilities for the paired
detection outcomes depend only on the product of the two measurements. For
both outcomes $++$ and $--$ yield a product of $+1$ and both outcomes $+-$ and
$-+$ yield a product of $-1$. Thus, the QM-motivated distribution for the
experimental value of the polarization product $A({\bf a^{*}})B({\bf b^{*}})$
is specified by $P[A({\bf a^{*}})B({\bf b^{*}})=+1]\ =\ cos^{2}({\bf
a^{*}},{\bf b^{*}})$ and $P[A({\bf a^{*}})B({\bf b^{*}})=-1]\ =\ sin^{2}({\bf
a^{*}},{\bf b^{*}})$.

As will be important to recognize in what follows, the expected value (first
moment) of this distribution for the detection product is
$E[A({\bf a^{*}})B({\bf b^{*}})]\ =\ (+1)cos^{2}({\bf a^{*}},{\bf
b^{*}})+(-1)sin^{2}({\bf a^{*}},{\bf b^{*}})\vspace{.3cm}\\\ \hskip
160.75789pt=\ cos^{2}({\bf a^{*}},{\bf b^{*}})-sin^{2}({\bf a^{*}},{\bf
b^{*}})\ =\ cos\,2({\bf a}^{*},{\bf b}^{*})\;$ (2)
according to standard double angle formulas. It is worthwhile reminding right
here that “the expected value of a probability distribution” is the “first
moment” of the distribution. Geometrically, it is the point of balance of the
probability mass function weights when they are positioned in space at the
places where the possible observations to which they pertain might occur. It
is a property of a probability distribution for the outcome of a specific
single observable variable. A final peculiarity of equation $(2)$ which will
be useful far down the road in this explication is that the expectation value
$E[A({\bf a^{*}})B({\bf b^{*}})]$ can also be represented as
$E[A({\bf a^{*}})B({\bf b^{*}})]\ \ =\ \ 2\,cos^{2}({\bf a}^{*},{\bf
b}^{*})-1\ \ =\ \ 4\,P_{++}({\bf a^{*}},{\bf b^{*}})-1$ . $(3)$
For the value of $sin^{2}({\bf a^{*}},{\bf b^{*}})$ appearing in the final
line of equation $(2)$ can also be written as $1-cos^{2}({\bf a^{*}},{\bf
b^{*}})$. Enough of this for now.

Secondly, again no matter what the relative angle $({\bf a}^{*},{\bf b}^{*})$
may be, the marginal probabilities that the detection observation of the
photon equals $+1$ at either angle ${\bf a}^{*}$ or ${\bf b}^{*}$ is equal to
$1/2$. For the standard margining equation for the result of a paired
experiment yields
$P[(A({\bf a}^{*})=+1)\ \ =\ \ P[(A({\bf a}^{*})=+1)(B({\bf b}^{*})=+1)]\ +\
P[(A({\bf a}^{*})=+1)(B({\bf b}^{*})=-1)]\vspace{.2cm}\\\ \hskip 102.43008pt=\
\ \frac{1}{2}\;cos^{2}({\bf a}^{*},{\bf b}^{*})\;+\frac{1}{2}\;sin^{2}({\bf
a}^{*},{\bf b}^{*})\;\ =\ \ 1/2$. (4)

This result codifies a touted feature of physical processes at quantum scales
of magnitude, that the photon behaviours of particle pairs are understood to
be entangled. Since the probability for the joint photon behaviour $P[(A({\bf
a}^{*})=+1)(B({\bf b}^{*})=+1)]$ does not factor into the product of their
marginal probabilities $P[(A({\bf a}^{*})=+1)]$ and $P[(B({\bf b}^{*})=+1)]$,
the conditional distribution for either one of these events depends on the
context of the conditioning behaviour:
$P[(A({\bf a}^{*})=+1)|(B({\bf b}^{*})=+1)]\ \ =\ \ cos^{2}({\bf a}^{*},{\bf
b}^{*})\ \ \neq\ \ P[(A({\bf a}^{*})=+1)\ \ =\ \ 1/2\vspace{.2cm}$ , (5)
and $P[(A({\bf a}^{*})=+1)|(B({\bf b}^{*})=-1)]\ =\ sin^{2}({\bf a}^{*},{\bf
b}^{*})$ , which is different still.

We have concluded what we need to say at the moment about the prescriptions of
quantum theory relevant to physical quantum behaviour of a single pair of
prepared photons. Before proceeding to the specification of Bell’s inequality,
we need to address what quantum theory professes not to say.

##  3 The uncertainty principle: what quantum theory disavows

Made famous as what is called “Heisenberg’s uncertainty principle”, the theory
of quantum mechanics explicitly disavows claims to what might happen in
physical situations that are impossible to instantiate. Here is an example
relevant to the classical scale of everyday observation. A dairy farmer may
choose to treat a milking cow with injections of bovine somatatropine (BST) or
may choose not to use such treatment. However, one cannot follow both programs
on the same dairy cow. One could treat some cows with BST and some other cows
without BST, but one cannot both treat and not treat the same cow with BST.
Now what will be the daily weight of the milk yield from this cow? No one
knows for sure. Of course you could well assess conditional distributions for
the weight of the cow’s milk yield, conditional on each of the two treatment
strategies. However, a distribution for the joint yields from following both
of these strategies on the same cow (which would be impossible) is
meaningless. This is just common sense. There are well known statistical
procedures for studying the yields from alternative treatment strategies,
applying one strategy to one group of cows and the other strategy to another
group. This is a classic statistical problem of agricultural statistics
codified as problems in the design of experiments. This subject does not
concern us here just now.

The problem of quantum physics relevant to Bell’s inequality concerns this
very same issue. We have identified a physical experiment on a pair of
photons, polarising the two of them at an array of possible exclusive angle
pairings, $({\bf a},{\bf b})$, $({\bf a},{\bf b^{\prime}})$, $({\bf
a^{\prime}},{\bf b})$, and $({\bf a^{\prime}},{\bf b^{\prime}})$. We could
perform our polarization experiment on a specific pair of photons at any one
of these angle pairings. But we cannot perform all four experiments on the
same pair of photons. The theory of quantum mechanics recognizes this fact
explicitly and loudly! Not only do the experimental physicists recognize that
this cannot be done, but the theoretical algebraic mechanism that is used to
identify the quantum probabilities we have specified in equations $(1)$ embed
this impossibility into its protocol. I will state how this recognition is
embedded into quantum theory, without deriving its application here in
complete detail.

Relevant to the polarization product detection in the case we are formalizing,
the theory of quantum mechanics characterizes the situation of a quantum
experiment in terms of a two-dimensional vector which resides in one of
several possible states of possibility. Let’s call the state vector ${\bf
s}=(s_{1},s_{2})^{T}$. We cannot observe what state the photon pair is in
without performing a measurement. The measurement process is characterised
algebraically by a matrix, call it $H$, which operates on the state vector by
multiplication. This matrix, on account of its form, identifies the
observation values that might arise when the measurement is performed. In our
case it identifies that the polarization product value we observe, in whatever
paired polarization experiment we perform, might equal $+1$ or $-1$. When this
matrix multiplies the state vector in the algebraic form of $H{\bf s}$, the
result of the product is a pair of probabilities specified for these two
possible observation values. In our polarization experiment on the photon pair
observed at $A({\bf a}^{*},{\bf b}^{*})$ and $B({\bf a}^{*},{\bf b}^{*})$,
these are the probabilities specified in our equation (1). (These two
probabilities have been split in half there, to account for the fact that both
of the paired polarization results $++$ and $--$ yield a product of $+1$, and
both of the paired polarization results $+-$ and $-+$ yield a product of
$-1$.) The matrix $H$ is said to be a “Hadamard matrix”.

At any rate, this is the mathematical formalism by which the probabilistic
specifications of quantum theory are derived for the various possible results
of a quantum experiment. Each observation possibility is characterised by its
own matrix $H$. Since we have four possible experimental designs under
consideration, codified by the paired angle settings $({\bf a},{\bf b})$,
$({\bf a},{\bf b^{\prime}})$, $({\bf a^{\prime}},{\bf b})$, and $({\bf
a^{\prime}},{\bf b^{\prime}})$, there are four distinct matrices, denoted by
$H_{({\bf a},{\bf b})}$, $H_{({\bf a},{\bf b^{\prime}})}$, $H_{({\bf
a^{\prime}},{\bf b})}$, and $H_{({\bf a^{\prime}},{\bf b^{\prime}})}$ which
codify our experimental measurement possibilities.

Now, is it possible to perform two observational measurements on a quantum
experimental situation? The answer is “in some cases yes, and in some cases
no!” Happily, there is a very simple way to determine whether two distinct
measurements can be performed on the same situation of the state of the
photons. Algebraically, a measurement codified by a matrix $H$ is compatible
with a simultaneous second measurement $G$ on the same experimental situation
if and only if the product of the two matrices commutes! That is to say, if
and only if the products of the matrices are identical no matter which be the
order of the multiplication: $HG=GH$. To check whether this is true or not is
a simple matter of performing the algebra of multiplying the matrices. If so,
the two matrices are said to be “Hermitian”.

An example of commuting operators has already been broached, without my having
mentioned it. There is an operator matrix $H_{{\bf a}}$ that codifies the
detection observation of the photon at station $A$ with polarization direction
${\bf a}$, and another which codifies a detection observation at station $B$
with polarization direction ${\bf b}$. Call it $H_{{\bf b}}$. Now it is a mere
matter of mathematical derivation to find that the product of these two
operator matrices does not depend of which one multiplies the other: that is,
$H_{{\bf a}}H_{{\bf b}}$ = $H_{{\bf b}}H_{{\bf a}}$. This lets us know
formally that indeed we can measure the detection of the two photons at both
of the stations $A$ and $B$. That is why we denote this operator by $H_{({\bf
a},{\bf b})}\equiv H_{{\bf a}}H_{{\bf b}}$. We could easily have denoted it
just as well by $H_{({\bf b},{\bf a})}=H_{{\bf b}}H_{{\bf a}}$.

On the contrary, the result in the case of the paired photon experiments under
consideration is that none of the four $H_{({\bf a}^{*},{\bf b}^{*})}$
matrices commute! That is, for example, $H_{({\bf a},{\bf b})}H_{({\bf a},{\bf
b^{\prime}})}\ \neq\ H_{({\bf a},{\bf b^{\prime}})}H_{({\bf a},{\bf b})}$. All
this is to say that the technical manipulations of mathematical quantum theory
instantiate formally just what we knew to begin with … that we cannot
simultaneously perform the measurement observation of the polarization
products at both angle settings $({\bf a},{\bf b})$ and $({\bf a},{\bf
b^{\prime}})$ on the same pair of photons.

Well, who would want to? We shall now find out.

##  4 The principle of local realism and its relevance to Bell

A feature crucial to the touted violation of Bell’s inequality is that it
pertains to experimental results supposedly conducted with a single photon
pair at all four angle settings. Sound unusual? When the probabilistic
pronouncements of quantum theory were formalized, Einstein among others was
puzzled by the fact that the conditional probability for the outcome of the
experiment at station $A$ depends on both the angle at which the experiment is
conducted at station $B$ and on the outcome of that experiment. This matter is
codified by the conditional probabilities we have seen in equations (5). This
entanglement of seemingly unrelated physical processes was deemed to be a
matter of “spooky action at a distance”. Well, Einstein proposed a solution to
this enigma, positing that there must be some other factors relevant to what
might be happening at the polarizer stations $A$ and $B$ that would account
for the photon detections found to arise. As yet unspecified in the theory, he
considered such factors to identify unknown values of “supplementary
variables”. It was proposed that the probabilities inherent in the results of
quantum theory must be representations of scientific uncertainty about the
action of these other variables on the two photons at their respective
stations. This was his way of accounting for the spooky action at a distance.

However, there was one aspect of the matter upon which Einstein wanted to
insist: this was termed “the principle of local realism”. Fair enough, quantum
theory does stipulate the probability for the photon detection at angle
setting ${\bf a}$ as depending on whether the polarizer direction at $B$ is
set at ${\bf b}$ or ${\bf b^{\prime}}$ and on what happens there. However, in
any specific instance of the joint experiment at a relative polarization angle
${(\bf a},{\bf b})$, if the measurement observation at $A$ happened to equal
$A({\bf a},{\bf b})=+1$, say, then in this instance the measurement at $A$
would have to be the same no matter whether the direction setting at station B
were ${\bf b}$ or ${\bf b^{\prime}}$. That is to say, if the polarization
observation $A({\bf a})=+1$ in a particular experiment on a pair of photons
measure in the paired angle design $({\bf a},{\bf b})$, then the value of
$A({\bf a})$ would also have to equal $+1$ in a companion experiment on the
same pair of photons when the polarization directions would be set in the
angle pairing $({\bf a},{\bf b^{\prime}})$.

Actually, in a way we have already deferred to such an understanding. We have
been denoting the photon detection value at station $A$ merely by $A({\bf a})$
rather than denoting it by $A({\bf a},{\bf b})$, even before we have now
introduced consideration of this principle of local realism. In the context of
locality, the importance of such simplification of the notation was stressed
by Aspect (2004, p 13/34). In fact there was no need to denote the paired
direction at $B$ in our notation earlier, because we can only do the
experiment on a specific photon at one specific possibility pair determining
the angle pairing $({\bf a}^{*},{\bf b}^{*})$. So we have merely denoted the
measurements as $A({\bf a})$ and $B({\bf b})$, or $B({\bf b^{\prime}})$.
Nonetheless, the QM probabilities of equations $(1)$ stipulated that each of
the paired results of the experiments does depend jointly on the relative
angle between the two polarization directions.

Despite this notational deference, we should now recognize explicitly and
declare loudly that this principle of local realism is based upon a claim that
lies outside the bounds of matters addressed by the theory of quantum physics.
For, as we have noted, it is impossible to make a measurement of both the
photon detection product $A({\bf a})B({\bf b})$ and the product $A({\bf
a})B({\bf b^{\prime}})$ on the same pair of photons. So quantum theory
explicitly disavows addressing this matter directly.

We are ready to conclude this Section by proposing an experimental measurement
that lies at the heart of Bell’s inequality. We are not yet ready to assess
it, nor to explain its relevance to the principle of local realism, but we
shall merely air it now for viewing. Peculiar, it is considered to be the
result of a gedankenexperiment.

Consider a pair of photons to be ejected toward stations $A$ and $B$ at which
the pair of polarizers can be directed in any of the four relative angles we
have described. According to the detection of whether the photons pass through
the polarizers or are deflected by them, Bell’s inequality pertains to an
experimental quantity defined by the equation
$\ \ \ \ \ \ \ \ s\ \equiv\ A({\bf a})\,B({\bf b})\;-\;A({\bf a})\,B({\bf
b}^{\prime})\;+\;A({\bf a}^{\prime})\,B({\bf b})\;+\;A({\bf
a}^{\prime})\,B({\bf b}^{\prime})\,\ \ .\vspace{.2cm}\hskip 65.44142pt(6)$
Mathematically, we would refer to this quantity $s$ as a linear combination of
four polarization detection products. Any one of the four terms that determine
the value of $s$ could be observed in an experiment on a pair of prepared
photons. Before we explain why this quantity is of interest, we should
recognize right here only that we can observe the value of this quantity $s$
if we are to conduct four component experiments on four distinct pairs of
photons, each ejected toward stations $A$ and $B$ with the polarizers directed
at a different relative angle pairing. However, we cannot observe the value of
$s$ if it were meant to pertain to all four experiments being conducted on the
same pair of photons. It just cannot be done, and quantum theory is very
explicit about having nothing directly to say about its value. If we are to
consider the value of $s$ in such an experimental design, it could only be as
the result of a “thought experiment”. Enough said for now.

Why would we even be interested by such a “gedankenexperiment” as its
perpetrators called it, and what does the supposition of “hidden variables”
have to do with the matter?

##  5 Einstein’s proposal of hidden variables relevant to the matter

Puzzled by the standing probabilistic conclusions of quantum theory which he
helped to formulate, Einstein wondered what could be the meaning of these
“probabilities” involved in its prescriptions. Others were proclaiming that
the experimental and theoretical discoveries of QM support the view that at
its fundamental level of particulate matter, the behaviour of Nature is
random, and that quantum theory had identified its probabilistic structure.
Convinced that “he (the old one) does not play dice with the universe”,
Einstein formulated another proposal: that the analysis of quantum theory is
incomplete. There must be some other matters involved in quantum level
experimentation that we do not know about; and these other unknown
“supplementary” variables would conceivably be distinguishable in the
observable outcome of any particular experimental result if only we knew how
to distinguish their measurable states. The probabilities of quantum
theoretical specifications must formalize our uncertain knowledge of the
situation, our uncertainty distribution concerning the conceivable
instantiation of these hidden variables in any particular experimental
setting.

Enter the necessity to formulate a gedankenexperiment to assess the matter. A
paper by Einstein, Podolsky, and Rosen (1935) presented this argument which
became known subsequently as the EPR proposal. It stimulated a fury of healthy
discussion and argument that I shall not summarize here. Well documented both
in the professional journals of physics and in literature of popular science,
the discussion featured considerations of the collapse of a quantum system
when subject to observation that disturbs it, the non-locality of quantum
processes, and esoteric formulations of the “many worlds” view of quantum
theory. What matters for my presentation here is that Einstein’s views were
widely relegated as a quirky peculiar sideline, and the recognition of
randomness as a fundamental feature of quantum activity came to the forefront
of theoretical physics.

Enter John Bell. Interested in a reconsideration of Einstein’s view, he began
his research with an idea to re-establish its validity as a contending
interpretation of what we know. However, he was surprised to find this
programme at an impasse when he discovered that the probabilistic
specifications of quantum theory which we have described above seem to defy a
simple requirement of mathematical probabilities, if the principle of local
realism is valid. In the context of a hidden variables interpretation of the
matter, this seemed to require that the principle of local realism must be
rejected. Reported in a pair of articles (Bell, 1964, 1966), these results too
stimulated a continuation of the flurry which has lasted through the 2015
publication in Nature of their apparently definitive substantiation by the
research at the Delft University of Technology.

The specification of Bell’s inequality can take many forms. The context in
which it is addressed in the remainder of my exposition here was presented in
an article by Clauser, Horne, Shimony, and Holt (1969), commonly referred to
as the CHSH formulation. This was the form that attracted still another
principal investigator in this story, Alain Aspect. A young experimentalist,
he wondered how could such a monumental result of quantum physics pertain only
to a thought experiment, devoid of actual physical experimental confirmation.
He thought to have devised an experimental method that could confirm or deny
the defiance of Bell’s inequality. My assessment of his empirical work follows
directly from his explanation of the situation (Aspect, 2002) reported to a
conference organized to memorialize Bell’s work. My notation is largely the
same as Aspect’s. I adjust only the notation for expectation of a random
variable to the standard form of $E(X)$, replacing his notation of $<X>$ which
has become standard in mathematical physics in the context of bra-ket notation
which I avoid. Here is how it works.

###  5.1 Explicit construction of $s$ with hidden variables

Hidden variables theory proposes that the quantity $s$ which we have
introduced in equation $(6)$ should be considered to derive from a physical
function of unobserved and unknown hidden variables, whose values might be
codified by the vector $\lambda$, viz.,
$\hskip 28.45274pts(\lambda)\ \equiv\ A(\lambda,{\bf a})\,B(\lambda,{\bf
b})\;-\;A(\lambda,{\bf a})\,B(\lambda,{\bf b}^{\prime})\;+\;A(\lambda,{\bf
a}^{\prime})\,B(\lambda,{\bf b})\;+\;A(\lambda,{\bf
a}^{\prime})\,B(\lambda,{\bf b}^{\prime})\,,\ $ (7)
for $\lambda\ \in\Lambda$ . The variable designated by $\lambda$ here could be
a vector of any number of components identifying unknown features of the
experimental setup that are relevant to the outcome of the experiment in any
specific instantiation. The set designated by $\Lambda$ is meant to represent
the space of possible values of theses hidden variables. The status of these
variables in the context of any particular experiment is supposed only to
depend on the state of the photon pair and its surrounds, independent of the
angle setting $({\bf a}^{*},{\bf b}^{*})$ at which the polarizers are
directed. According to the deterministic outlook underlying the physical
theory relying on recourse to the hidden variables, if we could only know the
values of these hidden variables at the time of any experimental run and have
a complete theoretical understanding of their relevance to the polarization
behaviour of the photon pair, then we would know what would be the values of
the polarization incidence detection of the photon pair at any one or all of
possible angle settings.

Now the personalist subjective theory of probability (apparently subscribed to
by Einstein, and surely by Bruno de Finetti and by me) specifies that any
individual’s uncertain knowledge of the values of observable but unknown
quantities could be representable by a probability density over its space of
possibilities. Aspect denotes such a density in this situation by
$\rho(\lambda)$. For any proponent of the quantum probabilities it might well
be presumed to be “rotationally invariant” over the full $360^{\circ}$ of
angles at which the photon may be fluttering toward the polarizer. That is to
say, the probabilities for the possible values of the supplementary variables
do not depend on the angular direction in $(x,y)$ dimensions of the photons
along ${\bf z}$ axes heading toward stations $A$ and $B$.

Since we avowedly have no idea of what these hidden variables might be, much
less what their numerical values may be relevant to any specific experimental
run, we can only ponder the “expected value” of $s(\lambda)$ with respect to
the distribution specified by $\rho(\lambda)$. The feature of rotational
invariance implies that this expectation is the same no matter what be the
rotational angle at which the photons flutter relative to their $(x,y)$ plane
detections. Let’s write this expectation equation down:
$E[s(\lambda)]\,\vspace{.3cm}=\,E[A(\lambda,{\bf a})B(\lambda,{\bf
b})]-E[A(\lambda,{\bf a})B(\lambda,{\bf b}^{\prime})]+E[A(\lambda,{\bf
a}^{\prime})B(\lambda,{\bf b})]+E[A(\lambda,{\bf a}^{\prime})B(\lambda,{\bf
b}^{\prime})].\ \ \ \ (8)$
This equation follows directly from equation $(7)$ because of the fact that a
rule of probability says that the expectation of any linear combination of
random quantities equals the same linear combination of their expectations.
Fair enough. Now fortunately, we have already reported in equation $(5)$ that
the probabilities of quantum theory identify the expected value of any
polarization product at the variable relative polarization angle $({\bf
a}^{*},{\bf b}^{*})$ as $E[A({\bf a}^{*})B({\bf b}^{*})]=cos\ 2({\bf
a}^{*},{\bf b}^{*})$. So we are ready to proceed.

###  5.2 Finally, Bell’s inequality

We have now arrived at a place we can state precisely what Bell’s inequality
says. There is just a little more specificity to detail before we soon will
have it. However, I should alert you that there is a little tic in the
understanding of equation $(8)$ to which we shall return after we learn how
the inequality is currently understood to be defied by quantum theory. But on
the face of it, the validity of equation $(8)$ is plain as day.

Now re-examining equation $(7)$, it is apparent that it can be factored into a
simplified form:
$\ \ \ \ \ \ \ \ s(\lambda)\ \equiv\ A(\lambda,{\bf a})\,B(\lambda,{\bf
b})\;-\;A(\lambda,{\bf a})\,B(\lambda,{\bf b}^{\prime})\;+\;A(\lambda,{\bf
a}^{\prime})\,B(\lambda,{\bf b})\;+\;A(\lambda,{\bf
a}^{\prime})\,B(\lambda,{\bf b}^{\prime})\,,\ $ for $\lambda\ \in\Lambda$ ,
$\hskip 28.45274pt=\ A(\lambda,{\bf a})\;[B(\lambda,{\bf b})-B(\lambda,{\bf
b}^{\prime})]\ +\ A(\lambda,{\bf a}^{\prime})\;[B(\lambda,{\bf
b})+B(\lambda,{\bf b}^{\prime})]\vspace{.2 cm}$ , and alternatively
$\hskip 28.45274pt=\ B(\lambda,{\bf b})\;[A(\lambda,{\bf a})+A(\lambda,{\bf
a}^{\prime})]\ -\ B(\lambda,{\bf b}^{\prime})\;[A(\lambda,{\bf
a})-A(\lambda,{\bf a}^{\prime})]\ .\ $ $(9)$
It is important to notice that once again, in performing this simple
factorization of the components $A(\lambda,{\bf a})$ and $A(\lambda,{\bf
a^{\prime}})$ in this second line, we have implicitly presumed the principle
of local realism. For when we consider the first two summands of the first
line, $A(\lambda,{\bf a})\,B(\lambda,{\bf b})$, and $A(\lambda,{\bf
a})\,B(\lambda,{\bf b}^{\prime})$, we should notice that the value of
$A(\lambda,{\bf a})$ in that first term is evaluated in an experiment at which
the paired polarization angle is $({\bf a},{\bf b})$, whereas in the second
term from which it is factored it is evaluated in an experiment at the
relative polarization angle $({\bf a},{\bf b^{\prime}})$. It is the principle
of local realism, extraneous to any claims of quantum theory, that provides
the observed value of $A(\lambda,{\bf a})$ must be identical in these two
conditions which are impossible to instantiate together. It is only under the
condition of this assertion that we would be able to factor this term out of
the two expressions. The same goes for the factorization of $A(\lambda,{\bf
a^{\prime}})$. This is not a source of any worry. I am merely mentioning this
so that we are all aware of what is going on. The same feature of supposition
is pertinent to the alternative factorization of the terms $B(\lambda,{\bf
b})$ and $B(\lambda,{\bf b^{\prime}})$ in the third line from the terms of the
first line.

Having arrived at this factorization, it will now take just a little thought
to recognize that if the value of the quantity $s$ is supposed to be
determined from a thought experiment on a single pair of photons, then the
numerical value of $s$ can equal only either $+2$ or $-2$. Of course, if we
were to calculate the value of $s$ from performing four component experiments
with four different pairs of photons (something we can actually do), then the
four component product values might each then equal either $-1$ or $+1$, so
the value of $s$ might equal any of $\\{-4,-2,0,+2,+4\\}$. However, in such a
case the factorization we performed in equation $(9)$ would not be permitted.
For each of the observed detection products appearing in the first line would
pertain to a different pair of photons whose multiplicands would be free to
equal either $+1$ or $-1$ as prescribed by experiment. The same possibilities
would be accessible if the principle of local realism were not valid. However,
if the value of $s$ is to be calculated from the results of a thought
experiment on the same pair of photons, then its possibilities would be
limited according to local realism merely to $\\{-2,+2\\}$. Here is how to
recognize this.

Suppose the values of $B(\lambda,{\bf b})$ and $B(\lambda,{\bf b}^{\prime})$
were both observed to equal $+1$. Then the first term in the factored form of
the second line must equal $A(\lambda,{\bf a})\;[B(\lambda,{\bf
b})-B(\lambda,{\bf b}^{\prime})]\ =\ 0$; and furthermore, the second term in
the factored representation would then be $A(\lambda,{\bf
a^{\prime}})\;[B(\lambda,{\bf b})+B(\lambda,{\bf b}^{\prime})]$. The factor
$A(\lambda,{\bf a^{\prime}})$ equaling either $+1$ or $-1$ would then be
multiplied by the factor $[B(\lambda,{\bf b})+B(\lambda,{\bf b}^{\prime})]$
which would equal the number $+2$. Thus, the value of $s$ could equal only
either $-2$ or $+2$. Alternatively, suppose that the values of $B(\lambda,{\bf
b})$ and $B(\lambda,{\bf b}^{\prime})$ are both observed to equal $-1$. Then
by a similar argument the value of the first factored expression would again
equal $0$ and the second expression would equal either $-1$ or $+1$ multiplied
now by $-2$. Again the computed result of the value of $s$ could equal only
$-2$ or $+2$. I leave it to the reader to confirm the same result for the
possible values of $s$ if the values of $B(\lambda,{\bf b})$ and
$B(\lambda,{\bf b}^{\prime})$ were observed to equal either $-1$ and $+1$
respectively, or $+1$ and $-1$ respectively.

The conclusion is indisputable. If the principle of local realism holds, then
the value of $s$ that would be instantiated as a result of a thought
experiment on the same pair of photons in all four polarization angle settings
can equal only $-2$ or $+2$. Thus, the expected value $E(s)$ deriving from any
coherent probability distribution over the four values of the component paired
polarization experiments would have to be a number between $-2$ and $+2$.
Stated algebraically and simply, without all the provisos explaining its
content, Bell’s inequality is the requirement that $-2\leq E(s)\leq+2$.

Well, what do the probabilities of quantum theory imply for the value of
$E(s)$ ? The answer universally presumed to be correct by proponents of the
Bell violation is that when the design of the four experiments on a single
pair of photons is constructed at a particular array of angle settings that we
shall soon identify, then $E(s)\ =\ 2\sqrt{2}\ =\ 2.8284$ to four decimal
places, a number that exceeds $+2$. (I shall show you why in the next
paragraph.) Moreover, the experimental results of Aspect, as well as the more
sophisticated experimentation of succeeding decades, is understood to
corroborate this result to many decimal places. I will soon explain how this
result is derived as well. However, I will insist on also showing you that not
only is this theoretical derivation wrong, but that the calculations used to
corroborate this result from experimental evidence are mistaken. Nonetheless,
there is nothing at all wrong with the experimental results, which are what
they are.

###  5.3 The mistaken violation of Bell’s inequality

It turns out that Bell’s inequality is not deemed to be defied at every four-
plex of possible experimental angle settings that we have characterised
generically as $({\bf a},{\bf b})$, $({\bf a},{\bf b^{\prime}})$, $({\bf
a^{\prime}},{\bf b})$, or $({\bf a^{\prime}},{\bf b^{\prime}})$. At some
paired directional settings of the polarizers it seems not to be defied at
all. Among other pairings at which it seems to be defied, it is apparently
defied more strongly at some pairings than at others. Aspect had thought that
if we were to find experimental evidence of the defiance, we should try to
find it at the angle pairings for which the theoretical defiance is the most
extreme. It is a matter of simple calculus of extreme values to discover that
the most extreme violation of the equality should occur at the angle settings
$({\bf a},{\bf b})=-\pi/8$, $({\bf a},{\bf b^{\prime}})=-3\pi/8$, $({\bf
a^{\prime}},{\bf b})=\pi/8$, or $({\bf a^{\prime}},{\bf b^{\prime}})=-\pi/8$.
(The angle measurements are expressed here in terms of their polar
representations. In terms of degrees, the angle $-\pi/8=-22.5^{\circ}$, while
$-3\pi/8=-67.5^{\circ}$, and $+\pi/8=+22.5^{\circ}$.) You may wish to examine
our Figure 2 and notice that the angles between the various polarization
directions we depicted there correspond to these relative angles. For the
record, doubling these angles yields the values of $\pm\pi/4=\pm 45^{\circ}$
and $-3\pi/4=-135^{\circ}$ in these instances. And why does that matter? …

Recall equation $(8)$ and the ensuing sentences. Evaluating $E(s)$ according
to this equation at the four angle settings just mentioned requires evaluating
the summand component expectations. Each of them in the form $E[A({\bf
a}^{*})B({\bf b}^{*})]=cos\ 2({\bf a}^{*},{\bf b}^{*})$, these would then be

$\displaystyle E[A(\lambda,{\bf a})B(\lambda,{\bf b})]\ $ $\displaystyle=\
cos\ 2({\bf a},{\bf b})$ $\displaystyle=\ cos(-\pi/4)$ $\displaystyle=\ \
1/\sqrt{2}\ ,\vspace{.2cm}$ $\displaystyle E[A(\lambda,{\bf a})B(\lambda,{\bf
b^{\prime}})]\ $ $\displaystyle=\ cos\ 2({\bf a},{\bf b^{\prime}})$
$\displaystyle=\ cos(-3\pi/4)$ $\displaystyle=-1/\sqrt{2}\ ,\vspace{.2cm}$
$\displaystyle E[A(\lambda,{\bf a^{\prime}})B(\lambda,{\bf b})]\ $
$\displaystyle=\ cos\ 2({\bf a^{\prime}},{\bf b})$ $\displaystyle=\
cos(\pi/4)$ $\displaystyle=\ \ 1/\sqrt{2}\ ,\ \ {\rm and}\vspace{.2cm}$
$\displaystyle E[A(\lambda,{\bf a^{\prime}})B(\lambda,{\bf b^{\prime}})]\ $
$\displaystyle=\ cos\ 2({\bf a^{\prime}},{\bf b^{\prime}})$ $\displaystyle=\
cos(-\pi/4)$ $\displaystyle=\ \ 1/\sqrt{2}\ ,\vspace{.3cm}$

apparently yielding $E[s(\lambda)]\ \ \vspace{.3cm}=\ \ 1/\sqrt{2}\ -\
(-1/\sqrt{2})\ +\ 1/\sqrt{2}\ +\ 1/\sqrt{2}\ \ =\ \ 4/\sqrt{2}\ =\ \
2\sqrt{2}\ .$
Voila! The expected value of $s$ apparently equals $2\sqrt{2}\approx 2.8284$,
a real number outside of the interval $[-2,+2]$, defying Bell’s inequality!
What could be more simple, direct, and stunning?

Answer: . . . the truth! . . . OK, what is wrong, if anything?

The answer is seen most simply by constructing and then examining a matrix,
which in the jargon of the operational subjective theory of probability is
called “the realm matrix of possible observation values” that could result
from the performance of the gedankenexperiment in CHSH form. I will display
this entire matrix on the next page, in a partitioned form of its full
extension as it pertains to every aspect of the problem we shall discuss. Then
we shall discuss it, piece by piece. I should mention here that while the name
“realm matrix of possibilities” has arisen from within the operational
subjective construction of the theory of probability, the matrix itself is
merely a well-defined matrix of numbers that can be understood and appreciated
by any experimentalist, no matter what may be your personal views about the
foundations of probability. In the jargon of quantum physics it might be
called the ensemble matrix of possible observation vectors.

##  6 A neglected functional dependence

In specifying the QM motivated expectation $E[s(\lambda)]$ as they do in our
equation $(8)$, Aspect/Bell fail to recognize a symmetric functional
dependence among the values of the four proposed polarization products
composing $s(\lambda)$ as defined in equation $(7)$, when it is meant to
correspond to the result of the 4-ply thought-experiment on the same pair of
photons. Perhaps surprisingly, the achieved values of any three products of
the paired polarization indicators imply a unique value for the fourth
product. We now engage to substantiate this claim.

###  6.1 The realm matrix of experimental quantities

Consider the realm matrix of all quantities relevant to the observations that
might be made in the proposed 4-ply gedankenexperiment on a pair of photons
under investigation. On the left side of the realm equation is written the
name ${\bf R}({\bf X})$, where ${\bf X}$ is a partitioned vector of names of
every quantity that will be relevant to the outcome of the experiment and what
quantum theory asserts about it. You will already recognize those in the first
two partitioned blocks. On the right side of the realm equation appears a
matrix whose columns exhaustively identify the values of these partitioned
quantities that could possibly result from conducting the gedankenexperiment.
We shall discuss them in turn.

${\bf R}\left(\begin{array}[]{c}A({\bf a})\\\ B({\bf b})\\\ A({\bf
a}^{\prime})\\\ B({\bf b}^{\prime})\\\ ****\\\ A({\bf a})B({\bf b})\\\ A({\bf
a})B({\bf b}^{\prime})\\\ A({\bf a}^{\prime})B({\bf b})\\\ A({\bf
a}^{\prime})B({\bf b}^{\prime})\\\ ****\\\ \mathcal{A}({\bf
a}^{\prime})\mathcal{B}({\bf b}^{\prime})\\\ ****\\\ \Sigma_{/({\bf a},{\bf
b})}\\\ \Sigma_{/({\bf a},{\bf b}^{\prime})}\\\ \Sigma_{/({\bf
a}^{\prime},{\bf b})}\\\ \Sigma_{/({\bf a}^{\prime},{\bf b}^{\prime})}\\\
****\\\ s(\lambda)\\\ s_{\mathcal{A/B}({\bf a}^{\prime},{\bf b}^{\prime})}\\\
1\end{array}\right)\ \ =\ \
\left(\begin{array}[]{cccccccccccccccc}1&1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1\\\
1&1&1&1&1&1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1\\\ 1&1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1\\\
1&\hbox to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&1&\hbox
to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&1&\hbox
to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1\\\
&&&&&&&&&&&&&&&\\\ 1&1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&1&1\\\ 1&\hbox to0.0pt{\hss$-$}1&1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&1&1&\hbox
to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&\hbox
to0.0pt{\hss$-$}1&1\\\ 1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1\\\ 1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1\\\
&&&&&&&&&&&&&&&\\\ 1&1&1&1&1&1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1\\\ &&&&&&&&&&&&&&&\\\ 3&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&1&\hbox
to0.0pt{\hss$-$}3&1&1&\hbox to0.0pt{\hss$-$}3&1&1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&3\\\
3&1&\hbox to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}3&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}3&\hbox to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&1&3\\\
3&\hbox to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}3&1&1&\hbox to0.0pt{\hss$-$}3&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&3\\\
3&1&1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&\hbox
to0.0pt{\hss$-$}3&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}3&1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&1&3\\\
&&&&&&&&&&&&&&&\\\ 2&2&\hbox to0.0pt{\hss$-$}2&2&2&\hbox
to0.0pt{\hss$-$}2&\hbox to0.0pt{\hss$-$}2&\hbox to0.0pt{\hss$-$}2&\hbox
to0.0pt{\hss$-$}2&\hbox to0.0pt{\hss$-$}2&\hbox to0.0pt{\hss$-$}2&2&2&\hbox
to0.0pt{\hss$-$}2&2&2\\\ 2&4&0&2&2&0&0&\hbox to0.0pt{\hss$-$}2&\hbox
to0.0pt{\hss$-$}4&\hbox to0.0pt{\hss$-$}2&\hbox to0.0pt{\hss$-$}2&0&0&\hbox
to0.0pt{\hss$-$}2&2&0\\\ 1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\end{array}\right)$

The sixteen columns of four-dimensional vectors in the first partitioned block
exhaustively list all the speculative $4\times 1$ vectors of observation
values that could possibly arise among the four experimental detections of
photons at the four angles of polarizer pairings. In order to observe the
detection products at the four relative angles $A({\bf a})B({\bf b}),\,A({\bf
a})B({\bf b}^{\prime}),\,A({\bf a}^{\prime})B({\bf b}),\,$ and $A({\bf
a}^{\prime})B({\bf b}^{\prime})\,$, we would surely have to observe each of
the four multiplicands involved in their specification: $A({\bf a}),\,B({\bf
b}),\,A({\bf a^{\prime}})$, and $B({\bf b^{\prime}})$. Since each of these
observation values might equal only either $-1$ or $+1$, there are sixteen
possibilities of the 4-dimensional result of the 4-ply experiment. There are
no presumptions made about these prospective quantity values: neither whether
they “exist” or not prior to the conduct of the experiment at all, nor even
whether they exist in any form after the experiment is conducted. We have
merely made a list of what we could possibly observe if indeed we were capable
of conducting the proposed gedankenexperiment on the same pair of photons. The
observation vector would have to equal one of the 16 columns appearing in the
top bank of the partitioned realm matrix.

Every other component quantity in the columns displayed in subsequent blocks
of the realm matrix is computed via some function of these possibilities.
Notice once again that the “exhaustiveness” of this list presupposes the
principle of local realism, specifying for example that the value of $A({\bf
a})$ identifying whether the photon passes through the polarizer at $A$ or
not, would be the same no matter whether the polarizer at which the paired
photon engages station $B$ is set at direction ${\bf b}$ or at${\bf
b^{\prime}}.$

To begin the completion of the realm matrix, the second block of components
identifies the four designated products of the paired polarization indicators
that yield the value of the quantity $s$ as it is simply defined in equation
$(6)$. The first row of this second block, identifying the product $A({\bf
a})B({\bf b})$, is the componentwise product of the first two rows of the
first block. The second row of this block, identifying the product $A({\bf
a})B({\bf b}^{\prime})$, is the componentwise product of the first and fourth
rows of the first block, and so on. This second block lists exhaustively all
the combinations of polarization products that we could possibly observe in
the conduct of our gedankenexperiment. Examine any one of these columns of
products, checking that in fact the value of each product in that column is
equal to the product of the corresponding multiplicands appearing in the
column directly above it.

The first item to notice about this realm matrix is that, whereas the sixteen
columns of the first block of polarization observations are distinct, the
second block contains only eight distinct column vectors. Columns 9 through 16
in block two of the realm matrix reproduce columns 1 through 8 in reverse
order. Moreover, examining the first three rows of this second block more
closely, it can be recognized that the first eight columns of these rows
exclusively exhaust the simultaneous measurement possibilities for the three
product quantities they identify. These are the eight vectors of the cartesian
product $\\{+1,-1\\}^{3}\,$, which are repeated in columns nine through
sixteen in reverse order. Together, what these two observations mean is that
the fourth product quantity in this second block of vector components is
derivable as a function of the first three. What is more, any one of the
product quantities identified in block two is determined by the same
computational function of the other three! This is what I meant earlier when
alluding that the photon detection products in the gedankenexperiment have
embedded within them four symmetric functional relations. This can be seen by
examining the columns of the fourth block of the matrix, which we shall do
shortly.

The third block of the realm matrix contains only a single row, corresponding
to a quantity we designate as $\mathcal{A}({\bf a}^{\prime})\mathcal{B}({\bf
b}^{\prime})$. This quantity takes values only of $\pm 1$, but it is logically
independent of the product quantities appearing in the first three rows of
block two. This is the quantity that Aspect/Bell think they are assessing when
they freely specify the quantum expectations for all four angle settings as
they do, seemingly defying Bell’s inequality. We denote its name with
calligraphic type to distinguish it from the actual polarization product
$A({\bf a}^{\prime})B({\bf b}^{\prime})$ whose functional relation to the
other three products we are now identifying. Peculiar, this singular component
of the fourth partition block is not an “Alice and Bob” observation quantity,
but rather an “Aspect/Bell” imagined quantity. It is logically independent of
the first three “Alice and Bob” products. This is to say that whatever values
these products may be, the value of $\mathcal{A}({\bf
a}^{\prime})\mathcal{B}({\bf b}^{\prime})$ may equal $+1$ in the appropriate
row among the first eight columns, or it may equal $-1$ in the corresponding
column among the second eight. However, it does not represent the photon
detection product $A({\bf a}^{\prime})B({\bf b}^{\prime})$ in the four
imagined experiments on a single photon pair.

###  6.2 Specifying the functional form via block four

Quantities in the fourth block of the realm matrix are designated with the
names $\Sigma_{/({\bf a},{\bf b})},\Sigma_{/({\bf a},{\bf b}^{\prime})}$,
$\Sigma_{/({\bf a}^{\prime},{\bf b})}$, and $\Sigma_{/({\bf a}^{\prime},{\bf
b}^{\prime})}$. These quantities are defined by sums of column elements in
those rows of the second block that are not marked behind the slash in the
notational subscript. For examples,
$\Sigma_{/({\bf a},{\bf b})}\ \equiv\ A({\bf a})B({\bf b}^{\prime})\;+\;A({\bf
a}^{\prime})B({\bf b})\;+\;A({\bf a}^{\prime})B({\bf b}^{\prime})$, and
$\Sigma_{/({\bf a},{\bf b}^{\prime})}\ \equiv\ \,A({\bf a})B({\bf
b})\,\;+\;A({\bf a}^{\prime})B({\bf b})\;+\;A({\bf a}^{\prime})B({\bf
b}^{\prime})\ .$
The quantities $\Sigma_{/({\bf a}^{\prime},{\bf b})}$, and $\Sigma_{/({\bf
a}^{\prime},{\bf b}^{\prime})}$ are defined similarly.

Next to notice is that the fourth row of the second matrix block,
corresponding to $A({\bf a}^{\prime})B({\bf b}^{\prime})$, has an entry of $1$
if and only if the fourth row of the fourth block, corresponding to
$\Sigma_{/({\bf a}^{\prime},{\bf b}^{\prime})}$, has an entry of $-1$ or $+3$
in the same column. When that entry is $+1$ or $-3$, the corresponding entry
of the second block is $-1$. What this recognition does is to identify the
functional relation of the fourth polarization product to the first three
polarization products, viz.,
$A({\bf a}^{\prime})B({\bf b}^{\prime})\ =\ {\bf G}[A({\bf a})B({\bf
b}),\,A({\bf a}^{\prime})B({\bf b}),\,A({\bf a})B({\bf b}^{\prime})]\\\ \hskip
135.15059pt\ \equiv\ \left(\Sigma_{/({\bf a}^{\prime},{\bf
b}^{\prime})}=-1\;or\;+3\right)\ -\ \left(\Sigma_{/({\bf a}^{\prime},{\bf
b}^{\prime})}=+1\;or\,-3\right)\ $. $(10)$
Here and throughout this note I am using notation in which parentheses
surrounding a mathematical statement that might be true and might be false
signifies the number $1$ when the interior statement is true, and signifies
$0$ when it is false.

Some eyeball work is required to recognize functional relationship $(10)$ by
examining the final row of block two and of block four together. It may take
even more concentration to recognize that this very same functional rule
identifies each of the other three polarization products as a function of the
other three as well! The four product quantities $A(^{.})B(^{.})$ are related
by four symmetric functional relationships, each of them being calculable via
the same functional rule applied to the other three! This surprising
recognition identifies the source of the Aspect/Bell error in assessing the
QM-motivated expectation for $s(\lambda)$ in the way they do.

It is surely true that $E[s(\lambda)]$ equals a linear combination of four
expectations of polarization products, as specified in equation $(8)$.
Moreover, if the definition of $s(\lambda)$ in equation $(6)$ were understood
to represent the combination of observed products from experiments on four
distinct pairs of photons, then the possible values of $s(\lambda)$ would span
the integers $\\{-4,-2,0,2,4\\}$; the expectation of each product $E[A({\bf
a}^{*})B({\bf b}^{*})]$ would equal $-1/\sqrt{2}$ or $+1/\sqrt{2}$ as
appropriate to the angle $({\bf a}^{*},{\bf b}^{*})$; and $E[s(\lambda)]$
would equal $2\sqrt{2}$ as proposed by Aspect/Bell. This involves no violation
of any probabilistic inequality at all, and there is no suggestion of
mysterious activity of quantum mechanics.

However, when it is proposed that the paired polarization experiments at all
four considered angles pertain to the same photon pair, then each of the
products is restricted to equal the specified function value of the other
three that we identified explicitly for $A({\bf a}^{\prime})B({\bf
b}^{\prime})$ in equation $(10)$ as $\Sigma_{/({\bf a}^{\prime},{\bf
b}^{\prime})}$ via the function ${\bf G}[A({\bf a})B({\bf b}),\,A({\bf
a}^{\prime})B({\bf b}),\,A({\bf a})B({\bf b}^{\prime})]$. In this context,
Aspect’s expected quantity would be representable equivalently by any of the
following equations: $E[s(\lambda)]\vspace{.3cm}=E[A({\bf a})B({\bf
b})]-E[A({\bf a})B({\bf b}^{\prime})]+E[A({\bf a}^{\prime})B({\bf
b})]+E\\{{\bf G}[A({\bf a})B({\bf b}),A({\bf a})B({\bf b}^{\prime}),A({\bf
a}^{\prime})B({\bf b})]\\}$
$\hskip 19.91684pt\vspace{.3cm}=\,E[A({\bf a})B({\bf b})]-E[A({\bf a})B({\bf
b}^{\prime})]+E[A({\bf a}^{\prime})B({\bf b}^{\prime})]+E\\{{\bf G}[A({\bf
a})B({\bf b}),A({\bf a})B({\bf b}^{\prime}),A({\bf a}^{\prime})B({\bf
b}^{\prime})]\\}$
$\hskip 18.49411pt\vspace{.3cm}=\,E[A({\bf a})B({\bf b})]+E[A({\bf
a}^{\prime})B({\bf b})]+E[A({\bf a}^{\prime})B({\bf b}^{\prime})]-E\\{{\bf
G}[A({\bf a})B({\bf b}),A({\bf a}^{\prime})B({\bf b}),A({\bf
a}^{\prime})B({\bf b}^{\prime})]\\}$
$\hskip 17.07182pt\vspace{.3cm}=\,-E[A({\bf a})B({\bf b}^{\prime})]+E[A({\bf
a}^{\prime})B({\bf b})]+E[A({\bf a}^{\prime})B({\bf b}^{\prime})]+E\\{{\bf
G}[A({\bf a})B({\bf b}^{\prime}),A({\bf a}^{\prime})B({\bf b}),A({\bf
a}^{\prime})B({\bf b}^{\prime})]\\}$.
(11)
The symmetries imposed on this problem would yield an identical result in each
case, which would surely not yield $2\sqrt{2}$ at all. This is the
mathematical error of neglect to which the title of this current exposition
alludes. What might it yield?

The functional relation we have exposed in $(10)$ is not linear. If it were,
then the specification of an expectation for its arguments would imply the
expectation value for the function value. As it is not, the specification of
expectation values for the arguments only imply bounds on any cohering
expectation value for the fourth. These numerical bounds can be computed using
a theorem due to Bruno de Finetti which he first presented at his famous
lectures at the Institute Henri Poincaré in 1935. He named it only in his
swansong text (de Finetti, 1974). It was first characterized in the form of a
linear programming problem by Bruno and Giglio (1980), and has appeared in
various forms in recent decades. Among them are presentations in dual form by
Whittle (1970, 1971) using standard formalist notation and objectivist
concepts. We shall review the content of de Finetti’s theorem shortly, and
then examine its relevance to assessing the expectation of $s(\lambda)$
motivated by considerations of quantum mechanics. We need first to air some
further brief remarks about the final block of the realm matrix.

###  6.3 The remaining block of quantities and their realm components

The first row of block five of the realm matrix merely identifies the values
of $s(\lambda)$ associated with the polarization observation possibilities
enumerated in the columns of block one. Each component of this row is computed
from the corresponding column of block two according to the defining equation
$(1)$. It is evident that every entry of this row is either $-2$ or $+2$. This
corresponds to the argument we have made following the factorization equation
$(9)$ in Section 5. The second row of this block pertains to a quantity
denoted as $s_{\mathcal{A/B}({\bf a}^{\prime},{\bf b}^{\prime})}$. Its value
is defined similarly to equation $(6)$, but its final summand is specified as
the Aspect/Bell quantity $\mathcal{A}({\bf a}^{\prime})\mathcal{B}({\bf
b}^{\prime})$ rather than the actual polarization product quantity $A({\bf
a}^{\prime})B({\bf b}^{\prime})$ that appears in this equation defining
$s(\lambda)$. Again peculiar, its realm can be seen to include the elements
$\\{-4,-2,0,2,4\\}$ whereas the realm of $s(\lambda)$ includes only
$\\{-2,2\\}$. The fact that the possibilities for $s_{\mathcal{A/B}({\bf
a}^{\prime},{\bf b}^{\prime})}$ include both $-4$ and $+4$ is what makes it
not surprising that the expectation of this quantity is $2\sqrt{2}$ as
pronounced by proponents of the Aspect/Bell analysis.

The third row of block five is merely an accounting device, denoting that the
“sure” quantity, $1$, is equal to $1$ no matter what the observed results of
the four imagined optic experiments of Aspect/Bell might be. Its relevance
will become apparent when the need arises to apply de Finetti’s fundamental
theorem to quantum assertions.

It is time for a rest and an interlude. It is a mathematical interlude whose
complete understanding relies only on your knowledge of some basic methods of
linear algebra. If you would like a slow didactic introduction to the subject,
my best suggestion is to look at Chapter 2.10 of my book, Lad (1996). You may
even wish to start in Section 2.7. Another purely computational presentation
appears in the article of Capotorti et al (2007, Section 4). I will make
another attempt here in a brief format, merely to keep this current exposition
self-contained. What does the fundamental theorem of probability say?

##  7 The relevance of the fundamental theorem of probability

In brief, the fundamental theorem says that if you can specify expectation
values for any vector of quantities whatsoever, then the rules of probability
provide numerical bounds on a cohering expectation for any other quantity you
would like to assess. These can be computed from the compilation of a linear
programming routine. If the expectations you have specified are incoherent
(meaning self-contradictory) among themselves, then the linear programming
problems they motivate have no solution. This theorem is immediately relevant
to our situation here in which we have identified quantum-theory-motivated
expectations for any three of the four detection products that determine the
value of $s$ for the gedankenexperiment. We wish to find the bounds on the
cohering expectation for the fourth detection product which is restricted to
equal a function value determined by these three. A discursive pedagogical
introduction is available in Lad (1996, 3.10). In brief, here is how the
theorem works.

Suppose you have identified the expectations for $N$ quantities, and you are
wondering what you might assert as the expectation for another one, call it
the $(N+1)^{st}$. What you should do to assess your sensible possibilities is
firstly to construct the realm matrix of possible values for the vector of all
$(N+1)$ quantities. Let’s call the vector ${\bf
X}_{N+1}=(X_{1},X_{2},...,X_{N},X_{N+1})^{T}$, and call its realm matrix then
${\bf R}({\bf X}_{N+1})$. In general it will look something like the realm
matrix we have just constructed for various aspects of our gedankenexperiment.
It will have $N+1$ rows, and some number $K$ columns. Just as an example, the
realm matrix we have already constructed happens to have $(N+1)=16$ rows and
$K=16$ columns. (Mind you, we have not yet specified expectation for the first
N components of the quantity vector to which this realm applies, but let’s not
let that deter us. I am merely suggesting here an example of a realm matrix
that could be considered to have $(N+1)$ rows. Let’s continue with the general
abstract specification.)

Now any such vector of quantities can be expressed as the product of its realm
matrix with a particular vector of events. The matrix equation, displayed in a
form that partitions the final row, would look like this:
$\left(\begin{array}[]{c}X_{1}\\\ X_{2}\\\ X_{3}\\\ \hskip
5.69046pt{}^{\bf.}\\\ \hskip 5.69046pt{}^{\bf.}\\\ \hskip
5.69046pt{}^{\bf.}\\\ X_{N}\\\ **\\\ X_{N+1}\vspace{.2cm}\end{array}\right)\
=\ \left(\begin{array}[]{cccccc}x_{1,1}&x_{1,2}&\hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&x_{1,(K-1)}&x_{1,K}\\\
x_{2,1}&x_{2,2}&\hskip 5.69046pt{}^{\bf.}&\hskip
5.69046pt{}^{\bf.}&x_{2,(K-1)}&x_{2,K}\\\ x_{3,1}&x_{3,2}&\hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&x_{3,(K-1)}&x_{3,K}\\\ \hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}\\\
\hskip 5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip
5.69046pt{}^{\bf.}\\\ \hskip 5.69046pt{}^{\bf.}&\hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&\hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}\\\ x_{N,1}&x_{N,2}&\hskip
5.69046pt{}^{\bf.}&\hskip 5.69046pt{}^{\bf.}&x_{N,(K-1)}&x_{N,K}\\\
**&**&**&**&**&**\\\ x_{(N+1),1}&x_{(N+1),2}&\hskip 5.69046pt{}^{\bf.}&\hskip
5.69046pt{}^{\bf.}&x_{(N+1),(K-1)}&x_{(N+1),K}\end{array}\right)\
\left(\begin{array}[]{c}({\bf X}_{N+1}={\bf x}_{{}^{.}1})\\\ ({\bf
X}_{N+1}={\bf x}_{{}^{.}2})\\\ ({\bf X}_{N+1}={\bf x}_{{}^{.}3})\\\ \hskip
5.69046pt{}^{\bf.}\\\ \hskip 5.69046pt{}^{\bf.}\\\ \hskip
5.69046pt{}^{\bf.}\\\ \hskip 5.69046pt{}^{\bf.}\\\ ({\bf X}_{N+1}={\bf
x}_{{}^{.}(K-1)})\\\ ({\bf X}_{N+1}={\bf x}_{{}^{.}K})\end{array}\right).$
On the left of this equation is the column vector of the quantity observations
under consideration. To the right of the equality comes firstly the
$(N+1)\times K$ realm matrix whose $K$ columns list all the possible columns
of numbers that could possibly result as the observation vector. These $K$
columns, each of which has $(N+1)$ components, correspond to vectors denoted
as ${\bf x}_{{}^{.}1},{\bf x}_{{}^{.}2},{\bf x}_{{}^{.}3},...,{\bf
x}_{{}^{.}(K-1)}$, and ${\bf x}_{{}^{.}K}$. (The initial subscripted dot
denotes that this is a whole column of numbers. The number that follows the
dot denotes which of the columns of the matrix it is we are talking about.)
This matrix is multiplied by the final $K\times 1$ column vector of events
that identify whether the quantity vector ${\bf X}_{N+1}$ turns out upon
observation to be the first, the second, …, or the $K^{th}$ of these listed
columns. We shall denote this vector by ${\bf Q}({\bf X}_{N+1})$, and call it
“the partition vector generated by ${\bf X}_{N+1}$”. One and only one of its
component events will equal $1$ and the rest will equal $0$. But we do not
know which of them is the $1$, because we do not know which column of
possibilities in the realm matrix will be the one that represents the observed
outcome of the vector of quantities ${\bf X}_{N+1}$.

We can represent this matrix equation more concisely and in a useful form by
writing it in an abbreviated partitioned form:

$\left(\begin{array}[]{c}{\bf X}_{N}\\\ X_{N+1}\end{array}\right)\ \ =\ \ {\bf
R}\left(\begin{array}[]{c}{\bf X}_{N}\\\
X_{N+1}\end{array}\right)\vspace{.3cm}\ \ {\bf Q}({\bf X}_{(N+1)})$ .

The payoff from constructing this matrix structure is that now every row of
this partitioned equation has on its left-hand side the unknown value of a
quantity, $X_{i}$. On the right-hand side in that row appears a list of the
possible values of that quantity, each multiplied in a linear combination with
the events that denote whether each of them is indeed the value of this
quantity (in the context of the observed values of the other quantities shown
in that column as well). Each row of this equation specifies how a different
one of the quantities under consideration equals a linear combination of
events. We have heard of that before. The expectation of a linear combination
equals the same linear combination of expectations for those events, which
would be their probabilities if we could specify values for them. This tells
us that we can evaluate an expectation operator on this partitioned equation
to yield the result that

$E\left(\begin{array}[]{c}{\bf X}_{N}\\\ X_{N+1}\end{array}\right)\ \ =\ \
{\bf R}\left(\begin{array}[]{c}{\bf X}_{N}\\\
X_{N+1}\end{array}\right)\vspace{.3cm}\ \ P[{\bf Q}({\bf X}_{(N+1)})]$ .

Well, we have not mentioned anything about probability specifications
appearing in the vector $P[{\bf Q}({\bf X}_{(N+1)})]$ on the right-hand side
of this equality. The only restrictions of probability are that these must be
non-negative numbers that sum to $1$, since the vector ${\bf Q}({\bf
X}_{(N+1)})$ constitutes a partition. We have mentioned only that expectations
have been identified for the first $N$ components of the vector on the left-
hand side, $E({\bf X}_{N})$. Yet we can compute something important on the
basis of this realization. The linearity of this equation ensures that the
implied value for the expectation of the final unspecified component
$E(X_{N+1})$ must lie within a specific interval. It is computable as the
minimum and maximum values of ${\bf R}(X_{N+1}){\bf q}_{K}$
subject to the linear restrictions ${\bf R}({\bf X}_{N}){\bf q}_{K}=E({\bf
X}_{N})$ ,
as required of the expectations that we have presumed to be specified,
and where the components of ${\bf q}_{K}$ must be non-negative and must sum to
$1$.

Such a computation is provided by the procedures of a linear programming
problem. The “solutions” to these linear programming problems are the vectors
${\bf q}_{min}$ and ${\bf q}_{max}$ that yield these minimum and maximum
values for $E(X_{N+1})$ subject to these constraints. The final row vector
identifying $E(X_{N+1})$ whose extreme values we seek is called “the objective
function” of the problems. Its coefficients are the partitioned final row of
the general realm matrix we identify as ${\bf R}(X_{N+1})$. Notice that that
$X$ is not bold. It represents merely the final quantity in the column vector
${\bf X}_{N+1}$. The coefficients vector of the objective function is the
final row vector of the realm matrix.

Here are the specific details appropriate to our gedankenexperiment.

${\bf E}\left(\begin{array}[]{c}1\\\ A({\bf a})B({\bf b})\\\ A({\bf a})B({\bf
b^{\prime}})\\\ A({\bf a}^{\prime})B({\bf b})\\\ A({\bf a}^{\prime})B({\bf
b}^{\prime})\end{array}\right)\ \ \ \ =\ \ \ \
\left(\begin{array}[]{cccccccc}1&1&1&1&1&1&1&1\\\ 1&1&1&1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1\\\ 1&\hbox to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&\hbox to0.0pt{\hss$-$}1&1\\\ 1&1&\hbox
to0.0pt{\hss$-$}1&\hbox to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1\\\ 1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1&1&\hbox to0.0pt{\hss$-$}1&\hbox
to0.0pt{\hss$-$}1&1\end{array}\right)\ \ \ {\bf q}_{8}$ $(12)$

I have listed the order of the quantities in the vector at left to begin with
the sure quantity, $1$, which equals $1$ no matter what happens in the
gedankenexperiment. There follow the four summands of the CHSH quantity $s$,
of which we have noticed that each one of them is restricted in the
gedankenexperiment to equal a function value of the other three. That is why
there are only eight columns in their realm matrix, as opposed to sixteen
columns in the expansive realm matrix we have already examined. As to the
components of the vector ${\bf q}_{8}$ at the right of the right-hand side,
notice that quantum theory says nothing at all about these, individually. Each
of them would equal the probability that the 4-ply gedankenexperiment would
yield detection products designated by a specific column of the realm matrix.
However, these would involve the joint detection of photon products in four
distinct measurements that are known to be incompatible. On account of the
generalised uncertainty principle, quantum theory eschews specification of
such probabilities. Nonetheless, for any individual photon detection product
in a specific experimental design, denoted on the left-hand side of the
equation, quantum theory does specifies an expectation value of either
$1/\sqrt{2}$ or $-1/\sqrt{2}$, as we have recognized. Since these four
products are not all free to equal $+1$ or $-1$ at the same time, we may
assert expectation values for any three of them, and use linear programming
computations to find the cohering bounds on the expectation of the fourth that
would accompany them, yielding bounds on the expectation equations $(11)$.

###  7.1 The result: quantum theory identifies restrictions on the valuation
of ${\bf q}_{8}$

This is what we find. The columns of the matrix below display the computed
results of the paired ${\bf q}_{min}$ and ${\bf q}_{max}$ vectors
corresponding to four linear programming problems. Each of them determines a
bound on an expected function value that appears in one of the four forms of
the expectation value $E[s(\lambda)]$ which we displayed in equation $(11)$.
The first pair of columns, for example, identify the fifth row of the matrix
in equation $(12)$ as the objective function, $E[A({\bf a^{\prime}},{\bf
b^{\prime}})]$, constrained by QM-specified values of the expectations of the
first four rows. The second pair of columns identify the fourth row of $(12)$
as the objective function constrained by QM specifications of expectations for
rows $1,2,3,$ and $5$, and display the appropriate solution vectors; and so
on.

$\left(\begin{array}[]{ccccccccc}&{}_{min}({\bf a^{\prime}},{\bf
b^{\prime}})&{}_{max}({\bf a^{\prime}},{\bf b^{\prime}})&{}_{min}({\bf
a^{\prime}},{\bf b})&{}_{max}({\bf a^{\prime}},{\bf b})&{}_{min}({\bf a},{\bf
b^{\prime}})&{}_{max}({\bf a},{\bf b^{\prime}})&{}_{min}({\bf a},{\bf
b})&{}_{max}({\bf a},{\bf b})\\\
q_{1}&0&0.1464&0&0.1464&0.5607&0.7803&0&0.1464\\\
q_{2}&0.7803&0.5607&0&0.1464&0.1464&0&0&0.1464\\\
q_{3}&0.0732&0&0.0732&0&0&0.0732&0&0\\\
q_{4}&0&0.1464&0.7803&0.5607&0.1464&0&0&0.1464\\\
q_{5}&0&0.1464&0&0.1464&0.1464&0&0.7803&0.5607\\\
q_{6}&0.0732&0&0&0&0&0.0732&0.0732&0\\\
q_{7}&0.0732&0&0.0732&0&0&0&0.0732&0\\\
q_{8}&0&0&0.0732&0&0&0.0732&0.0732&0\end{array}\right)$

Each of these column vectors resides in 8-dimensional space, providing a
coherent assessment of probabilities for the constituent event vector ${\bf
Q}({\bf X}_{8})$, without specifying precise probabilities for any of them. In
fact, quantum theory denies itself the capability of identifying such
probabilities precisely. We will discuss this feature further, below. However
the results of the linear programming computations can and do specify
possibilities for what might be specified in a way that would cohere with what
quantum theory can and does tell us. The columns of this matrix identify some
of them. In fact, these columns display extreme values of what are possible.
Any convex (linear) combination of them would cohere with quantum theory as
well. Thus, geometrically the columns constitute vertices of a polytope of
quantum-theory-supported possibilities for $P[{\bf Q}({\bf X}_{8})]$. This
polytope is called “the convex hull” of these vectors. However, although we
have found eight of them, the rank of the matrix of all of them is only four!
That is, these eight-dimensional vectors all reside within a four-dimensional
subspace of a unit-simplex. Why is quantum theory not more specific in
specifying the expectation of Bell’s quantity $E(s)$? We shall delay this
discussion until we have clarified what we have learned from these results of
${\bf q}_{min}({\bf a}^{*},{\bf b}^{*})$ and ${\bf q}_{max}({\bf a}^{*},{\bf
b}^{*})$.

###  7.2 Implied bounds on expected detection products and on
$E(s)\in(1.1213,2]$

According to the prescription of equation $(12)$, each of these ${\bf q}_{8}$
vectors appearing in Section $7.1$ would identify a vertex of another polytope
of cohering expectation vectors for the components of the CHSH quantity $s$.
Followed at bottom by the implied expectation $E(s)$, these are

$\left(\begin{array}[]{ccccccccc}E[A({\bf a})B({\bf
b})]&0.7071&0.7071&0.7071&0.7071&0.7071&0.7071&\hbox
to0.0pt{\hss$-$}1.0000&\hbox to0.0pt{\hss$-$}0.1213\\\ E[A({\bf a})B({\bf
b^{\prime}})]&\hbox to0.0pt{\hss$-$}0.7071&\hbox to0.0pt{\hss$-$}0.7071&\hbox
to0.0pt{\hss$-$}0.7071&\hbox to0.0pt{\hss$-$}0.7071&0.1213&1.0000&\hbox
to0.0pt{\hss$-$}0.7071&\hbox to0.0pt{\hss$-$}0.7071\\\ E[A({\bf
a^{\prime}})B({\bf b})]&0.7071&0.7071&\hbox to0.0pt{\hss$-$}1.0000&\hbox
to0.0pt{\hss$-$}0.1213&0.7071&0.7071&0.7071&0.7071\\\ E[A({\bf
a^{\prime}})B({\bf b}^{\prime})]&\hbox to0.0pt{\hss$-$}1.0000&\hbox
to0.0pt{\hss$-$}0.1213&0.7071&0.7071&0.7071&0.7071&0.7071&0.7071\\\
E[s]&1.1213&2.0000&1.1213&2.0000&2.0000&1.1213&1.1213&2.0000$
$\end{array}\right)$

In any of these columns appear three values of $E[A({\bf a^{*}})B({\bf
b^{*}})]$ specifications supported by quantum theory, and a fourth value which
is either a lower bound or upper bound on any cohering expectation for the
fourth. (By the way, $0.7071$ is the value of $1/\sqrt{2}$ to four decimal
places.) At the bottom of the column is the value of $E(s)$ that would
correspond to these four. The vectors of the four $E[A({\bf a^{*}})B({\bf
b^{*}})]$ values are the vertices of the four-dimensional space of QM-
supported expectation values of the gedankenexperiment, and the value of
$E(s)$ listed at bottom would be a quantum-theory-permitting assessment of
$E(s)$, Bell’s quantity. All of their convex combinations lie within Bell’s
reputed bounds of $[-2,+2]$. There is more to be said about this, but let us
first address the question of why quantum theory leaves four dimensions of
freedom unaccounted for in its prescriptions.

###  7.3 Why are there four free dimensions to the QM specification of $E(s)$
?

Let’s just get down to it, without any prelude. Quantum theory specifies
precise values for outcome probabilities of the photon pair detections at any
choice of three angle settings of the gedankenexperiment. Consider the QM
identifications of detection probabilities at the angles $({\bf a},{\bf
b}),({\bf a},{\bf b^{\prime}})$, and $({\bf a^{\prime}},{\bf b})$. These have
been identified in our equations $(1)$, and the corresponding expectations of
the detection products have appeared in equation $(2)$. If quantum theory were
to specify a complete distribution for the outcome of this gedankenexperiment,
it would have to specify eight probabilities. These would involve three
corresponding to detection events at any one of the polarizations angles, also
jointly at any two of the three detection angles, and also at all three of the
detection angles. But according to the uncertainty principle discussed in
Section 3, the theory eschews commitments regarding the latter four of these
probabilities: neither
$P\\{[(A({\bf a})=+1)(B({\bf b})=+1)][(A({\bf a})=+1)(B({\bf
b^{\prime}})=+1)]\\}$ , nor
$P\\{[(A({\bf a})=+1)(B({\bf b})=+1)][(A({\bf a^{\prime}})=+1)(B({\bf
b^{\prime}})=+1)]\\}\vspace{.2cm}$ , nor
$P\\{[(A({\bf a^{\prime}})=+1)(B({\bf b})=+1)][(A({\bf a^{\prime}})=+1)(B({\bf
b^{\prime}})=+1)]\\}\vspace{.2cm}$ , nor
$P\\{[(A({\bf a})=+1)(B({\bf b})=+1)][(A({\bf a})=+1)(B({\bf
b^{\prime}})=+1)][(A({\bf a^{\prime}})=+1)(B({\bf b^{\prime}})=+1)]\\}$.
For each of these would amount to claims regarding the joint outcomes of
incompatible measurements, characterised by Hadamard matrix operators that do
not commute. Quantum theory explicitly avoids such claims. That leaves four
dimensions of the eight-dimensional pmf over the four detection products
unspecified … explicitly! That is why quantum theory allows four unspecified
dimensions to the expectations it provides regarding the four polarization
products on the same pair of photons.

Perhaps this comment does need a little bit more explication. You will need to
view equation $(12)$ while reading the following remarks. They concern
assertions that quantum theory does allow us to make, and those that it
doesn’t. Recall that we are considering a linear programming problem in which
quantum expectations are asserted for the polarization products at the angle
settings $({\bf a},{\bf b}),({\bf a},{\bf b^{\prime}})$, and $({\bf
a^{\prime}},{\bf b})$, and investigating coherent bounds for expectation of
the product at the setting $({\bf a^{\prime}},{\bf b^{\prime}})$. Notice
firstly that quantum theory does allow us to, and indeed insists that we
assert
$E[A({\bf a})B({\bf b})]\ \ =\ \
q_{1}+q_{2}+q_{3}+q_{4}-q_{5}-q_{6}-q_{7}-q_{8}\ \ =\ \ 1/\sqrt{2}$
Examining the corresponding columns of the realm matrix seen in $(12)$, it is
evident that these involve assertions regarding the outcomes of $(A({\bf
a})B({\bf b})=+1)$ and $(A({\bf a})B({\bf b})=-1)$ irrespective of the values
of $A({\bf a})B({\bf b^{\prime}})$ and $A({\bf a^{\prime}})B({\bf b})$. For
each of these events involve an outcome of the product $A({\bf a})B({\bf b})$
summed over all four possible joint outcomes of the products $A({\bf a})B({\bf
b^{\prime}})$ and $A({\bf a^{\prime}})B({\bf b})$. So these latter two
incompatible observations would be irrelevant to the assertion of this
expectation. The same feature would pertain to the required assertions of
$E[A({\bf a})B({\bf b^{\prime}})]$ and $E[A({\bf a^{\prime}})B({\bf b})]$
which are involved in the first LP problem. Neither of these involves any
concomitant assertions regarding observations incompatible with them. On the
other hand, an assertion of a probability for the joint occurrence of two
pairs of polarization observations, such as $P\\{[(A({\bf a})=+1)(B({\bf
b})=+1)][(A({\bf a})=+1)(B({\bf b^{\prime}})=+1)]\\}$ for example, would
require specifications of the sum $q_{1}+q_{3}$. Examining equation $(12)$
makes clear that it is only columns $1$ and $3$ of the matrix in which this
joint event is instantiated. Asserting a specific value for the sum
$q_{1}+q_{3}$ would necessarily entail assessments of joint probabilities for
incompatible events. The same would be true of any of the other three
probabilities regarding joint events for which quantum theory eschews
assessment.

If one were to claim, as do the reigning proponents of Bell violations, that
the probabilities of quantum theory support the valuation of
$E[s(\lambda)]=2\sqrt{2}$ according to the derivation that concluded our
Section 5, that would be just plain wrong. Full stop.

Our next project is an amusing one, of actually envisaging the 4-dimensional
polytope of quantum probabilities relevant to the gedankenexperiment. This
will be achieved by passing this 4-dimensional quantum polytope through our
3-dimensional space, so to view it, just as the inhabitants of 2-dimensional
space in Flatland (Abbott, 1884) viewed the sphere passing through their lower
dimensional world. It suddenly appeared as a point, which gradually expanded
to circles of increasing diameter, and then diminished until they suddenly
disappeared again. Let’s view what we can of our 4-dimensional quantum
polytope in this way.

###  7.4 Transforming the expectation polytope into quantum probabilities

The expected photon detection products displayed in Section $7.2$ can be
transformed into $P_{++}$ probabilities by applying the transformation
$P_{++}({\bf a}^{*},{\bf b}^{*})=[\,E({\bf a}^{*},{\bf b}^{*})+1]/4$ of
equation $(3)$ to the eight vertices. This yields the vertices of another
polytope in the space of the probability vector $[P_{++}({\bf a},{\bf
b}),P_{++}({\bf a},{\bf b^{\prime}}),P_{++}({\bf a^{\prime}},{\bf
b}),P_{++}({\bf a^{\prime}},{\bf b^{\prime}})]$ displayed below:

$\left(\begin{array}[]{ccccccccc}P_{++}({\bf a},{\bf
b})&0.4268&0.4268&0.4268&0.4268&0.4268&0.4268&0&0.2197\\\ P_{++}({\bf a},{\bf
b^{\prime}})&0.0732&0.0732&0.0732&0.0732&0.2803&0.5000&0.0732&0.0732\\\
P_{++}({\bf a^{\prime}},{\bf
b})&0.4268&0.4268&0&0.2197&0.4268&0.4268&0.4268&0.4268\\\ P_{++}({\bf
a^{\prime}},{\bf
b^{\prime}})&0&0.2197&0.4268&0.4268&0.4268&0.4268&0.4268&0.4268$
$\end{array}\right)\;.$

###  7.5 And now viewing it ! … as it passes through our space

The convex hull of the 4-D column vectors shown in Section $7.4$ can be
visualized through a sequence of 3-D intersections it affords with slices
perpendicular to any one of its axes. Figure $3$ on the next page displays
such a sequence, by slices perpendicular to the $P_{++}({\bf a}^{\prime},{\bf
b}^{\prime})$ axis at values increasing from $0$ to $.4268$. When $P_{++}({\bf
a}^{\prime},{\bf b}^{\prime})=0$, the intersection of the slice identifies
only a single vertex point $(.0732,.4268,.4268)$ which appears in the subplot
$(1,1)$. See also column one of the matrix in Section $7.4$. As the value of
$P_{++}({\bf a}^{\prime},{\bf b}^{\prime})$ for the slice level increases to a
.1098 in subplot $(2,1)$, the intersection appears as a tetrahedron. The size
of the intersecting tetrahedron increases further at the probability level
$.2197$ in subplot $(3,1)$. The tetrahedrons continue to increase in size as
the level of the $P_{++}({\bf a}^{\prime},{\bf b}^{\prime})$ increases still
further to $.2561$ in subplot (1,2), but a corner of their intersections
begins to be cut off there. This clipped portion is cut more severely from the
enlarging polytope as $P_{++}({\bf a}^{\prime},{\bf b}^{\prime})$ increases
further, displayed in subplots $(2,2)$ through $(3,2)$ which is our view of
the polytope when it suddenly disappears.

Figure 3: Sequential intersections of the 4-D convex hull of vectors
$[P_{++}({\bf a},{\bf b}),P_{++}({\bf a},{\bf b}^{\prime}),P_{++}({\bf
a}^{\prime},{\bf b}),P_{++}({\bf a}^{\prime},{\bf b}^{\prime})]$ with slices
perpendicular to the $P_{++}({\bf a}^{\prime},{\bf b}^{\prime})$ axis, at
levels increasing from $0$ to $.4268$.

The symmetry of the configuration implies that slices along the other axes
would create identical intersection sequences. This Figure was produced by my
colleague Rachael Tappenden who has also produced a moving sequence of this
progression as the intersections proceed along the $P_{++}({\bf
a}^{\prime},{\bf b}^{\prime})$ axis in more refined stages. I will make this
available online when I get to it.

##  8 What to make of Aspect’s and subsequent empiricism

Taken in by the alluring derivation of Section $5.3$ which ignores the
symmetric functional relations among the polarization products of the
gedankenexperiment, Aspect and followers were convinced that Bell’s inequality
has been defied, and that the theory of hidden variables must be rejected.
This conclusion would support the assertion that quantum theory has identified
the structure of randomness which supposedly inheres in Nature at its finest
resolution. The behaviour of the photons is considered to be governed purely
by a probability distribution. It remained only to devise some physical
experiments that could verify the defiance of the inequality.

According to the tenets of objective probability theory and its statistical
programme, probabilities are not observable quantities. What are observable
are outcomes of random variables which are generated by them. It is a matter
of statistical theory to devise methods for estimating the unobservable
probabilities and their implied expectations from carefully observed outcomes
of the random variables they generate. Understood in this way, equation $(8)$
which I repeat here constitutes a structure requiring estimation if the
violation of Bell’s inequality is to be verified:
$E[s(\lambda)]\,\vspace{.15cm}\ \ =\ \ \,E[A({\bf a})B({\bf b})]\ -\ E[A({\bf
a})B({\bf b}^{\prime})]\ +\ E[A({\bf a}^{\prime})B({\bf b})]\ +\ E[A({\bf
a}^{\prime})B({\bf b}^{\prime})].\ \ \ \ \ \ \ \ \ $
According to long respected statistical procedures, the unobservable
expectations of detection products on the right-hand-side of this equation can
be estimated by the generally applicable non-parametric method of moments.
Supported by the probabilistic law of large numbers, its validity as an
estimating procedure stems from the 1930’s.

The programme for estimating equation $(8)$ would proceed as follows. To
estimate the first component of $E[s(\lambda)]$, which is $E[A({\bf a})B({\bf
b})]$, one would conduct $N$ independent polarization experiments at the angle
setting $({\bf a},{\bf b})$, and record the value of the polarization products
$A({\bf a})B({\bf b})$ observed in each case, these being either $-1$ or $+1$.
The average of these values would provide a method of moments estimate of the
expectation $E[A({\bf a})B({\bf b})]$ which is common to all of these random
experiments. A similar programme would be followed in estimating the other
three components of $E[s(\lambda)]$.

Using the notation of Aspect (2002) we would conduct $N$ repetitions of the
CHSH/Bell experiment with the relative polarizing angles set at $({\bf a},{\bf
b})$, resulting in $N_{++}({\bf a},{\bf b})$ observations of $(A({\bf
a}),B({\bf b}))=(+,+)$, $N_{+-}({\bf a},{\bf b})$ observations of $(+,-)$,
$N_{-+}({\bf a},{\bf b})$ observations of $(-,+)$, and $N_{--}({\bf a},{\bf
b})$ observations of $(-,-)$. An estimated version of equation $(8)$ would
then be expressed as
$\displaystyle\hat{E}[s(\lambda)]\,\vspace{.2cm}\ \ \equiv\ \
\,\displaystyle\hat{E}[A({\bf a})B({\bf b})]\ -\ \displaystyle\hat{E}[A({\bf
a})B({\bf b}^{\prime})]\ +\ \displaystyle\hat{E}[A({\bf a}^{\prime})B({\bf
b})]\ +\ \displaystyle\hat{E}[A({\bf a}^{\prime})B({\bf b}^{\prime})]\,,\ \ \
\ \ \ (13)$
where the component estimator $\displaystyle\hat{E}[A({\bf a})B({\bf b})]$ is
defined by

$\displaystyle\hat{E}[A({\bf a})B({\bf b})]\ \equiv\ \frac{[N_{++}({\bf
a},{\bf b})-N_{+-}({\bf a},{\bf b})-N_{-+}({\bf a},{\bf b})+N_{--}({\bf
a},{\bf b})]}{[N_{++}({\bf a},{\bf b})+N_{+-}({\bf a},{\bf b})+N_{-+}({\bf
a},{\bf b})+N_{--}({\bf a},{\bf b})]}\ \ $, $(14)$
with a similar specification for the components of $\hat{E}[s(\lambda)]$
pertaining to the relative angles $({\bf a}^{\prime},{\bf b})$, $({\bf a},{\bf
b}^{\prime})$, and $({\bf a}^{\prime},{\bf b}^{\prime})$. The denominator of
$(14)$ is equal to $N$, the number of experiments run at this angle, merely
displayed as the sum of its four component counts of $N_{\pm\pm}({\bf a},{\bf
b})$.

The momentous results were published in Aspect et al (1982), confirming the
apparent defiance of Bell’s inequality to several decimal places.

###  8.1 Examining and reassessing Aspect’s empirical results

What are we to make of Aspect’s and subsequent empirical results ?

Aspect (2002, page 15, and 1982) reports the estimation $\hat{E}[s(\lambda)]$
from experimental data, using the method of moments as defined in equations
$(13)$ and $(14)$. Of course actually, it is impossible to conduct an
experiment on a single pair of photons at all four angle settings, much less
conduct a sequence of such experiments. Instead, experimental sequences of
observations using different photon pairs were generated at each of four angle
settings. These were presumed to provide independent estimates of the four
expectations as they appear in equation $(14)$. These independent estimates
were then inserted into equation $(13)$, yielding Aspect’s touted estimate
$\hat{E}[s(\lambda)]$ near to $2\sqrt{2}$.

Although experimentation protocols have subsequently been improved to account
for the challenges of possible loopholes during the following thirty years,
the estimation procedures using the improved data have been the same. Results
from several of the improved protocols have been reported only in the form of
so-called p-values of significance for hypothesis tests posed as to whether
$E[s(\lambda)]$ exceeds $2$ or not. The results have been lionized, apparently
quite impressive, and deemed to be decisive.

We can now recognize the fault in Aspect’s estimation procedure which allows
complete liberty in all four polarization product estimations $\hat{E}[A({\bf
a}^{*})B({\bf b}^{*})]$, using experimental incidence values of
$N_{\pm\pm}({\bf a}^{*},{\bf b}^{*})$ from many experimental runs with
different photon pairs. Each of his experimental observations may be whatever
value it happens to be at its experimental angle setting, identifying whatever
value of polarization product that it does. However, if the estimation were
meant to apply to the ontological understanding of $s(\lambda)$ in the
gedankenexperiment within which he and Bell couch their theoretical claims, he
would have to adjust this methodology. One might well pick experimental runs
using three different photon pairs at any three angles one wishes, to simulate
the behaviours $A({\bf a}^{*})B({\bf b}^{*})$ for any three polarization
products of a single pair of photons. However, to be consistent with the
Aspect/Bell problem as posed for this single pair of photons at all four
relative angle settings, one then would need to compute the implied value of
the polarization product observation for the fourth angle according to the
functional form that we have identified in equation $(10)$. The same
functional form connects the detection product at any one of the four angle
settings to the other three.

Statistical estimation values reported by Aspect as well as those by
subsequent research groups over the past thirty years have no relevance to the
estimation of $E[s(\lambda)]$ as it is understood to pertain to four spin
products on a single pair of photons. It is perfectly reasonable to find
estimation values exceeding the bounds of $[-2,+2]$ as they have. For although
these results could reasonably pertain to an estimate of $E[s(\lambda)]$ with
$s(\lambda)$ defined as a combination of polarization products on four
different pairs of photons, they do not pertain to Bell’s inequality which is
relevant to a 4-ply gedankenexperiment on the same pair of photons at all four
angle settings. In the context to which their experimental results are
appropriate, $E[s(\lambda)]$ is not bound by the Bell bounds of $[-2,\,+2]$,
but rather by the interval $[-4,\,+4]$ which is unchallenged in this context.

Nonetheless, Aspect’s empirical estimation programme might be adjusted to
account for the symmetric functional relations that would necessarily
characterise the imagined results of the gedankenexperiment. In the next
subsection I shall display the unsurprising results of such an adjusted
methodology. They do not suggest any defiance of Bell’s inequality at all. The
simulation I construct will mimic the way Aspect’s data needs to be treated,
recognizing his data as the result of conditionally independent experiments on
distinct pairs of photons at each of the four relative angle settings of the
polarizers.

###  8.2 Exposition by simulation

Because Aspect’s experimental observation data is not available in full, a
method for correcting his estimation procedure shall now be displayed, along
with its numerical implications. To begin, four columns of one million
$(10^{6})$ pseudo random numbers, uniform on $[0,1]$, were generated with a
MATLAB routine. These were then transformed into simulated observations of
paired photon polarization experiments at the four relative angles we have
been studying. These transformations were performed using the QM probabilities
based on calculations of $\frac{1}{2}\;cos^{2}({\bf a}^{*},{\bf b}^{*})$ and
$\frac{1}{2}\;sin^{2}({\bf a}^{*},{\bf b}^{*})$ as described in our equations
$(1)$. Each resulting simulated polarization pair was then multiplied together
to yield a polarization product. In this way were created four columns of
simulated observations corresponding to polarization products from one million
experiments at each of the four angles: $({\bf a}^{\prime},{\bf
b}^{\prime}),\;({\bf a},{\bf b}^{\prime}),\;({\bf a},{\bf b}),\;({\bf
a}^{\prime},{\bf b})$. We shall refer to this matrix of simulated polarization
products below as the SIMPROD matrix.

Aspect’s estimation equation $(14)$ was applied to each of these columns,
yielding estimates of the expected polarization product pertinent to that
column, $\hat{E}[A({\bf a}^{*})B({\bf b}^{*})]$. These appear in the first row
of Table $1$. These four estimates were then inserted into equation $(13)$
appropriately to yield an Aspect estimate $\hat{E}[s(\lambda)]=2.827738$,
appearing in the second row of the Table under each of these columns. This
number is quite near to $2\sqrt{2}\approx 2.828427$, as was Aspect’s reported
empirical estimate, proposed as an evidential violation of Bell’s inequality.
As we now know, the problem is that when the product observations are supposed
to apply to the same photon pair, the observed value of the polarization
product at any angle is required to be related to the product at the other
three angles via the functional equation we exemplified our equation $(10)$.
The four of them may not all range freely, as they may in real experiments on
different pairs of photons. Rather, they are required to be bound by the
symmetric functional relation $G(^{.},^{.},^{.})$ that we have identified. The
rows of the matrix SIMPROD do not respect this requirement, so the Aspect
estimate $\hat{E}[s(\lambda)]$ which they produce cannot be used to estimate
the expected value of $s(\lambda)$ for the gedankenexperiment. We shall now
endeavor to correct this error.

Table 1: Corrections to Aspect’s estimate of $E[s(\lambda)]$
$({\bf a}^{*},{\bf b}^{*})$ $({\bf a},{\bf b})$ $({\bf a},{\bf b}^{\prime})$
$({\bf a}^{\prime},{\bf b})$ $({\bf a}^{\prime},{\bf b}^{\prime})$
$\hat{E}[A({\bf a}^{*})B({\bf b}^{*})]$ $0.707232$ $-$$0.706186$ $0.706840$
$0.707480$ Aspect $\hat{E}[s]$ $2.827738$ $2.827738$ $2.827738$ $2.827738$
Functional $\hat{E}[A({\bf a}^{*})B({\bf b}^{*})]$ $-$$0.353078$ $0.354348$
$-$$0.354766$ $-$$0.353934$ Corrected $\hat{E}[s]$ $1.767180$ $1.767204$
$1.765740$ $1.766964$

The third row of Table $1$ has been generated then by first applying the
function $G(^{.},^{.},^{.})$ to each choice of three components of the rows of
the SIMPROD matrix. Each result was entered into the same row of a companion
matrix of the same size, but placed into the column corresponding to the
column entry that was not used in the evaluation of the $G$ function. Let’s
call this matrix by the name SIMGEN. Next, Aspect’s estimation equation $(14)$
was applied to each of the four columns of SIMGEN, and the result is printed
in the third row of Table $1$, labeled “Functional $\hat{E}[A({\bf
a}^{*})B({\bf b}^{*})]$”. These display estimates of $E[G(^{.},^{.},^{.})]$
required for estimation of the four alternative expectation equations $(11)$.
In this way we can be considered to have generated $4$ times $10^{6}$
simulated versions of the Aspect/Bell gedankenexperiment. Their component
results can be taken to be any three simulation results from a row of SIMPROD
along with the fourth result being the functionally generated result found in
the same row and the appropriate fourth column of SIMGEN. Finally, the last
row of Table $1$ presents the estimated values of $E[s(\lambda)]$ deriving
from these simulated experiments. They appear as “corrected estimates”, column
by column, for each of which the $\hat{E}[G(^{.},^{.},^{.})]$ is the one
appropriate to that column while the other three expected polarization
products are those appropriate to the other three columns of row $1$ of the
Table. The elements of this row display corrected estimates of $E[s(\lambda)]$
as they should be calculated with the simulated Aspect data. Each of these
four estimates is slightly different from the others. Averaging them over the
four ways of generating a column of polarization products from the other three
columns of simulated products would yield a “Corrected estimate” of
$E[s(\lambda)]$ as $1.766772\,$, well within the Bell bounds of $[-2,\,+2]$.

Based on Aspect’s report of his experimental data, I feel quite sure that
applying this same estimation procedure to his experimental data, considered
as a simulation of the impossible gedankenexperiment, would yield a similar
result.

Results on the order of this peculiar number are quite stable over repeated
runs of this simulation as described. Since the theoretical analysis reported
in this article yields only an interval of cohering possibilities for
$E[s(\lambda)]$, this simulation leaves us with a tantalizing problem of how
to account for this stable result, which is quite near to
$[3/\sqrt{2}-1/(2\sqrt{2})]\approx 1.767766952966369$. Most likely, this
specific result is a construct of the independence feature embedded in the
simulation results across angle pairings. Such a feature would be highly
suspect in Nature, given what we know now about quantum entanglement itself in
a single experiment. I should mention that among all distributions in the
polytope cohering with the prescriptions of quantum theory, the maximum
entropy distribution inheres an expectation value $E(s)=1.1522$. Discussion of
its assessment and related issues must await another forum. However, there can
be no real empirical evidence on the issue, since it is impossible in
principle to activate the setup of the four imagined simultaneous experiments
on a single pair of photons. Thus, the physicists’ long interest in the fabled
gedankenexperiment.

###  8.3 A comment on empirical work and statistical estimation

While Aspect’s conception of statistical estimates appropriate to the photon
detection problem is understandable, and corrections can be made to improve
its relevance to the Aspect/Bell problem, developments of statistical theory
and practice during the past fifty years have surely generated superior
methods for evaluating the physical theory of quantum behavior. These rely on
the subjective theory of probability which, under the leadership of Bruno de
Finetti and researchers adhering to his viewpoint, has gained substantial
credibility from the past half-century of research in the foundations of
probability and statistics. There are even some prominent physicists among its
proponents, though not many. Proclaimers of inherent randomness in the physics
of quantum behaviour have won the day for now, largely on the basis of the
mistaken violation of Bell’s inequality that we have debunked in this essay.
In the very least, it is apparent that calls for open access to raw data
(Khrennikov, 2015) from several well-known research programs that publish
summary results, usually in the form of p-values, need to be heeded.

##  9 Concluding comments

The mathematical structure of the Aspect/Bell problem and its resolution align
well with the theory of subjective probability. This viewpoint is in keeping
with Einstein’s interpretation of quantum mechanics, known by his famous adage
that the old one does not roll dice. However, readers more comfortable with
the standard realist interpretation of quantum mechanics may also consider the
probabilities as ontic properties of the photons themselves without disturbing
the mathematical issues we have engaged. Anyone who professes uncertain
knowledge about the possible values of a quantum optical gedankenexperiment
may assert whatever probabilities are deemed appropriate for the sixteen
possible observation vectors displayed in block one of our experimental realm
matrix. This may involve as many or as few expectations as one wishes, and
whether based on the theory of quantum mechanics or not. These of course need
to be assessed scientifically in the light of what evidence can be brought to
bear. Similarly, realist proponents of quantum theory may hypothesize whatever
probability values they think it prescribes. However, since the sixteen
vectors of possible polarization observations listed in the realm matrix
provide an exclusive and exhaustive list of possible gedankenexperiment
results, the sum of these probabilities must equal $1$ for anyone who makes
coherent assertions. This understanding is what resolves the conundrum posed
by apparent violations of Bell’s inequality.

As to the characterization of the theory of hidden variables, this is another
endeavour that has been misconstrued in accepted literature, largely on the
basis of the mistaken understanding of the defiance of Bell’s inequality which
we have corrected here. I have examined this matter in a separate article,
entitled “Resurrection of the principle of local realism and the prospects for
supplementary variables.” For now I shall merely state that mathematically,
the theory of supplementary variables specifies the form of a mixing density
$\rho(\lambda)$ that can be made isomorphic to any coherent distribution over
the empirical observations of polarization experiments whatsoever. It matters
not whether they are the prescriptions of quantum theory or not. At any rate,
no coherent distribution over observable quantities, whether considered to be
a formalization of hidden variables theory or not, supports the defiance of
Bell’s inequality.

Virtually all discussion of quantum probabilities since the original work of
Bell has supported the conclusion that probabilities pertinent to quantum
behaviour can violate the seemingly innocuous inequality that he identified.
The mathematical error that has been discovered and reported here
substantiates the end of an era of accepting this conclusion. The results we
have aired will have ramifications for many published estimations based on
more sophisticated experimentation as well. There are further consequences for
a host of theoretical issues that have been studied and discussed in the
context of a mistaken understanding. These include related notions of hidden
variables, entangled particles, and information transfer. Discussion of these
topics do require philosophical attention to a variety of conceptual
constructs in which they are imbedded. However, the analysis of Aspect/Bell
presented here has nothing to do with philosophical distinctions. It has
identified a mathematical error in accepted work that must be recognized no
matter what might be the philosophical positions of interested parties.
Probabilistic forecasts motivated by quantum theory do not violate any laws of
probability theory. Full stop.

Discussions of related issues proceeding henceforth will need to begin with
this new recognition. Interestingly, this resolution was suspected in some way
by Bell himself, though not the analytical detail. This was clearly evident in
his musings on the hidden variables question in Bell (1971) which he himself
had reprinted in a collection of his publications, Bell (1987). I have
recently learned of an article of Adenier (2001) which seems to have been
printed only by arXiv. It recognises quite clearly the relevance of Bell’s
inequality exclusively to a gedankenexperiment on the same pair of photons at
all four relative polarization angles. My discovery of the functional
relations involved and the 4-D polytope of cohering quantum theoretic
distributions for the components of the 4-ply quantity $S$ are truly novel.

A final reference relevant to this analysis is the article of Romano
Scozzafava (2000) on the role of probability in statistical physics. He
discusses several issues that clarify fundamental matters in the context of
the constructive mathematics of Bruno de Finetti’s operational subjective
statistical method.

## Acknowledgments

Thanks especially to Mike Ulrey for stimulating me to work on this problem,
and for very helpful discussions; also to Duncan Foley who has been a keen
supporter of my investigations over the years, with many helpful comments.
Neither of them should be presumed to concur with every aspect of my analysis.
Thanks also to my colleague Rachael Tappenden for programming the 3-D slices
of the QM-motivated coherent prevision polytope using MATLAB, and for
discussions concerning the linear algebraic structure of the problem. The
University of Canterbury provided computing and research facilities. Thanks to
Paul Brouwers, Steve Gourdie, and Allen Witt for IT service and consultation.

## References

Adeneir, G. (2001) A refutation of Bell’s theorem, arXiv:quant-ph/0006014v3.
Aspect, A. (2002) Bell’s Theorem: the naive view of an experimentalist,
Quantum [Un]speakables
– from Bell to Quantum Information, R.A. Bertlmann and A. Zeilinger (eds.),
Springer.
Aspect, A., Grangier, P., and Gérard R (1981) Experimental tests of realistic
local
theories via Bell’s theorem, Physical Revue Letters 47 (8): 460–3.
Aspect, A., Dalibard, J., and Gérard R (1982) Experimental tests of Bell’s
inequalities
using time-Varying analyzers, Physical Revue Letters 49 (25): 1804-7.
Bell, J.S. (1964) On the Einstein Podolsky Rosen paradox, Physics,1(3),
195-200.
Bell, J.S. (1966) On the problem of hidden variables in quantum mechanics,
Reviews of Modern
Physics, 38(3), 447-452.
Bell, J.S. (1971) Introduction to the hidden variables question, reprinted in
Bell, J.S. (1987),
29-39.
Bell, J.S. (1987) Speakable and Unspeakable in Quantum Mechanics, Cambridge
Univ Press.
Clauser, J.F., Horne, M.A., Shimony, A., and Holt, R.A. (1969) Proposed
experiment
to test local hidden-variable theories, Physical Review Letters, 23, 880\.
Einstein, A., Podolsky, B, and Rosen, N. (1935) Can quantum mechanical
description of physical reality be considered complete? The Physical Review,
47, 777-780.
de Finetti, B. (1974, 1975) Theory of Probability, 2 volumes, A.M.F. Smith and
A. Machi
(trs.), New York: John Wiley.
Khrennikov, A. (2015) Unuploaded experiments have no result, arXiv
1505.04293v2.
Lad, F. (1996) Operational Subjective Statistical Methods: a mathematical,
philosophical and
historical introduction, New York: John Wiley.
Lad, F., Dickey, J.M., and Rahman, M. (1990) The fundamental theorem of
prevision, Statistica, 50, 19-38.
Scozzafava, R. (2000) The role of probability in statistical physics,
Transport Theory and Stat-
istical Physics, 29, 107-123.
Whittle, P. (1970) Probability, Harmondsworth, Middlesex: Penguin.
Whittle, P. (1971) Optimization under Constraints, London: Wiley.