# Violation of Bell’s Inequality in the Clauser-Horne-Shimony-Holt Form with
Entangled Quantum States Revisited

Carlo Cafaro1, Christian Corda2,3,4, Philip Cairns1, Ayhan Bingolbali5,6 1SUNY
Polytechnic Institute, 12203 Albany, New York, USA  2SUNY Polytechnic
Institute, 13502 Utica, New York, USA  3Istituto Livi, 59100 Prato, Prato,
Italy  4International Institute for Applicable Mathematics and Information
Sciences, 500063 Adarsh Nagar, Hyderabad, India  5Yildiz Technical University,
34220 Istanbul, Turkey  6University at Albany-SUNY, 12222 Albany, New York,
USA

###### Abstract

Scientific imagination and experimental ingenuity are at the heart of physics.
One of the most known instances where this interplay between theory (i.e.,
foundations) and experiments (i.e., technology) occurs is in the discussion of
Bell’s inequalities. In this paper, we present a revisitation of the violation
of Bell’s inequality in the Clauser-Horne-Shimony-Holt (CHSH) form with
entangled quantum states. First, we begin with a discussion of the 1935
Einstein-Podolski-Rosen (EPR) paradox (i.e., incompleteness of quantum
mechanics) that emerges from putting the emphasis on Einstein’s locality and
the absolute character of physical phenomena. Second, we discuss Bell’s 1971
derivation of the 1969 CHSH form of the original 1964 Bell inequality in the
context of a realistic local hidden-variable theory (RLHVT). Third,
identifying the quantum-mechanical spin correlation coefficient with the RLHVT
one, we follow Gisin’s 1991 analysis to show that quantum mechanics violates
Bell’s inequality when systems are in entangled quantum states. For
pedagogical purposes, we show how the extent of this violation depends both on
the orientation of the polarizers and the degree of entanglement of the
quantum states. Fourth, we discuss the basics of the experimental verification
of Bell’s inequality in an actual laboratory as presented in the original 1982
Aspect-Grangier-Roger (AGR) experiment. Finally, we provide an outline of some
essential take home messages from this wonderful example of physics at its
best.

###### pacs:

Entanglement and quantum nonlocality (03.65.Ud), Foundations of Quantum
Mechanics: Measurement Theory (03.65.Ta), Quantum Computation (03.67.Lx),
Quantum Information (03.67.-a), Quantum Mechanics (03.65.-w).

##  I Introduction

Information theory, quantum theory, and relativity theory are essential
ingredients of theoretical physics peres2004 . Theoretical physics aims at
describing and, to a certain extent, comprehending natural events.
Unfortunately, issues emerge when one tries to combine general relativity (GR)
theory with quantum theory, two main pillars of modern physics. During the
last fifty years, quantum theory and general relativity have been the object
of an important scientific discussion by theoretical physicists Corda2 . For
details on a general discussion on the quantum gravity problem, see Corda2 .
Here we limit ourselves to release some general considerations. On the one
hand, general relativity well-describes gravitational phenomena at large
scales, starting from observations from cosmological distances to millimeter
scales. On the other hand, quantum mechanics and quantum field theory well-
describe phenomena at small scales from a fraction of a millimeter down to
$10^{-19}$ meters. Such scales are dominated by strong and electroweak
interactions. From a historical point of view, general relativity obtained
great success. For instance, the famous Soviet physicist Lev Davidovich Landau
used to say that Einstein’s theory of gravity was the world’s best scientific
theory on a par with quantum mechanics Landau . While GR has correctly
predicted theoretical results that are consistent with experiments and
observations, it has shown a variety of limitations and weaknesses as well.
Therefore, gravitational physicists can question whether nowadays GR has a
definitive behavior Corda . Unlike other theories (such as, for instance,
electromagnetism) general relativity has not been quantized so far. This issue
does not allow to treat gravity like other quantized theories. This, in turn,
constitutes a strong obstacle for the path leading to the unification of
gravitation with other forces. So far, none has been able to realize a
consistent quantum gravity theory capable of leading to the unification of
gravity with the other fundamental interactions. A key point to understand
seems to be the role played by black holes in a quantum gravity setting.
Indeed, researchers in quantum gravity have nowadays the general conviction
that black holes should be the fundamental bricks of quantum gravity, just
like atoms have been the fundamental bricks of quantum mechanics at the outset
of quantum theory. This conviction originated from an original intuition of
Bekenstein Bekenstein .

Historically and while seeking a unifying theory of all fundamental
interactions, Einstein had the conviction that quantum mechanics could not be
considered as being complete. His opinion was that scientists should attempt
to “test” quantum mechanics via a final deterministic theory which he labelled
_Generalized Theory of Gravitation_ as pointed out in Ref. Pais . Indeed,
Einstein attempted to follow such a path. However, he failed in obtaining the
final equations of such a unified field theory Pais . Nowadays, Einstein’s
opinion is still partially endorsed by some scientists Corda2 . The problem of
a missing quantization of gravity is present and issues emerge when one tries
to combine GR with quantum theory. For instance, in classical Newtonian
gravity the mass does not enter Newton’s equation of motion,

$\frac{d^{2}\vec{x}}{dt^{2}}=\vec{g}\text{,}$ (1)

with $\vec{g}$ denoting the (constant) acceleration of gravity and
$\vec{x}=\vec{x}\left(t\right)$ being the particle’s position vector. The lack
of the mass term in Eq. (1) is justified by the equality of the gravitational
and inertial masses (Equivalence Principle). The scenario is different in
quantum mechanics where, focusing on a particle in an external gravitational
potential field $\Phi_{\text{gravity}}$, the mass $m$ no longer cancels in
Schrödinger’s quantum-mechanical wave equation sakurai ,

$\left[-\frac{\hbar^{2}}{2m}\vec{\nabla}_{\vec{x}}^{2}+m\Phi_{\text{gravity}}\right]\psi\left(\vec{x}\text{,
}t\right)=i\hbar\frac{\partial}{\partial t}\psi\left(\vec{x}\text{,
}t\right)\text{,}$ (2)

with $\psi=\psi\left(\vec{x}\text{, }t\right)$ being the particle’s wave
function. Unlike Eq. (1), the mass $m$ does not disappear in Eq. (2). It
manifests itself in the combination $\hbar/m$, where
$\hbar\overset{\text{def}}{=}h/2\pi$ and $h$ denotes the Planck constant. From
Eq. (2), is seems that $m$ is expected to appear when $\hbar$ appears. It
seems reasonable to think that the challenges in merging a classical theory of
gravity and quantum theory in a single unified theory are not simply technical
and mathematical, but rather conceptual and fundamental colella1975 . In fact,
from the above example one could expect a breakdown of the Equivalence
Principle in the quantum framework, despite there is no experimental evidence
in that sense, because the Equivalence Principle is today tested with a very
high precision Corda2 ; MTW . In recent years, it was proposed in Ref.
brukner14  that _quantum causality_ might help shedding some light on
foundational issues emerging from the general relativity-quantum mechanics
combination problem. This is in agreement with the above cited point of view
of Einstein Pais .

When characterizing physical phenomena at the quantum level, the interaction
between the mechanical object under study and the observer (or, alternatively,
observer’s measuring equipment) is not negligible and cannot be predicted
bohr35 ; bohr37 ; bohr50 . The presence of such a nonnegligible interaction
yields the impossibility of unambiguously discriminating between the object
and the measuring instruments. However, the classical concept of causality
demands the possibility of sharply distinguishing between the subject and the
object. Therefore, the above mentioned impossibility of unambiguous
discrimination in a quantum-mechanical setting is not logically compatible
with the classical notion of causality. While trying to bring consistency in
science, Bohr proposed to substitute the classical ideal of causality with a
more general concept termed _complementarity_. In rather simple words, it is
clear that an individual cannot bow in front of somebody without showing one’s
back to somebody else. This simplifying statement is at the roots of one of
the most revolutionary scientific concepts of the twentieth century, namely
Bohr’s complementarity principle wheeler63 . More specifically, this principle
encodes an essential feature of quantum physics and represents the dichotomy
between the corpuscular (i.e., particle) and ondulatory (i.e., wave) nature of
quantum-mechanical objects (both matter and light). Within this dichotomy
descriptive setting, particle and wave properties are specified by well-
defined position and momentum, respectively vaccaro10 . In the famous 1935
_EPR paradox_ epr , Einstein, Podolsky, and Rosen arrived at the conclusion
that the quantum-mechanical description of physical reality provided by wave
functions was incomplete based on a line of reasoning that relied on a
gedankenexperiment (i.e., thought experiment). Then, employing the
complementarity principle along with providing a different interpretation of
the notion of locality bohr35 , Bohr criticized the EPR argumentation.

It is worth emphasizing that it is possible to formulate many different
versions of the original 1935 EPR paradox omnes . Given its historical
relevance, we would like to briefly mention Bohm’s 1951 version of the EPR
paradox Bohm ; bohm57 ; Laloe . In Bohm’s version of the paradox, pairs of
discrete variables (such as, spin components $\left\\{s_{i}\right\\}_{1\leq
i\leq 3}$ along distinct directions with $\left[s_{i}\text{,
}s_{j}\right]=i\hslash\varepsilon_{ijk}s_{k}$) replace pairs of continuous
variables (such as, position $\left\\{x_{i}\right\\}_{1\leq i\leq 3}$ and
momentum $\left\\{p_{i}\right\\}_{1\leq i\leq 3}$ with $\left[x_{i}\text{,
}p_{j}\right]=i\hslash\delta_{ij}$). Bohm considered the decay of a spin-$0$
particle into a pair of spin-$1/2$ particles, such as the decay of a neutral
pion $\pi^{0}$ into a positron-electron pair (i.e., $\pi^{0}\rightarrow
e^{+}e^{-}$). As a preliminary remark, we recall that quantum mechanics
requires that at most one spin component of each particle may be definite and
be predictable with no uncertainty, since spin components along distinct
directions constitute incompatible observables. Assume that after the decay
the two particles (i.e., $e^{+}$ and $e^{-}$) are very far apart. Assume also
to measure the $x$-component $s_{x}^{e^{-}}$ of the electron and find
$s_{x}^{e^{-}}=+\hslash/2$. Then, since the total spin is assumed to be
conserved and since the measurement on $e^{-}$ is supposed to not affect the
real factual situation of $e^{+}$, one knows with certainty that
$s_{x}^{e^{+}}=-\hslash/2$. Additionally, one could have measured
$s_{y}^{e^{-}}$ and, using the same line of reasoning, $s_{y}^{e^{+}}$ would
have been definite; and similarly for $s_{z}^{e^{+}}$. This scenario is
paradoxical since it seems that one can predict with no uncertainty the values
of all three spin components $s_{x}^{e^{+}}$, $s_{y}^{e^{+}}$, and
$s_{z}^{e^{+}}$, despite the fact that they are non-commuting observables. We
shall see that the solution to this inconsistency lies in an unsound line of
reasoning. First, one is not taking into account the fact the there are no
quantum measurement outcomes without actually performing an experiment peres78
. Second, one is neglecting the fact that the choice of the experiment
performed on $e^{-}$ dictates the prediction that can be made for the
observations of the experiment performed on $e^{+}$, being the composite
quantum system $\left(e^{-}\text{, }e^{+}\right)$ entangled. Interestingly,
quantum entanglement seems to play a key role in a possible solution to
Hawking’s information paradox in the context of black holes physics Hawking .
Indeed, the original paradox relies on the (faulty) factorization assumption
of the radiation. However, as reported in Ref. calmet22 , black hole
information is encoded in entangled macroscopic superposition states of the
radiation.

In 1964, John Bell solved the EPR paradox in Ref. bell64 . The concept of
causality played a crucial role in the EPR paradox, Bohr’s complementarity
principle and, finally, Bell’s theorem.

The literature on Bell’s inequality is vast and we shall not try to cite it
all here. However, as students, teachers, and researchers, we really enjoyed
the presentation of the 1991 Gisin work in Ref. gisin91 . For this reason, we
presented our first mathematical reconsideration of Gisin’s work in Ref.
cafaro16 . There, we reexamined Gisin’s 1991 proof regarding the violation of
Bell’s inequality for any pure entangled state of two-particle systems. Our
investigation was motivated by didactic reasons and permitted to straighten a
few mathematical points in the original proof that in no way modified the
physical content provided by Gisin’s work. In recent years elena , it was
brought to our attention that in both Gisin’s 1991 work and our 2016 work,
there was an incorrect mathematical inequality that remained to be fixed. The
work that we present in this paper is mainly motivated by our desire of fixing
this mathematical imperfection along with the will of providing a better
physical interpretation of our own 2016 work cafaro16 . In short, our goals
can be summarized as follows:

  1. [i]

Present a correct mathematical treatment of Gisin’s 1991 proof of Bell’s
inequality following the works in Refs. gisin91 ; cafaro16 .

  2. [ii]

Enhance the self-sufficiency of the presentation in Refs. gisin91 ; cafaro16
by providing a derivation of Bell’s inequality in the so-called Clauser-Horne-
Shimony-Holt (CHSH) form clauser69 , the one originally used by Gisin in his
1991 work gisin91 .

  3. [iii]

Clarify, within a unifying setting with both conceptual and experimental
insights, the comparison between theoretical predictions that emerge from
quantum mechanics (QM) and a realistic hidden variable theory (RLHVT) with
actual experimental observations carried out in the well-known 1982 Aspect-
Grangier-Roger (AGR) experiment aspect82 .

The rest of this paper is organized as follows. In Section II, we discuss the
1935 Einstein-Podolski-Rosen (EPR) paradox. In Section III, we revisit Bell’s
1971 derivation of the 1969 Clauser-Horne-Shimony-Holt (CHSH) form of the
original 1964 Bell inequality in the framework of realistic local hidden-
variable theories. In Section IV, after identifying the quantum-mechanical
spin correlation coefficient with the RLHVT one and, in addition, following
the path traced by Gisin’s 1991 analysis gisin91 , we verify in an explicit
manner that quantum mechanics violates Bell’s inequality when systems are in
entangled quantum states. Furthermore, we illustrate with a simple set of
examples how the extent of this Bell violation depends both on the orientation
of the polarizers and the degree of entanglement of the quantum states for the
physical systems being considered. In Section V, we present the essential
features of the experimental verification of Bell’s inequality as originally
proposed by Aspect-Grangier-Roger in their 1982 experiment. Finally, we
provide in Section VI an outline of some essential take home messages that
originate from this wonderful example of physics at its best.

##  II The EPR paradox

Property | Meaning
---|---
Realism | Systems have local objective properties, independent of observation
Einstein locality | Superluminal communication is impossible
Bell locality | Factorization of the joint probability distribution of the properties of a composite system
Quantum nonlocality | Entanglement allows instantaneous propagation of correlations among properties
Table 1: Schematic description of the meaning of key concepts that appear in
the Bell inequality discussion: Realism, Einstein locality, Bell locality and,
finally, quantum nonlocality.

Before discussing the EPR paradox, let us introduce some terminology. When
focusing on a realistic local hidden-variable theory, we emphasize that: i)
_realism_ means that physical systems have objective properties that are
defined prior to, and independent of, observation; ii) _locality_ means
Einstein locality, that is, physical influences do not propagate faster than
light (i.e., superluminal communication is not possible); iii) _hidden-
variables_ are variables that are not accessible from an empirical standpoint.
For completeness, we also remark that _quantum nonlocality_ denotes the
possibility that, thanks to entanglement, correlations among properties of a
physical system can propagate instantaneously. Finally, _Bell locality_ is a
property that means the factorization of the joint probability distribution of
the properties of the two subsystems that specify a composite physical system.
The coexistence of realism and locality within the same theoretical construct
is at the roots of the EPR paradox. In Table I, we report a schematic
description of the meaning of key concepts that appear in the Bell inequality
discussion: Realism, Einstein locality, Bell locality and, finally, quantum
nonlocality.

In what follows, we discuss the EPR paradox epr  by following to a great
extent the pedagogical presentation in Ref. peres95 . In 1935, Einstein,
Podolsky, and Rosen focused their attention on the physics of a composite
quantum-mechanical system specified by two distant particles and described by
an entangled wave function $\psi$ defined as epr

$\psi=\psi\left(x_{1}\text{, }x_{2}\text{; }p_{1}\text{,
}p_{2}\right)\overset{\text{def}}{=}\delta\left(x_{1}-x_{2}-L\right)\delta\left(p_{1}+p_{2}\right).$
(3)

The quantity $\delta$ in Eq. (3) is a normalizable function characterized by
an arbitrarily high and narrow peak. The quantity $L$, instead, denotes a
distance that is much larger than the range of mutual interaction between two
the particles. The wave function $\psi$ in Eq. (3) has a clear physical
interpretation. It specifies two particles that have been prepared in such a
way that their total momentum is arbitrarily close to zero, and their relative
distance is arbitrarily close to $L$. Observe that the operators $x_{1}-x_{2}$
and $p_{1}+p_{2}$ in Eq. (3) commute, $\left[x_{1}-x_{2}\text{,
}p_{1}+p_{2}\right]=0$, since the canonical quantum-mechanical commutation
relation is given by $\left[x_{k}\text{, }p_{l}\right]=i\hbar\delta_{kl}$. In
the state $\psi$ in Eq. (3) , one does not know anything about the positions
of the individual particles. Instead, one only knows their distance from each
other. Furthermore, one does not know anything of their individual momenta and
only has proper knowledge of their total momentum. However, one shall be
capable of predicting with certainty the value of $x_{2}$ if one measures
$x_{1}$, without having disturbed particle $2$ in any way. Therefore, EPR
argue that no real change can happen in the second system in consequence of
anything that may be performed on the first system since at the time of
measurement the two systems do not interact any longer. Therefore, according
to EPR, $x_{2}$ corresponds to an element of physical reality. Alternatively,
if one desires to perform a measurement of $p_{1}$ rather than $x_{1}$, one
shall then be capable of predicting with certainty the value of $p_{2}$
without having perturbed particle $2$ in any way. Therefore, by the same
argument as above, $p_{2}$ also is an element of reality according to EPR.
From this reasoning, it appears that $x_{2}$ and $p_{2}$ are equally elements
of reality. However, since $\left[x_{2}\text{, }p_{2}\right]\neq 0$, these
operators do not commute and quantum mechanics forbids the simultaneous
assignment of precise values to both $x_{2}$ and $p_{2}$. For this reason, EPR
are forced to arrive at the conclusion that the quantum-mechanical description
of physical reality in terms of wave functions is incomplete. At the same
time, EPR cautiously leave open the possibility of the existence of a complete
theory. According to EPR, reality can be described as follows: If one can
predict with certainty the value of a physical quantity without perturbing the
system in any way, then there is an element of reality that corresponds to
such a physical quantity. The EPR article was not incorrect and, arguably, it
had been written too early as pointed out by Peres in Ref. peres04 . The EPR
argument did not consider that, like any other physical object, the observer’s
information was localized. Specifically, in addition to being an abstract
notion, information needs as approximately localized physical carrier. The
problem is that Einstein’s locality principle states that events happening in
a given spacetime region do not depend on external parameters that may be
tuned, at the same moment, by agents placed in distant spacetime regions.
However, in quantum physics, one has to accept that a measurement on a part of
the system is to be considered as a measurement on the whole system. If one
wishes to keep Einstein’s locality principle, alternative theories that
incorporate such a principle lead to a testable inequality constraint among
certain observables that violates quantum-mechanical predictions. This is the
so-called Bell’s inequality bell64 . From an experimental standpoint, several
violations of Bell’s inequality have been observed peres95 . Therefore,
despite that fact that this situation may appear psychologically
uncomfortable, quantum mechanics has prevailed over alternative theories
because of the experimental verdict. Quantum-mechanical predictions violate
Bell’s inequality. Moreover, alternative theories fulfilling Einstein’s
locality principle yield experimentally verifiable differences with respect to
quantum mechanics. To a certain extent, it is ironic that Bell’s theorem can
be regarded as one of the most profound discovery of science because it is not
obeyed by experimental facts stapp75 . We emphasize that Bell’s work in Ref.
bell64  is not about quantum mechanics. Rather, it is a general proof where it
is shown the existence of an upper limit to the correlation of distant events
in any physical theory that assumes the validity of Einstein’s principle of
local causes. In particular, Bell showed that in a theory where parameters are
introduced to obtain the results of individual measurements, without modifying
the statistical predictions, there must exist a mechanism whereby the setting
of one measuring apparatus can affect the reading of another instrument,
however remote. Furthermore, such a theory could not be Lorenz invariant since
the signal being considered must propagate in an instantaneous fashion. In
particular, given an arbitrary entangled quantum state
$\left|\psi\right\rangle$, one can find pairs of observables whose
correlations violate Bell’s inequality (i.e., Eq. (18)). This implies that,
for such a state $\left|\psi\right\rangle$, quantum mechanics makes
statistical predictions that are not compatible with Einstein’s locality
(i.e., the principle of local causes). Locality requires that the outcomes of
experiments carried out at a given location in space are not dependent on
arbitrary choices of other experiments that can be performed, simultaneously,
at distant locations in space. Bell’s theorem leads to the conclusion that
quantum mechanics is not compatible with the viewpoint that physical
observables have pre-existing values that do not depend on the measurement
process. A hidden variable theory that incorporates Einstein’s locality
principle would predict individual events violating the canonical principle of
special relativity. Namely, there would exist no covariant discrimination
between cause and effect. It turns out that the EPR paradox is settled in a
manner which Einstein would have disliked the most. This point shall be
further addressed in the next sections. For the time being, we report two
different realizations of the EPR gedankenexperiment in Table II. Having
introduced the EPR paradox, we are ready to discuss the derivation of Bell’s
inequality in a RLHVT.

Scientists | Particle pairs | Experimental scheme | Quantity of interest
---|---|---|---
Bell, 1962 | Electrons | Stern-Gerlach magnet | Spin correlation coefficient
Aspect-Grangier-Roger, 1982 | Photons | Two-channel polarizer | Polarization correlation coefficient
Table 2: Different realizations of the Einstein-Podolski-Rosen (EPR)
gedankenexperiment. The spin and polarization correlation coefficients can be
experimentally determined in terms of coincidence counting rates of spin and
polarization configurations, respectively.

##  III Derivation of Bell’s inequality in a RLHVT

In this section, we present the derivation of the Bell inequality in the 1969
Clause-Horne-Shimony-Holt (CHSH) form (see Eq. (1a) in Ref. clauser69 ) as
originally discussed in 1971 by Bell (see Eqs. (9) and (20) in Chapters 4 and
16, respectively, of Ref. bell04 ). It was derived within a theoretical
framework of a realistic logical hidden variable theory. Before entering the
discussion of the derivation, let us explain in a clear manner the
significance of _Bell’s locality_. Let us consider two distant systems $S_{A}$
and $S_{B}$. Let $\rho_{AB}\left(x_{A}\text{, }x_{B}\left|X_{A}\text{,
}X_{B}\text{, }\lambda\right.\right)$ be the joint probability distribution of
the measurement outcomes $x_{A}$ and $x_{B}$ of the properties $X_{A}$ and
$X_{B}$ corresponding to systems $A$ and $B$, respectively

$\rho_{AB}\left(x_{A}\text{, }x_{B}\left|X_{A}\text{, }X_{B}\text{,
}\lambda\right.\right)=\rho_{A}\left(x_{A}\left|x_{B}\text{, }X_{A}\text{,
}X_{B}\text{, }\lambda\right.\right)\rho_{B}\left(x_{B}\left|X_{A}\text{,
}X_{B}\text{, }\lambda\right.\right)\text{.}$ (4)

The quantity $\lambda$ specifies the classical hidden-variable model. Then,
the Einstein locality condition implies that the measurement outcomes $x_{A}$
($x_{B}$) at system $S_{A}$ ($S_{B}$) cannot be influenced by the selected
property $X_{B}$ ($X_{A}$) being measured on the second system $S_{B}$
($S_{A}$). Therefore, the Einstein locality implies that
$\rho_{A}\left(x_{A}\left|x_{B}\text{, }X_{A}\text{, }X_{B}\text{,
}\lambda\right.\right)=\rho_{A}\left(x_{A}\left|X_{A}\text{,
}\lambda\right.\right)$ and $\rho_{B}\left(x_{B}\left|X_{A}\text{,
}X_{B}\text{, }\lambda\right.\right)=\rho_{B}\left(x_{B}\left|X_{B}\text{,
}\lambda\right.\right)$. Therefore, the Einstein locality implies that the
joint probability distribution in Eq. (4) can be recast in a separable form as

$\rho_{AB}\left(x_{A}\text{, }x_{B}\left|X_{A}\text{, }X_{B}\text{,
}\lambda\right.\right)=\rho_{A}\left(x_{A}\left|X_{A}\text{,
}\lambda\right.\right)\rho_{B}\left(x_{B}\left|X_{B}\text{,
}\lambda\right.\right)\text{.}$ (5)

Eq. (5) is known as the Bell locality condition. As a final remark, we observe
that the Bell locality condition is a weaker form of the Einstein locality
condition since the latter implies the former (and not the converse). We are
now ready to discuss the derivation.

Consider a system of two spin-$1/2$ particles prepared in a state such that
they proceed in distinct directions towards two measuring apparatuses. Assume
these apparatuses are used to measure spin components along unit directions
$\hat{a}$ and $\hat{b}$. Furthermore, assume the hypothetical complete
direction of the initial state of the composite quantum system can be
expressed by means of a hidden variable $\lambda$ distributed according to a
probability distribution $\rho\left(\lambda\right)$ satisfying the
normalization condition $\int\rho\left(\lambda\right)d\lambda=1$. From our
previous discussion, we note that locality requires that observables $A$ and
$B$ of the composite quantum system are such that $A=A\left(\hat{a}\text{,
}\lambda\right)$ and $B=B\left(\hat{b}\text{, }\lambda\right)$. However, $A$
($B$) does not depend on $\hat{b}$ ($\hat{a}$) and, in addition, the
measurable results of $A$ ($B$) are supposed to be $\pm 1$. Assume the spin
correlation coefficient $E\left(\hat{a}\text{, }\hat{b}\right)$ in this local
realistic theory is given by,

$E\left(\hat{a}\text{, }\hat{b}\right)\overset{\text{def}}{=}\int
A\left(\hat{a}\text{, }\lambda\right)B\left(\hat{b}\text{,
}\lambda\right)\rho\left(\lambda\right)d\lambda\text{.}$ (6)

More generally, the measuring devices themselves could depend on hidden
variables which could affect the experimental results. Averaging first over
these variables, $E\left(\hat{a}\text{, }\hat{b}\right)$ in Eq. (6) can be
represented as bell04

$E\left(\hat{a}\text{,
}\hat{b}\right)\overset{\text{def}}{=}\int\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\rho\left(\lambda\right)d\lambda\text{,}$ (7)

with $\left|\bar{A}\right|\leq 1$ and $\left|\bar{B}\right|\leq 1$. Let
$\hat{a}^{\prime}$ and $\hat{b}^{\prime}$ be alternative orientations of the
apparatuses. Then, we have

$E\left(\hat{a}\text{, }\hat{b}\right)-E\left(\hat{a}\text{,
}\hat{b}^{\prime}\right)=\int\left[\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)-\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{,
}\lambda\right)\right]\rho\left(\lambda\right)d\lambda\text{,}$ (8)

that is, adding and subtracting suitably chosen identical terms,

$E\left(\hat{a}\text{, }\hat{b}\right)-E\left(\hat{a}\text{,
}\hat{b}^{\prime}\right)=\int\left\\{\begin{array}[c]{c}\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{, }\lambda\right)\right]+\\\
-\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{,
}\lambda\right)\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\right]\end{array}\right\\}\rho\left(\lambda\right)d\lambda\text{.}$
(9)

Recalling that $\left|x-y\right|\leq\left|x\right|+\left|y\right|$ for any
$x$, $y\in\mathbb{R}$ and, in addition, $\left|x\right|=x$ when $x\geq 0$, Eq.
(9) yields

$\displaystyle\left|E\left(\hat{a}\text{,
}\hat{b}\right)-E\left(\hat{a}\text{, }\hat{b}^{\prime}\right)\right|$
$\displaystyle=\left|\int\left\\{\begin{array}[c]{c}\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{, }\lambda\right)\right]+\\\
-\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{,
}\lambda\right)\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\right]\end{array}\right\\}\rho\left(\lambda\right)d\lambda\right|$
(12) $\displaystyle\leq\left|\int\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{,
}\lambda\right)\right]\rho\left(\lambda\right)d\lambda\right|+$
$\displaystyle+\left|\int\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{,
}\lambda\right)\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\right]\rho\left(\lambda\right)d\lambda\right|$
$\displaystyle\leq\left|\int\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}^{\prime}\text{,
}\lambda\right)\right]\rho\left(\lambda\right)d\lambda\right|+\left|\int\left[1\pm\bar{A}\left(\hat{a}^{\prime}\text{,
}\lambda\right)\bar{B}\left(\hat{b}\text{,
}\lambda\right)\right]\rho\left(\lambda\right)d\lambda\right|$
$\displaystyle=2\pm\left[E\left(\hat{a}^{\prime}\text{,
}\hat{b}^{\prime}\right)+E\left(\hat{a}^{\prime}\text{,
}\hat{b}\right)\right]\text{,}$ (13)

that is,

$\left|E\left(\hat{a}\text{, }\hat{b}\right)-E\left(\hat{a}\text{,
}\hat{b}^{\prime}\right)\right|\leq 2\pm\left[E\left(\hat{a}^{\prime}\text{,
}\hat{b}^{\prime}\right)+E\left(\hat{a}^{\prime}\text{,
}\hat{b}\right)\right]\text{.}$ (14)

Finally, an alternative and more symmetric way to rewrite the inequality in
Eq. (14) is bell04

$\left|E\left(\hat{a}\text{, }\hat{b}\right)-E\left(\hat{a}\text{,
}\hat{b}^{\prime}\right)\right|+\left|E\left(\hat{a}^{\prime}\text{,
}\hat{b}^{\prime}\right)+E\left(\hat{a}^{\prime}\text{,
}\hat{b}\right)\right|\leq 2\text{.}$ (15)

The inequality in Eq. (15), also reported in Table III, is the well-known CHSH
inequality that appears in Ref. clauser69 . Observe that identifying the
hidden variable theory quantity $E\left(\hat{a}\text{, }\hat{b}\right)$ in Eq.
(7) with the quantum-mechanical quantity $P\left(a\text{, }b\right)$ in Eq.
(19), violation of Eq. (15) yields Eq. (18). As a final remark, note that when
$\hat{a}^{\prime}=\hat{b}^{\prime}$ in Eq. (15), assuming
$E\left(\hat{b}^{\prime}\text{, }\hat{b}^{\prime}\right)=-1$, Eq. (15) reduces
to

$\left|E\left(\hat{a}\text{, }\hat{b}\right)-E\left(\hat{a}\text{,
}\hat{b}^{\prime}\right)\right|\leq 2-\left|-1+E\left(\hat{b}^{\prime}\text{,
}\hat{b}\right)\right|\text{,}$ (16)

that is,

$\left|E\left(\hat{a}\text{, }\hat{b}\right)-E\left(\hat{a}\text{,
}\hat{b}^{\prime}\right)\right|\leq 1+E\left(\hat{b}^{\prime}\text{,
}\hat{b}\right)\text{.}$ (17)

The inequality in Eq. (17) is the original result obtained by Bell in 1964
(see Eq. (15) in Ref. bell64 ). From Eq. (15), we note that the CHSH Bell
inequality is obtained within the framework of a RLHVT, applies to a pair of
two-state widely separated systems and, finally, constrains the statistics of
the measurement outcomes in terms of the value of a linear combination of four
correlation functions between the two systems. We are now ready to show from a
theoretical standpoint that QM violates the CHSH Bell inequality in Eq. (15).

The Bell inequality in the CHSH form
---
$\left|E\left(\hat{a}\text{, }\hat{b}\right)-E\left(\hat{a}\text{,
}\hat{b}^{\prime}\right)\right|+\left|E\left(\hat{a}^{\prime}\text{,
}\hat{b}^{\prime}\right)+E\left(\hat{a}^{\prime}\text{,
}\hat{b}\right)\right|\leq 2$
Table 3: The Bell inequality in the Clause-Horne-Shimony-Holt (CHSH) form.
Experimentally, the inequality demands four measurements, one for each and
every polarizaton correlation coefficient: $E\left(\hat{a}\text{,
}\hat{b}\right)$, $E\left(\hat{a}\text{, }\hat{b}^{\prime}\right)$,
$E\left(\hat{a}^{\prime}\text{, }\hat{b}^{\prime}\right)$, and
$E\left(\hat{a}^{\prime}\text{, }\hat{b}\right)$. In particular,
$E\left(\hat{a}\text{, }\hat{b}\right)$ requires an experimental setup with
two polarizers in orientations $\hat{a}$ and $\hat{b}$.

##  IV Violation of Bell’s inequality in QM

This section is partitioned in two subsections. In the first subsection, we
explicitly show that QM violates the CHSH Bell inequality when the physical
systems are in entangled quantum states. In the second subsection, we
illustrate with a simple set of examples how the degree of this violation
depends on the orientation of the polarizers and, in addition, on the degree
of entanglement of the quantum states specifying the physical systems being
investigated.

###  IV.1 Formal discussion

In this first subsection, we revisit Bell’s theorem as presented by Gisin in
Ref. gisin91 . Moreover, we refer to Ref. gisin92  for a more general
derivation of Bell’s theorem by Gisin and Peres. Gisin’s version of Bell’s
Theorem can be described as follows gisin91 : Let $\left|\psi\right\rangle$ be
a two-particles quantum state that belongs to the composite Hilbert space
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. Then, $\left|\psi\right\rangle$
violates Bell’s inequality if $\left|\psi\right\rangle$ is entangled (i.e.,
nonfactorable). Stated otherwise and using standard quantum-mechanical
notations, there exist quantum-mechanical projectors $a$, $a^{\prime}$, $b$,
and $b^{\prime}$ with
$a\overset{\text{def}}{=}\left(1/2\right)\left[\mathrm{I}+\hat{a}\cdot\vec{\sigma}\right]$
such that

$\left|P\left(a\text{, }b\right)-P\left(a\text{,
}b^{\prime}\right)\right|+P\left(a^{\prime}\text{,
}b\right)+P\left(a^{\prime}\text{, }b^{\prime}\right)>2\text{.}$ (18)

For completeness, we remark that $\mathrm{I}$ is the identity operator on the
single-qubit Hilbert space, $\hat{a}$ is a unit vector, and $\vec{\sigma}$ is
the three-dimensional vector of Pauli operators. Eq. (18) is Bell’s inequality
in the so-called Clauser-Horne-Shimony-Holt (CHSH) form clauser69 . The
quantity $P\left(a\text{, }b\right)$ in Eq. (18) is the quantum-mechanical
analog of $E\left(\hat{a}\text{, }\hat{b}\right)$ in Eq. (7) and is the spin
correlation coefficient given by

$P\left(a\text{,
}b\right)\overset{\text{def}}{=}\left\langle\left(2a-1\right)\otimes\left(2b-1\right)\right\rangle_{\psi}$
(19)

To verify Eq. (18), we introduce first some relevant preliminary remarks.

  1. [i]

First, the Schmidt decomposition theorem mosca  yields the following result.
If $\left|\psi\right\rangle\in$ $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, then
there is an orthonormal basis
$\left\\{\left|\varphi_{l}\right\rangle\right\\}_{1\leq l\leq
n_{\mathcal{H}_{1}}}$ of $\mathcal{H}_{1}$ with
$n_{\mathcal{H}_{1}}\overset{\text{def}}{=}\dim_{\mathbb{C}}\mathcal{H}_{1}$,
and an orthonormal basis $\left\\{\left|\phi_{m}\right\rangle\right\\}_{1\leq
m\leq n_{\mathcal{H}_{2}}}$ of $\mathcal{H}_{2}$ with
$n_{\mathcal{H}_{2}}\overset{\text{def}}{=}\dim_{\mathbb{C}}\mathcal{H}_{2}$,
and non-negative _real_ numbers $\left\\{c_{k}\right\\}$ so that

$\left|\psi\right\rangle=\sum_{k}c_{k}\left|\varphi_{k}\right\rangle\otimes\left|\phi_{k}\right\rangle\text{.}$
(20)

The quantities $c_{k}$ in Eq. (20) are termed Schmidt coefficients, and the
number of terms in the summation is at most the minimum of
$n_{\mathcal{H}_{1}}$ and $n_{\mathcal{H}_{2}}$. To consider an entangled
state of two-particle systems, one can assume with no loss of generality that
at least two $c_{k}$-coefficients are nonvanishing in Eq. (20). Assume, for
instance, $c_{1}\neq 0\neq c_{2}$ and $c_{k}=0$ for any $k>2$. For
completeness, we point out that in principle $c_{k}$-coefficients with $k>2$
could also be different from zero. However, with no loss of generality, one
can focus the discussion to the case in which $c_{k}=0$ for any $k>2$, since
it can be shown that the Bell inequality in the CHSH form clauser69  can be
violated in any two-dimensional Hilbert subspace with nonzero Schmidt
coefficients bell64 ; prlgisin . For completeness, we point out that the
projection of the joint state of n pairs of particles onto a subspace spanned
by states having a common Schmidt coefficients is a key step in the so-called
Schmidt projection method, a technique used to study entanglement
concentration in any pure state of a bipartite system fuck01 . Furthermore,
for a practical demonstration of the violation of the CHSH version of Bell’s
inequality in $2D$ subspaces of an higher-dimensional orbital angular momentum
Hilbert space, we refer to fuck02 . For completeness, we emphasize at this
point that an essential step in the so-called Schmidt projection method, a
technique employed to investigate entanglement concentration in arbitrary pure
states of bipartite quantum systems fuck01 , is the projection of the joint
state of $n$-pairs of particles onto a subspace spanned by states that share a
common set of Schmidt coefficients. Moreover, we refer to Ref. fuck02  for a
verification of the violation of the Bell inequality in the CHSH form in two-
dimensional subspaces of an higher-dimensional orbital angular momentum
Hilbert space.

  2. [ii]

Second, the measure of entanglement vanishes for any separable quantum state.
Moreover, the degree of entanglement of a composite quantum state does not
change under local unitary transformations vedral . A local unitary
transformation is essentially a change of basis with respect to which we
consider a given entangled state. Since, at any given time, one could just
reverse the basis change, a change of basis should not be responsible for
modifying the amount of entanglement accessible to an external observer.
Therefore, the amount of entanglement should be the same in both bases. For
convenience and with no change in the entanglement behavior of the quantum
state $\left|\psi\right\rangle$, given these before-mentioned comments, let us
apply a local unitary transformation $\left(U_{1}\otimes U_{2}\right)$ on
$\left|\psi\right\rangle$ such that

$\left|\psi\right\rangle\rightarrow\left(U_{1}\otimes
U_{2}\right)\left|\psi\right\rangle=\left(U_{1}\otimes
U_{2}\right)\left(c_{1}\left|\varphi_{1}\right\rangle\otimes\left|\phi_{1}\right\rangle+c_{2}\left|\varphi_{2}\right\rangle\otimes\left|\phi_{2}\right\rangle\right)\text{,}$
(21)

where,

$\displaystyle\left(U_{1}\otimes
U_{2}\right)\left(c_{1}\left|\varphi_{1}\right\rangle\otimes\left|\phi_{1}\right\rangle+c_{2}\left|\varphi_{2}\right\rangle\otimes\left|\phi_{2}\right\rangle\right)$
$\displaystyle=c_{1}U_{1}\left|\varphi_{1}\right\rangle\otimes
U_{2}\left|\phi_{1}\right\rangle+c_{2}U_{1}\left|\varphi_{2}\right\rangle\otimes
U_{2}\left|\phi_{2}\right\rangle$
$\displaystyle=c_{1}\left|01\right\rangle+c_{2}\left|10\right\rangle\text{.}$
(22)

In the specific scenario of spin-$1/2$ systems, the state
$\left|0\right\rangle$ and $\left|1\right\rangle$ in Eq. (22) are defined as
$\left|0\right\rangle\overset{\text{def}}{=}\binom{1}{0}$, and
$\left|1\right\rangle\overset{\text{def}}{=}\binom{0}{1}$, respectively.

  3. [iii]

Third, an arbitrary density matrix $\rho$ for a single-qubit quantum system
can be recast as
$\rho\overset{\text{def}}{=}(1/2)\left[\mathrm{I}+\vec{p}\cdot\vec{\sigma}\right]$
nielsen2000 ; cafaro2012 , where $\vec{p}\in\mathbb{R}^{3}$ with
$\left\|\vec{p}\right\|\leq 1$ is the Bloch vector corresponding to the state
$\rho$, $\mathrm{I}$ is the $\left(2\times 2\right)$-identity matrix , and
$\vec{\sigma}$ is the Pauli matrix vector given in an explicit fashion by
nielsen2000 ; cafaro2010 ; cafaro2014 ; cafaro2022 ,

$\vec{\sigma}=\left[\sigma_{x}\text{, }\sigma_{y}\text{,
}\sigma_{z}\right]\overset{\text{def}}{\mathbf{=}}\left[\left(\begin{array}[c]{cc}0&1\\\
1&0\end{array}\right)\text{, }\left(\begin{array}[c]{cc}0&-i\\\
i&0\end{array}\right)\text{, }\left(\begin{array}[c]{cc}1&0\\\
0&-1\end{array}\right)\right]\text{.}$ (23)

Note that a state $\rho$ is pure if and only if $\left\|\vec{p}\right\|=1$. In
addition, a density matrix for a pure state can be described as
$\rho=\left|\psi\right\rangle\left\langle\psi\right|$ with
$\left|\psi\right\rangle\left\langle\psi\right|$ being an orthogonal projector
since $\rho^{2}=\rho$ and $\rho^{\dagger}=\rho$. The symbol “$\dagger$” is
used to denote the usual Hermitian conjugation operation in quantum mechanics.

Taking into consideration the remarks presented in the three points [i], [ii],
and [iii], we aim at verifying the correctness of the inequality in Eq. (18)
for entangled states $\left|\psi\right\rangle$ in Eq. (20) and for a
convenient selection of projectors $a$, $a^{\prime}$, $b$, and $b^{\prime}$.
The projectors $a$, $a^{\prime}$, $b$, $b^{\prime}$ are given by,

$a\overset{\text{def}}{=}\frac{1+\hat{a}\cdot\vec{\sigma}}{2}\text{,
}a^{\prime}\overset{\text{def}}{=}\frac{1+\hat{a}^{\prime}\cdot\vec{\sigma}}{2}\text{,
}b\overset{\text{def}}{=}\frac{1+\hat{b}\cdot\vec{\sigma}}{2}\text{,
}b^{\prime}\overset{\text{def}}{=}\frac{1+\hat{b}^{\prime}\cdot\vec{\sigma}}{2}\text{,}$
(24)

with $\hat{a}$, $\hat{a}^{\prime}$, $\hat{b}$,
$\hat{b}^{\prime}\in\mathbb{R}^{3}$ being unit vectors so that $a^{2}=a$,
$a^{\prime 2}=a^{\prime}$, $b^{2}=b$, and $b^{\prime 2}=b^{\prime}$. Using Eq.
(24), the spin correlation coefficient $P\left(a\text{,
}b\right)\overset{\text{def}}{=}\left\langle\left(2a-1\right)\otimes\left(2b-1\right)\right\rangle_{\psi}$
in Eq. (19) becomes

$P\left(a\text{,
}b\right)=\left\langle\psi|\left(\hat{a}\cdot\vec{\sigma}\right)\otimes(\hat{b}\cdot\vec{\sigma})|\psi\right\rangle\text{,}$
(25)

where
$\left|\psi\right\rangle\overset{\text{def}}{=}c_{1}\left|10\right\rangle+c_{2}\left|01\right\rangle$
and $c_{1}$, $c_{2}\in\mathbb{R}\backslash\left\\{0\right\\}$. To evaluate
$P\left(a\text{, }b\right)$ in Eq. (25), let us find the explicit expression
of $\left(\vec{\sigma}\cdot\hat{a}\right)\otimes(\vec{\sigma}\cdot\hat{b})$.
Making use of Eq. (23), we get after some algebra

$\left(\hat{a}\cdot\vec{\sigma}\right)\otimes(\hat{b}\cdot\vec{\sigma})=\left(\begin{array}[c]{cccc}a_{z}b_{z}&a_{z}\left(b_{x}-ib_{y}\right)&b_{z}\left(a_{x}-ia_{y}\right)&\left(a_{x}-ia_{y}\right)\left(b_{x}-ib_{y}\right)\\\
a_{z}\left(b_{x}+ib_{y}\right)&-a_{z}b_{z}&\left(a_{x}-ia_{y}\right)\left(b_{x}+ib_{y}\right)&-b_{z}\left(a_{x}-ia_{y}\right)\\\
b_{z}\left(a_{x}+ia_{y}\right)&\left(a_{x}+ia_{y}\right)\left(b_{x}-ib_{y}\right)&-a_{z}b_{z}&-a_{z}\left(b_{x}-ib_{y}\right)\\\
\left(a_{x}+ia_{y}\right)\left(b_{x}+ib_{y}\right)&-b_{z}\left(a_{x}+ia_{y}\right)&-a_{z}\left(b_{x}+ib_{y}\right)&a_{z}b_{z}\end{array}\right)\text{,}$
(26)

where $\hat{a}\overset{\text{def}}{=}\left(a_{x}\text{, }a_{y}\text{,
}a_{z}\right)$ and $\hat{b}\overset{\text{def}}{=}\left(b_{x}\text{,
}b_{y}\text{, }b_{z}\right)$. Therefore, by means of Eqs. (26), (25), and the
relation
$\left|\psi\right\rangle\overset{\text{def}}{=}c_{1}\left|10\right\rangle+c_{2}\left|01\right\rangle$,
the explicit expression of $P\left(a\text{, }b\right)$ becomes

$P\left(a\text{,
}b\right)=-\left(c_{1}^{2}+c_{2}^{2}\right)a_{z}b_{z}+c_{1}c_{2}\left(2a_{x}b_{x}+2a_{y}b_{y}\right)\text{.}$
(27)

Observe that the normalization condition for the state vector
$\left|\psi\right\rangle$ requires $\left\langle\psi|\psi\right\rangle=1$,
that is $c_{1}^{2}+c_{2}^{2}=1$. Thus, the spin correlation coefficient
$P\left(a\text{, }b\right)$ in Eq. (27) reduces to

$P\left(a\text{,
}b\right)=+2c_{1}c_{2}\left(a_{x}b_{x}+a_{y}b_{y}\right)-a_{z}b_{z}\text{.}$
(28)

This straightforward mathematical calculation that yields Eq. (28) leads to
revise the incorrect sign that comes into view in Ref. gisin91 . Following
Gisin’s work in Ref. gisin91 , we adopt the same working assumption according
to which suitable expressions for the Bloch vectors $\hat{a}$ and $\hat{b}$
are given by,

$\hat{a}\overset{\text{def}}{=}\left(a_{x}\text{, }a_{y}\text{,
}a_{z}\right)=\left(\sin\alpha\text{, }0\text{, }\cos\alpha\right)\text{, and
}\hat{b}\overset{\text{def}}{=}\left(b_{x}\text{, }b_{y}\text{,
}b_{z}\right)=\left(\sin\beta\text{, }0\text{, }\cos\beta\right)\text{,}$ (29)

with $\alpha=0$, $\alpha^{\prime}=\pm\pi/2$ where the sign is the _same_ as
that of $c_{1}c_{2}$. Unit vectors $\hat{a}^{\prime}$ and $\hat{b}^{\prime}$
are assumed to be given by $\hat{a}^{\prime}=\left(\sin\alpha^{\prime}\text{,
}0\text{, }\cos\alpha^{\prime}\right)$ and
$\hat{b}^{\prime}=\left(\sin\beta^{\prime}\text{, }0\text{,
}\cos\beta^{\prime}\right)$, respectively. Observe that due to the previous
sign mistake that appears in Gisin’s analogue of our Eq. (28), in Ref. gisin91
it is stated that $\alpha^{\prime}=\pm\pi/2$ where the sign is the _opposite_
of that of $c_{1}c_{2}$. Recalling Eq. (18), let us focus on the quantity
$\left|P\left(a\text{, }b\right)-P\left(a\text{,
}b^{\prime}\right)\right|+P\left(a^{\prime}\text{,
}b\right)+P\left(a^{\prime}\text{, }b^{\prime}\right)$. Exploiting Eqs. (19),
(24), and (29), we get

$\left|P\left(a\text{, }b\right)-P\left(a\text{,
}b^{\prime}\right)\right|=\left|-\cos\beta+\cos\beta^{\prime}\right|=\left|-\left(\cos\beta-\cos\beta^{\prime}\right)\right|=\left|\cos\beta-\cos\beta^{\prime}\right|\text{,}$
(30)

and,

$P\left(a^{\prime}\text{, }b\right)+P\left(a^{\prime}\text{,
}b^{\prime}\right)=2c_{1}c_{2}\sin\alpha^{\prime}\left(\sin\beta+\sin\beta^{\prime}\right)-\cos\alpha^{\prime}\left(\cos\beta+\cos\beta^{\prime}\right)\text{.}$
(31)

Assuming $c_{1}c_{2}>0$ and $\alpha^{\prime}=+\pi/2$, Eq. (31) becomes

$P\left(a^{\prime}\text{, }b\right)+P\left(a^{\prime}\text{,
}b^{\prime}\right)=2\left|c_{1}c_{2}\right|\left(\sin\beta+\sin\beta^{\prime}\right)\text{.}$
(32)

Furthermore, taking $c_{1}c_{2}<0$ and $\alpha^{\prime}=-\pi/2$, we obtain

$P\left(a^{\prime}\text{, }b\right)+P\left(a^{\prime}\text{,
}b^{\prime}\right)=-2c_{1}c_{2}\left(\sin\beta+\sin\beta^{\prime}\right)\text{,}$
(33)

with $-2c_{1}c_{2}>0$. Therefore $2\left|c_{1}c_{2}\right|=-2c_{1}c_{2}>0$,
and $P\left(a^{\prime}\text{, }b\right)+P\left(a^{\prime}\text{,
}b^{\prime}\right)$ in Eq. (33) becomes

$P\left(a^{\prime}\text{, }b\right)+P\left(a^{\prime}\text{,
}b^{\prime}\right)=2\left|c_{1}c_{2}\right|\left(\sin\beta+\sin\beta^{\prime}\right)\text{.}$
(34)

Summing up, employing Eqs. (30) and (34), $\left|P\left(a\text{,
}b\right)-P\left(a\text{, }b^{\prime}\right)\right|+P\left(a^{\prime}\text{,
}b\right)+P\left(a^{\prime}\text{, }b^{\prime}\right)$ can be recast as

$\left|P\left(a\text{, }b\right)-P\left(a\text{,
}b^{\prime}\right)\right|+P\left(a^{\prime}\text{,
}b\right)+P\left(a^{\prime}\text{,
}b^{\prime}\right)=\left|\cos\left(\beta\right)-\cos\left(\beta^{\prime}\right)\right|+2\left|c_{1}c_{2}\right|\cdot\left[\sin\left(\beta\right)+\sin\left(\beta^{\prime}\right)\right]\text{.}$
(35)

Let $\beta$ and $\beta^{\prime}$ be such that $\sin\beta>0$ and
$\sin\beta^{\prime}>0$ so that
$2\left|c_{1}c_{2}\right|\left(\sin\beta+\sin\beta^{\prime}\right)>0$. Let us
also adopt the working condition given by

$-\cos\left(\beta^{\prime}\right)=\cos\left(\beta\right)=\left(1+4\left|c_{1}c_{2}\right|^{2}\right)^{-1/2}\text{,}$
(36)

which, after some algebra, yields

$\sin\left(\beta\right)=\sin\left(\beta^{\prime}\right)=2\left|c_{1}c_{2}\right|\left(1+4\left|c_{1}c_{2}\right|^{2}\right)^{-1/2}\text{.}$
(37)

Therefore, substituting Eqs. (36) and (37) into Eq. (35), we finally get

$\displaystyle\left|P\left(a\text{, }b\right)-P\left(a\text{,
}b^{\prime}\right)\right|+P\left(a^{\prime}\text{,
}b\right)+P\left(a^{\prime}\text{, }b^{\prime}\right)$
$\displaystyle=2\left(1+4\left|c_{1}c_{2}\right|^{2}\right)^{-1/2}+8\left|c_{1}c_{2}\right|^{2}\left(1+4\left|c_{1}c_{2}\right|^{2}\right)^{-1/2}$
$\displaystyle=2\left(1+4\left|c_{1}c_{2}\right|^{2}\right)^{-1/2}\left(1+4\left|c_{1}c_{2}\right|^{2}\right)$
$\displaystyle=2\left(1+4\left|c_{1}c_{2}\right|^{2}\right)^{1/2}$
$\displaystyle>2\text{,}$ (38)

that is,

$\left|P\left(a\text{, }b\right)-P\left(a\text{,
}b^{\prime}\right)\right|+P\left(a^{\prime}\text{,
}b\right)+P\left(a^{\prime}\text{, }b^{\prime}\right)\geq
2\left(1+4\left|c_{1}c_{2}\right|\right)^{-1/2}>2\text{.}$ (39)

The derivation of the inequality in Eq. (39) concludes our formal verification
concerning the fact that QM violates the CHSH Bell inequality.

###  IV.2 Illustrative examples

In this second subsection, we discuss three distinct scenarios leading to the
violation of Bell’s inequality expressed as

$\left|P\left(a\text{, }b\right)-P\left(a\text{,
}b^{\prime}\right)\right|+P\left(a^{\prime}\text{,
}b\right)+P\left(a^{\prime}\text{, }b^{\prime}\right)>2\text{,}$ (40)

where the spin correlation coefficient $P\left(a\text{, }b\right)$ is given by
$\left\langle\psi|\left(\hat{a}\cdot\vec{\sigma}\right)\otimes(\hat{b}\cdot\vec{\sigma})|\psi\right\rangle$.

We begin by observing that the quantities that can be manipulated in order to
obtain the above-mentioned inequality are essentially two, the set of
polarization vectors $\left\\{\hat{a}\text{, }\hat{b}\text{,
}\hat{a}^{\prime}\text{, }\hat{b}^{\prime}\right\\}$ and the entangled quantum
state $\left|\psi\right\rangle$. In our discussion, we focus our attention on
$\left|\psi\right\rangle=c_{1}\left|01\right\rangle+c_{2}\left|10\right\rangle$
with $c_{1}$, $c_{2}\in\mathbb{R}$ such that $c_{1}^{2}+c_{2}^{2}=1$ and
$2c_{1}c_{2}=\pm 1$. Furthermore, we propose three different sets of
polarization vectors $\left\\{\hat{a}\text{, }\hat{b}\text{,
}\hat{a}^{\prime}\text{, }\hat{b}^{\prime}\right\\}$. Note that, in general,
the polarization vectors $\hat{a}$, $\hat{b}$, $\hat{a}^{\prime}$,
$\hat{b}^{\prime}$ can be parametrized in terms of the spherical angles
$\theta\in\left[0\text{, }\pi\right]$ and $\varphi\in\left[0\text{,
}2\pi\right)$ as

$\displaystyle\hat{a}\overset{\text{def}}{=}\left(\sin\left(\theta_{1}\right)\cos\left(\varphi_{1}\right)\text{,
}\sin\left(\theta_{1}\right)\sin\left(\varphi_{1}\right)\text{,
}\cos\left(\theta_{1}\right)\right)\text{,
}\hat{b}\overset{\text{def}}{=}\left(\sin\left(\theta_{2}\right)\cos\left(\varphi_{2}\right)\text{,
}\sin\left(\theta_{2}\right)\sin\left(\varphi_{2}\right)\text{,
}\cos\left(\theta_{2}\right)\right)\text{,}$
$\displaystyle\hat{a}^{\prime}\overset{\text{def}}{=}\left(\sin\left(\theta_{3}\right)\cos\left(\varphi_{3}\right)\text{,
}\sin\left(\theta_{3}\right)\sin\left(\varphi_{3}\right)\text{,
}\cos\left(\theta_{3}\right)\right)\text{, and }\text{
}\hat{b}^{\prime}\overset{\text{def}}{=}\left(\sin\left(\theta_{4}\right)\cos\left(\varphi_{4}\right)\text{,
}\sin\left(\theta_{4}\right)\sin\left(\varphi_{4}\right)\text{,
}\cos\left(\theta_{4}\right)\right)\text{,}$ (41)

respectively. Clearly, considering Eq. (41) and Bell’s inequality, we see that
in the most general case one needs to handle an inequality with eight angular
parameters, once the entangles quantum state has been chosen. To gain physical
insights with simpler configurations, in what follows we shall deal only with
two-dimensional parametric regions (i.e., regions parametrized by a polar
angle $\theta$ and an azimuthal angle $\varphi$) where the Bell inequality is
violated. This way, we shall be able to visualize this regions with neat two-
dimensional parametric plots. In particular, we shall be able to comment on
the behavior of these regions when we modify the degree of entanglement of the
quantum state $\left|\psi\right\rangle$ while keeping fixed the set of
polarization vectors $\left\\{\hat{a}\text{, }\hat{b}\text{,
}\hat{a}^{\prime}\text{, }\hat{b}^{\prime}\right\\}$.

Before presenting our scenarios, we remark for completeness that the
projectors $\left\\{a\text{, }b\text{, }a^{\prime}\text{,
}b^{\prime}\right\\}$ in Eq. (40) need not be mutually orthogonal. Two
projectors
$a\overset{\text{def}}{=}\left(1/2\right)\left[\mathrm{I}+\hat{a}\cdot\vec{\sigma}\right]$
and
$b\overset{\text{def}}{=}\left(1/2\right)\left[\mathrm{I}+\hat{b}\cdot\vec{\sigma}\right]$
with $a^{2}=a$ and $b^{2}=b$ are orthogonal when $ab$ is the null operator
$\mathcal{O}$. Recalling that
$\left(\hat{a}\cdot\vec{\sigma}\right)(\hat{b}\cdot\vec{\sigma})=(\hat{a}\cdot\hat{b})\mathrm{I}+i(\hat{a}\times\hat{b})\cdot\vec{\sigma}$,
we get

$ab=\frac{1}{4}\mathrm{I}+\frac{1}{4}(\hat{a}+\hat{b})\cdot\vec{\sigma}+\frac{1}{4}(\hat{a}\cdot\hat{b})\mathrm{I}+\frac{1}{4}i(\hat{a}\times\hat{b})\cdot\vec{\sigma}\text{.}$
(42)

From Eq. (42), we note that $ab=\mathcal{O}$ when $\hat{b}=-\hat{a}$. Indeed,
in this case we have $\hat{a}+\hat{b}=0$, $\hat{a}\cdot\hat{b}=-1$, and
$\hat{a}\times\hat{b}=0$. Recasting the polarization vectors $\hat{a}$ and
$\hat{b}$ of orthogonal projectors in terms of spherical angles $\theta$ and
$\varphi$, we have $\hat{a}=\hat{a}\left(\theta\text{,
}\varphi\right)\overset{\text{def}}{=}\left(\cos\left(\varphi\right)\sin\left(\theta\right)\text{,
}\sin\left(\varphi\right)\sin\left(\theta\right)\text{,
}\cos\left(\theta\right)\right)$ and $\hat{b}=\hat{b}\left(\theta\text{,
}\varphi\right)\overset{\text{def}}{=}\hat{a}\left(\pi-\theta\text{,
}\varphi+\pi\right)=\left(-\cos\left(\varphi\right)\sin\left(\theta\right)\text{,
}-\sin\left(\varphi\right)\sin\left(\theta\right)\text{,
}-\cos\left(\theta\right)\right)$ so that $\hat{a}\cdot\hat{b}=-1$. In the
following three scenarios, the set of projectors $\left\\{a\text{, }b\text{,
}a^{\prime}\text{, }b^{\prime}\right\\}$ in Eq. (40) are not mutually
orthogonal. In particular, we shall focus on configurations specified by
$\hat{a}\cdot\hat{a}^{\prime}=0=\hat{b}\cdot\hat{b}^{\prime}$ and only two
tunable angular parameters.

Figure 1: Plot of the two-dimensional parametric region (black region) where
the Bell inequality is violated. The region is defined by the inequality
$f\left(\varphi_{1}\text{, }\varphi_{2}\right)>2\mathrm{C}\left(\psi\right)$,
where $f\left(\varphi_{1}\text{,
}\varphi_{2}\right)\overset{\text{def}}{=}\left|\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)\right|+\left[\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)\right]$,
$\mathrm{C}\left(\psi\right)\overset{\text{def}}{=}2\left|c_{1}c_{2}\right|\in\left[0\text{,
}1\right]$, and $c_{1}c_{2}>0$. We assume $\mathrm{C}\left(\psi\right)$ equals
$1$, $4/5$, and $8/11$ in plots (a), (b), and (c), respectively. Note that the
parametric region tends to shrink as the degree of entanglement
$\mathrm{C}\left(\psi\right)$ of the two-qubit quantum state
$\left|\psi\right\rangle\overset{\text{def}}{=}c_{1}\left|01\right\rangle+c_{2}\left|10\right\rangle$
becomes weaker.

####  IV.2.1 First scenario: $xz$-plane

In the first scenario, we assume that polarization vectors
$\left\\{\hat{a}\text{, }\hat{b}\text{, }\hat{a}^{\prime}\text{,
}\hat{b}^{\prime}\right\\}$ are in the $xz$-plane. We set
$\varphi_{1}=\varphi_{2}=0$, $\varphi_{3}=\varphi_{4}=0$,
$\theta_{3}=\theta_{1}+\pi/2$, and $\theta_{4}=\theta_{2}+\pi/2$. Then, Eq.
(41) yields

$\displaystyle\hat{a}\overset{\text{def}}{=}\left(\sin\left(\theta_{1}\right)\text{,
}0\text{, }\cos\left(\theta_{1}\right)\right)\text{,
}\hat{b}\overset{\text{def}}{=}\left(\sin\left(\theta_{2}\right)\text{,
}0\text{, }\cos\left(\theta_{2}\right)\right)\text{,}$
$\displaystyle\hat{a}^{\prime}\overset{\text{def}}{=}\left(\cos\left(\theta_{1}\right)\text{,
}0\text{, }-\sin\left(\theta_{1}\right)\right)\text{, and }\text{
}\hat{b}^{\prime}\overset{\text{def}}{=}\left(\cos\left(\theta_{2}\right)\text{,
}0\text{, }-\sin\left(\theta_{2}\right)\right)\text{.}$ (43)

Consider the inequality in Eq. (40), with $P\left(a\text{,
}b\right)=2c_{1}c_{2}\left(a_{x}b_{x}+a_{y}b_{y}\right)-a_{z}b_{z}$ and
$2c_{1}c_{2}=1$. Then, using Eq. (43), Eq. (40) becomes

$\left[f_{\mathrm{case}\text{-}1}\left(\theta_{1}\text{,
}\theta_{2}\right)\right]_{2c_{1}c_{2}=1}\overset{\text{def}}{=}\left|\cos\left(\theta_{1}+\theta_{2}\right)+\sin\left(\theta_{1}+\theta_{2}\right)\right|+\cos\left(\theta_{1}+\theta_{2}\right)+\sin\left(\theta_{1}+\theta_{2}\right)>2\text{.}$
(44)

If we set $2c_{1}c_{2}=-1$, $P\left(a\text{, }b\right)=-\vec{a}\cdot\vec{b}$.
Then, Eq. (40) becomes
$\left|\vec{a}\cdot(\vec{b}^{\prime}-\vec{b})\right|-\vec{a}^{\prime}\cdot(\vec{b}^{\prime}+\vec{b})>2$
and leads to the inequality

$\left[f_{\mathrm{case}\text{-}1}\left(\theta_{1}\text{,
}\theta_{2}\right)\right]_{2c_{1}c_{2}=-1}\overset{\text{def}}{=}\left|\cos\left(\theta_{1}-\theta_{2}\right)-\sin\left(\theta_{1}-\theta_{2}\right)\right|-\left[\cos\left(\theta_{1}-\theta_{2}\right)-\sin\left(\theta_{1}-\theta_{2}\right)\right]>2\text{.}$
(45)

Note that both inequalities in Eq. (44) and Eq. (45) could be visualized in a
two-dimensional parametric plot. Interestingly, we emphasize that in Gisin’s
1991 work as revisited in Section IV, the violation of Bell’s inequality was
presented with polarization vectors $\left\\{\hat{a}\text{, }\hat{b}\text{,
}\hat{a}^{\prime}\text{, }\hat{b}^{\prime}\right\\}$ in the $xz$-plane.
Specifically, Gisin selected
$\hat{a}\overset{\text{def}}{=}(\sin\left(\alpha\right)$, $0$,
$\cos\left(\alpha\right))$, $\hat{b}\overset{\text{def}}{=}$
$(\sin\left(\beta\right)$, $0$, $\cos\left(\beta\right))$,
$\hat{a}^{\prime}\overset{\text{def}}{=}(\sin\left(\alpha^{\prime}\right)$,
$0$, $\cos\left(\alpha^{\prime}\right))$, and
$\hat{b}^{\prime}\overset{\text{def}}{=}$ $(\sin\left(\beta^{\prime}\right)$,
$0$, $\cos\left(\beta^{\prime}\right))$. Furthermore, it was chosen
$\alpha=0$, $\alpha^{\prime}=\pm\pi/2$ (depending on the sign of
$c_{1}c_{2}$), and, finally, arbitrary $\beta$ and $\beta^{\prime}$.

####  IV.2.2 Second scenario: $xy$-plane

In the second scenario, we assume that polarization vectors
$\left\\{\hat{a}\text{, }\hat{b}\text{, }\hat{a}^{\prime}\text{,
}\hat{b}^{\prime}\right\\}$ are in the $xy$-plane. We set
$\theta_{1}=\theta_{2}=\pi/2$, $\theta_{3}=\theta_{4}=\pi/2$,
$\varphi_{3}=\varphi_{1}+\pi/2$, and $\varphi_{4}=\varphi_{2}+\pi/2$. Then,
Eq. (41) yields

$\displaystyle\hat{a}\overset{\text{def}}{=}\left(\cos\left(\varphi_{1}\right)\text{,
}\sin\left(\varphi_{1}\right)\text{, }0\right)\text{,
}\hat{b}\overset{\text{def}}{=}\left(\cos\left(\varphi_{2}\right)\text{,
}\sin\left(\varphi_{2}\right)\text{, }0\right)\text{,}$
$\displaystyle\hat{a}^{\prime}\overset{\text{def}}{=}\left(-\sin\left(\varphi_{1}\right)\text{,
}\cos\left(\varphi_{1}\right)\text{, }0\right)\text{, and }\text{
}\hat{b}^{\prime}\overset{\text{def}}{=}\left(-\sin\left(\varphi_{2}\right)\text{,
}\cos\left(\varphi_{2}\right)\text{, }0\right)\text{.}$ (46)

Consider the inequality in (40). Then, using Eq. (46) and setting
$2c_{1}c_{2}=1$, Eq. (40) yields

$\left[f_{\mathrm{case}\text{-}2}\left(\varphi_{1}\text{,
}\varphi_{2}\right)\right]_{2c_{1}c_{2}=1}\overset{\text{def}}{=}\left|\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)\right|+\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)>2\text{.}$
(47)

If we set $2c_{1}c_{2}=-1$, (47) is replaced by

$\left[f_{\mathrm{case}\text{-}2}\left(\varphi_{1}\text{,
}\varphi_{2}\right)\right]_{2c_{1}c_{2}=-1}\overset{\text{def}}{=}\left|\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)\right|-\left[\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)\right]>2\text{.}$
(48)

Again, observe that both inequalities in Eq. (47) and Eq. (48) could be
visualized in a simple two-dimensional parametric plot. Interestingly, since
$P\left(a\text{, }b\right)=2c_{1}c_{2}\left(a_{x}b_{x}+a_{y}b_{y}\right)$ in
this second scenario, the inequality in Eq. (47) can be extended to quantum
states
$\left|\psi\right\rangle=c_{1}\left|01\right\rangle+c_{2}\left|10\right\rangle$
(where $c_{1}$, $c_{2}\in\mathbb{R}$ so that $c_{1}c_{2}>0$) with arbitrary
degree of entanglement
C$\left(\psi\right)\overset{\text{def}}{=}2\left|c_{1}c_{2}\right|\in\left[0\text{,
}1\right]$ as

$f\left(\varphi_{1}\text{,
}\varphi_{2}\right)\overset{\text{def}}{=}\left|\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)\right|+\cos\left(\varphi_{1}-\varphi_{2}\right)-\sin\left(\varphi_{1}-\varphi_{2}\right)>2\mathrm{C}\left(\psi\right)\text{.}$
(49)

A plot of the two-dimensional region where the Bell inequality expressed in
terms of Eq. (49) is violated appears in Fig. $1$ where we assume
C$\left(\psi\right)=1$, $4/5$, and $8/11$ in plots (a), (b), and (c),
respectively. These three C$\left(\psi\right)$-choices correspond to three
distinct choices in terms of entangled quantum states, namely

$\left|\psi\right\rangle\overset{\text{def}}{=}\frac{\left|01\right\rangle+\left|10\right\rangle}{\sqrt{2}}\text{,
}\left|\psi\right\rangle\overset{\text{def}}{=}\sqrt{\frac{1}{2}+\sqrt{\frac{9}{100}}}\left|01\right\rangle+\sqrt{\frac{1}{2}-\sqrt{\frac{9}{100}}}\left|10\right\rangle\text{,
and
}\left|\psi\right\rangle\overset{\text{def}}{=}\sqrt{\frac{1}{2}+\frac{\sqrt{57}}{22}}\left|01\right\rangle+\sqrt{\frac{1}{2}-\frac{\sqrt{57}}{22}}\left|10\right\rangle\text{,}$
(50)

respectively. As expected, we observe in Fig. $1$ that as the degree of
entanglement decreases, the two-dimensional parametric region where the Bell
inequality is violated tends to become smaller. Eventually, this region
disappears completely when the two-qubit state is separable.

####  IV.2.3 Third scenario: $yz$-plane

In the third scenario, we assume that polarization vectors
$\left\\{\hat{a}\text{, }\hat{b}\text{, }\hat{a}^{\prime}\text{,
}\hat{b}^{\prime}\right\\}$ are in the $yz$-plane. We set
$\varphi_{1}=\varphi_{2}=\pi/2$, $\varphi_{3}=\varphi_{4}=\pi/2$,
$\theta_{3}=\theta_{1}+\pi/2$, and $\theta_{4}=\theta_{2}+\pi/2$. Then, Eq.
(41) yields

$\displaystyle\hat{a}\overset{\text{def}}{=}\left(0\text{,
}\sin\left(\theta_{1}\right)\text{, }\cos\left(\theta_{1}\right)\right)\text{,
}\hat{b}\overset{\text{def}}{=}\left(0\text{,
}\sin\left(\theta_{2}\right)\text{,
}\cos\left(\theta_{2}\right)\right)\text{,}$
$\displaystyle\hat{a}^{\prime}\overset{\text{def}}{=}\left(0\text{,
}\cos\left(\theta_{1}\right)\text{,
}-\sin\left(\theta_{1}\right)\right)\text{, and }\text{
}\hat{b}^{\prime}\overset{\text{def}}{=}\left(0\text{,
}\cos\left(\theta_{2}\right)\text{,
}-\sin\left(\theta_{2}\right)\right)\text{.}$ (51)

Consider the inequality in (40). Then, using Eq. (51) and putting
$2c_{1}c_{2}=1$, Eq. (40) reduces to

$\left[f_{\mathrm{case}\text{-}3}\left(\theta_{1}\text{,
}\theta_{2}\right)\right]_{2c_{1}c_{2}=1}\overset{\text{def}}{=}\left|\cos\left(\theta_{1}+\theta_{2}\right)+\sin\left(\theta_{1}+\theta_{2}\right)\right|+\cos\left(\theta_{1}+\theta_{2}\right)+\sin\left(\theta_{1}+\theta_{2}\right)>2\text{.}$
(52)

Finally, setting $2c_{1}c_{2}=1$, (52) becomes

$\left[f_{\mathrm{case}\text{-}3}\left(\theta_{1}\text{,
}\theta_{2}\right)\right]_{2c_{1}c_{2}=-1}\overset{\text{def}}{=}\left|\cos\left(\theta_{1}-\theta_{2}\right)-\sin\left(\theta_{1}-\theta_{2}\right)\right|-\left[\cos\left(\theta_{1}-\theta_{2}\right)-\sin\left(\theta_{1}-\theta_{2}\right)\right]>2\text{.}$
(53)

Once again, we point out that both inequalities in Eq. (52) and Eq. (53) can
be used to visualize the two-dimensional parametric region where the Bell
inequality is violated.

Having discussed the violation of Bell’s inequality in the CHSH form from a
theoretical standpoint, in next section we briefly discuss the violation of
Bell’s inequality from an experimental standpoint.

Type of experimental loophole | Problem
---|---
Locality loophole | Excluding any communication between the polarizers
Detection loophole | Guaranteeing efficient measurements with no fair-sampling assumption
Table 4: Schematic description of the two main experimental loopholes, the
locality and detection loopholes. The locality loophole requires being certain
there is no communication between the polarizers. It can be solved using
random and ultrafast switching of the orientations of the polarizers. The
detection loophole, instead, requires detecting a sufficiently large fraction
of the emitted photons in experiments with optical systems in order to avoid
the fair-sampling assumption. In experiments with electron spins, efficient
spin read-out helps avoiding the fair-sampling assumption.

##  V Violation of Bell’s inequality in a laboratory

In this section, after describing some of the most important challenges that
one finds when experimentally verifying the violation of Bell’s in a
laboratory, we briefly focus on the 1982 AGR experiment where the first
experimental realization of the EPR gedankenexperiment was proposed. Finally,
we outline the links connecting RLHVTs and QM to experiments.

###  V.1 Experimental challenges

The path towards an experimental verification, free of loopholes, of the Bell
inequality in an EPR-type experiment has been rather interesting and
challenging from a scientific standpoint. For a review on experimental aspects
related to Bell’s 1964 theorem, we refer to Ref. brunner14 . The two main
experimental loopholes in EPR-type experiments are the locality and detection
loopholes. The locality loophole demands being sure there exists no
communication between the polarizers. It can be solved employing random and
ultrafast switching of the orientations of the polarizers. The detection
loophole, instead, demands detecting a sufficiently large fraction of the
emitted photons in experiments with optical systems to avoid the fair-sampling
assumption. The solution of the detection loophole is important to obtain a
statistically significant rejection of the local-realistic null hypothesis.
Therefore, it is necessary to detect a sufficiently large fraction of the
emitted photons (i.e., avoid the fair sampling assumption) so that the
correlations of the detected photons are representative. This way, one can
completely rule out that the Bell inequality is satisfied. Solving the
locality loophole is important to make sure that the choice of setting at a
measurement site $A$ has no influence on the measurement result at site $B$.
To achieve this goal, one has to make sure that the time interval that passes
between the choice of measurement setting at site $A$ and the generation of a
measurement outcome at site $B$ is shorter than the time interval it takes for
a photon to travel between the two measurement sites. Relying on the
experience gained in several experiments and theoretical proposals wu50 ;
kocher67 ; clauser69 ; clauser72 ; aspect81 , the first experimental
realization of the EPR gedankenexperiment was performed by Aspect and
collaborators in Ref. aspect82  with a stationary switching of the
orientations of the polarizers. In Ref. aspect82B , Aspect and collaborators
improved their experimental apparatus by means of a periodic (yet,
predictable) switching of the orientations of the polarizers. In Ref. weihs98
, Zeilinger and collaborators observed a strong violation of Bell’s inequality
in an EPR-type experiment with independent observers. They solved the locality
loophole by using a random and ultrafast switching of the orientations of the
polarizers, separated by $400$ meters aspect99 . Finally, the first experiment
where both the detection and locality loopholes were closed simultaneously was
an experiment with electron spins separated by $1.3$ kilometres and was
performed by Hensen and collaborators in Ref. hensen15 . Finally, for a review
on loopholes in Bell inequality tests of local realism, we hint to Ref.
larsson14 . In Table IV, we show a schematic description of the two main
experimental loopholes, the locality and detection loopholes.

Figure 2: Schematic depiction of the experimental apparatus for measuring the
polarization coefficient $E\left(\hat{a}\text{, }\hat{b}\right)$ in the
Aspect-Grangier-Roger (AGR) experiment in Ref. aspect82 . Two photons
$\gamma_{1}$ and $\gamma_{2}$ generated by a source $\mathrm{S}$ are in a
singlet state $\left|\psi\right\rangle$ and propagate in opposite directions.
The linear polarizations $\left(\pm 1\right)$ of $\gamma_{1}$ and $\gamma_{2}$
are measured along the orientations $\hat{a}$ and $\hat{b}$ specifying a pair
of two-channel polarizers $\mathrm{P}_{\mathrm{1}}$ and
$\mathrm{P}_{\mathrm{2}}$. The polarization coefficient $E\left(\hat{a}\text{,
}\hat{b}\right)$ is experimentally determined by measuring, in a single run,
the four coincidence counting rates $R_{\pm\pm}\left(\hat{a}\text{,
}\hat{b}\right)$.

###  V.2 The 1982 AGR experiment

In Ref. aspect82 , Aspect and collaborators proposed the first experimental
realization of the EPR gedankenexperiment. In this experiment, they used the
optical analogs of the Stern-Gerlach filters (i.e., two-channel polarizers) to
measure the linear-polarization correlation of pairs of photons emitted in a
radiative cascade of calcium. Specifically, considering that the quantity
$S_{\mathrm{RLHVT}}$ $\in\left[-2\text{, }2\right]$ given within a RLHVT by

$S_{\mathrm{RLHVT}}=E(\hat{a},\hat{b})-E(\hat{a},\hat{b}^{\prime})+E(\hat{a}^{\prime},\hat{b})+E(\hat{a}^{\prime},\hat{b}^{\prime})\text{,}$
(54)

Aspect and collaborators found $S_{\mathrm{\exp}}=2.697\pm 0.015$ for their
chosen set of orientations $\left\\{\hat{a}\text{, }\hat{b}\text{,
}\hat{a}^{\prime}\text{, }\hat{b}^{\prime}\right\\}$. They compared
$S_{\mathrm{\exp}}$ with $S_{\mathrm{QM}}=2.70\pm 0.05$ predicted by QM. They
concluded that their experimental findings were in excellent agreement with
quantum-mechanical predictions and, thus, Bell’s inequality was significantly
violated. For completeness, we point out that the correlation coefficient
$E(\hat{a}$, $\hat{b})$ can be expressed, from an experimental standpoint, as

$E(\hat{a}\text{, }\hat{b})\overset{\text{def}}{=}P_{++}(\hat{a}\text{,
}\hat{b})+P_{--}(\hat{a}\text{, }\hat{b})-P_{+-}(\hat{a}\text{,
}\hat{b})-P_{-+}(\hat{a}\text{, }\hat{b})\text{,}$ (55)

where the probabilities $P_{ij}(\hat{a}$, $\hat{b})$ with $i$,
$j\in\left\\{+\text{, }-\right\\}$ are defined as

$P_{ij}(\hat{a}\text{,
}\hat{b})\overset{\text{def}}{=}\frac{R_{ij}(\hat{a}\text{,
}\hat{b})}{R_{ii}(\hat{a}\text{, }\hat{b})+R_{jj}(\hat{a}\text{,
}\hat{b})+R_{ij}(\hat{a}\text{, }\hat{b})+R_{ji}(\hat{a}\text{,
}\hat{b})}\text{.}$ (56)

Given that a measurement along $\hat{a}$ yields the result $+1$ ($-1$) if the
polarization of the photon is found parallel (perpendicular) to $\hat{a}$,
$P_{\pm\pm}(\hat{a}$, $\hat{b})$ in Eq. (55) denote the probabilities of
measuring outcomes $\pm 1$ along $\hat{a}$ for particle-$1$ and $\pm 1$ along
$\hat{b}$ for particle-$2$. Moreover, $R_{\pm\pm}(\hat{a}$, $\hat{b})$ in Eq.
(56) denote the coincidence counting rates of measuring outcomes $\pm 1$ along
$\hat{a}$ for particle-$1$ and $\pm 1$ along $\hat{b}$ for particle-$2$.

###  V.3 From RLHVT and QM to experiment

In Fig. $2$, we depict the experimental apparatus for measuring the
polarization coefficient (i.e., $E_{\mathrm{RLHVT}}(\hat{a}$, $\hat{b})$ in
Eq. (7)) in the Aspect-Grangier-Roger (AGR) experiment. We are now in a good
position to outline the logical steps connecting theoretical predictions that
emerge from a realistic logical hidden variable theory (RLHVT) and quantum
mechanics (QM) bell64 ; clauser69 ; bell04  to experimental observations in
the Aspect-Grangier-Roger experiment aspect82 . First, the Bell inequality is
initially expressed in terms of $E_{\mathrm{RLHVT}}(\hat{a}$, $\hat{b})$-like
coefficients with $E_{\mathrm{RLHVT}}(\hat{a}$,
$\hat{b})\overset{\text{def}}{=}\int\bar{A}\left(\hat{a}\text{,
}\lambda\right)\bar{B}(\hat{b}$, $\lambda)\rho\left(\lambda\right)d\lambda$ in
Eq. (7). Second, identifying $E_{{}_{\mathrm{QM}}}(\hat{a}$,
$\hat{b})\overset{\text{def}}{=}\left\langle\psi|\left(\vec{\sigma}\cdot\hat{a}\right)\otimes(\vec{\sigma}\cdot\hat{b})|\psi\right\rangle$
with $E_{\mathrm{RLHVT}}(\hat{a}$, $\hat{b})$, the Bell inequality can be
recast in terms of $E_{{}_{\mathrm{QM}}}(\hat{a}$, $\hat{b})$-like
coefficients. Third, one shows that for any entangled quantum state
$\left|\psi\right\rangle$ with $\left\langle\psi|\psi\right\rangle=1$, the
Bell inequality in terms of $E_{{}_{\mathrm{QM}}}(\hat{a}$, $\hat{b})$-like
coefficients can be violated gisin91 . Finally, it can be verified that
quantum mechanical predictions, unlike theoretical predictions arising from a
realistic logical hidden variable theory, agree with experimental observations
expressed in terms of $E_{\mathrm{AGR}\text{-{exp}}}(\hat{a}$, $\hat{b})$,
where aspect82

$E_{\mathrm{AGR}\text{-{exp}}}(\hat{a}\text{,
}\hat{b})\overset{\text{def}}{=}\frac{R_{++}(\hat{a}\text{,
}\hat{b})+R_{--}(\hat{a}\text{, }\hat{b})-R_{+-}(\hat{a}\text{,
}\hat{b})-R_{-+}(\hat{a}\text{, }\hat{b})}{R_{++}(\hat{a}\text{,
}\hat{b})+R_{--}(\hat{a}\text{, }\hat{b})+R_{+-}(\hat{a}\text{,
}\hat{b})+R_{-+}(\hat{a}\text{, }\hat{b})}\text{.}$ (57)

These logical steps can be outlined as follows:

  1. [i]

Derive the Bell inequality in the Clause-Horne-Shimony-Holt (CHSH) form within
the formalism of a realistic logical hidden variable theory (RLHVT),

$\left|E_{\mathrm{RLHVT}}(\hat{a}\text{,
}\hat{b})-E_{\mathrm{RLHVT}}(\hat{a}\text{,
}\hat{b}^{\prime})\right|+\left|E_{\mathrm{RLHVT}}(\hat{a}^{\prime}\text{,
}\hat{b})+E_{\mathrm{RLHVT}}(\hat{a}^{\prime}\text{,
}\hat{b}^{\prime})\right|\leq 2\text{.}$ (58)

  2. [ii]

Identify $P(\hat{a}$,
$\hat{b})\overset{\text{def}}{=}E_{{}_{\mathrm{QM}}}(\hat{a}$, $\hat{b})$ with
$E_{\mathrm{RLHVT}}(\hat{a}$, $\hat{b})$. Note that $P(\hat{a}$, $\hat{b})$
and $P\left(a\text{, }b\right)$ in the main text denote the same quantity.

  3. [iii]

Rewrite the Bell inequality in [i] by using the identification in [ii],

$\left|P(\hat{a}\text{, }\hat{b})-(P\hat{a}\text{,
}\hat{b}^{\prime})\right|+\left|P(\hat{a}^{\prime}\text{,
}\hat{b})+P(\hat{a}^{\prime}\text{, }\hat{b}^{\prime})\right|\leq 2\text{.}$
(59)

  4. [iv]

Show that there are quantum mechanical states $\left|\psi\right\rangle$ (i.e.,
entangled states) for which the Bell inequality in Eq. (59) is violated. For
instance,

$\left|P(\hat{a}\text{, }\hat{b})-P(\hat{a}\text{,
}\hat{b}^{\prime})\right|+\left|P(\hat{a}^{\prime}\text{,
}\hat{b})+P(\hat{a}^{\prime}\text{, }\hat{b}^{\prime})\right|>2\text{.}$ (60)

In particular, for a given entangled state $\left|\psi\right\rangle$, the
strength of the violation depends on the sets of orientations $(\hat{a}$,
$\hat{b})$, $(\hat{a}$, $\hat{b}^{\prime})$, $(\hat{a}^{\prime}$, $\hat{b})$,
and $(\hat{a}^{\prime}$, $\hat{b}^{\prime})$.

  5. [v]

Confirm, by means of the Aspect-Grangier-Roger experiment aspect82 , the
violation of the Bell inequality in the CHSH form once
$E_{\mathrm{AGR}\text{-{exp}}}(\hat{a}$, $\hat{b})$ in Eq. (57) is identified
with $E_{{}_{\mathrm{QM}}}(\hat{a}$, $\hat{b})$.

In Table III, we report the expressions of spin correlation coefficients in a
realistic local hidden variable theory (RLHVT), in quantum mechanics (QM) and,
finally, in the Aspect-Grangier-Roger experiment (AGR-exp).

Theory and experiment | Spin correlation coefficient symbol | Spin correlation coefficient expression
---|---|---
Realistic local hidden variable theory | $E_{\mathrm{RLHVT}}\left(\hat{a}\text{, }\hat{b}\right)$ | $\int\bar{A}\left(\hat{a}\text{, }\lambda\right)\bar{B}\left(\hat{b}\text{, }\lambda\right)\rho\left(\lambda\right)d\lambda$
Quantum mechanics | $E_{\mathrm{QM}}\left(\hat{a}\text{, }\hat{b}\right)$ | $\left\langle\psi|\left(\vec{\sigma}\cdot\hat{a}\right)\otimes\left(\vec{\sigma}\cdot\hat{b}\right)|\psi\right\rangle$
The Aspect-Grangier-Roger experiment | $E_{\mathrm{AGR}\text{-{exp}}}\left(\hat{a}\text{, }\hat{b}\right)$ | $\frac{R_{++}\left(\hat{a}\text{, }\hat{b}\right)+R_{--}\left(\hat{a}\text{, }\hat{b}\right)-R_{+-}\left(\hat{a}\text{, }\hat{b}\right)-R_{-+}\left(\hat{a}\text{, }\hat{b}\right)}{R_{++}\left(\hat{a}\text{, }\hat{b}\right)+R_{--}\left(\hat{a}\text{, }\hat{b}\right)+R_{+-}\left(\hat{a}\text{, }\hat{b}\right)+R_{-+}\left(\hat{a}\text{, }\hat{b}\right)}$
Table 5: Expressions of spin correlation coefficients in a realistic local
hidden variable theory (RLHVT), in quantum mechanics (QM) and, finally, in the
Aspect-Grangier-Roger experiment (AGR-exp).

##  VI Conclusions

In this paper, we presented a revisitation of the violation of Bell’s
inequality in the CHSH form with entangled quantum states. First, we discussed
the 1935 EPR paradox that emerges from putting the emphasis on Einstein’s
locality and the absolute character of physical phenomena. Second, we
discussed Bell’s 1971 derivation of the 1969 CHSH form of the original 1964
Bell inequality in the context of a RLHVT. Third, identifying the quantum-
mechanical spin correlation coefficient with the RLHVT one, we followed
Gisin’s 1991 analysis to show that QM violates Bell’s inequality when systems
are in entangled quantum states. For pedagogical purposes, we showed how this
violation depends both on the orientation of the polarizers and the degree of
entanglement of the quantum states (see Fig. $1$, for instance). Fourth, we
discussed the basics of the experimental verification of Bell’s inequality in
an actual laboratory as presented in the original 1982 AGR) experiment.

In what follows, we present a summary of take home messages from our
discussion on this beautiful piece of physical science with a deep interplay
between theory (i.e., scientific imagination and theoretical creativity) and
experiments (i.e., cleverly designed experimental set-ups and high statistical
accuracy):

  1. [i]

From the EPR paradox epr , one might intuitively think that the statistical
predictions of quantum mechanics emerge from a hidden underlying deterministic
substructure that encodes the natural assumption of locality.

  2. [ii]

John Bell derived Bell’s inequality within the framework of a realistic local-
hidden variable theory bell64 ; bell04 .

  3. [iii]

Quantum mechanics is a non-local theory and violates Bell’s inequality bell64
; bell04 .

  4. [iv]

Natural phenomena occur in a laboratory, not in a Hilbert space peres2004 .
The experiments with entangled photons by Aspect and collaborators aspect82 ;
aspect82B , Clauser and collaborators clauser69 ; clauser72  and, finally,
Zeilinger and collaborators pan98  experimentally establish the violation of
Bell’s inequalities and give evidence of one of the most striking aspects of
quantum physics (i.e., quantum entanglement).

  5. [v]

Entanglement is the key concept needed to understand the EPR paradox and the
violation of Bell’s inequality by quantum mechanics pan98 ; zeilinger98 ;
brukner01 .

  6. [vi]

Unlike what one might intuitively believe, quantum entanglement requires
neither the entangled particles to come from a common source nor to have
interacted in the past. This fact was experimentally verified by Zeilinger and
collaborators in Ref. pan98 .

  7. [vii]

As pointed out by Zeiliger in Ref. zeilinger98 , information about quantum
systems is more important than any possible so-called “ _real_ ” property
these systems might possess. Finally, there appears to exist no quantum world.
Instead, as pointed out by Bohr in Ref. bohr35 , there is only an abstract
quantum-mechanical description of natural phenomena.

This set of take home messages concludes our revisitation of the violation of
Bell’s inequality in the Clauser-Horne-Shimony-Holt form with entangled
quantum states. Interestingly, our idea of this revisitation started at the
end of 2021 and, with our gladness, it is a very nice coincidence that the
Nobel prize in Physics was assigned in 2022 to Aspect, Clauser, and Zeilinger
epjd22 . Clearly, we express our congratulations to these three physicists
along with their research Groups. Finally, we wholeheartedly believe this
whole chain of events- EPR, Bohr, Bell, entanglement, interplay between theory
and experiment- is the essence of physics at its best. We are aware that our
effort here is in no way representative of this wonderful scientific endeavor.
It would have been too big a task for us to accomplish in the present
scientific effort. However, being this paper written by two theorists, a young
undergraduate student, and an experimental physicist, we hope it will help
young students and future generations of researchers appreciate (via a
description of a sequence of actual scientific events) how important are the
processes of _scientific imagination_ and _experimental verification_ in
modern physics.

###### Acknowledgements.

C. Cafaro thanks Ariel Caticha, Domenico Felice, Nicholas Gisin, Yanjun He,
and Elena R. Loubenets for valuable comments and helpful electronic mail
exchanges throughout the last years that helped shaping this paper in its
current form.

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