# Entanglement in Cognition violating Bell Inequalities
Beyond Cirel’son’s Bound

Diederik Aerts, Jonito Aerts Arguëlles, Lester Beltran
Suzette Geriente111Center Leo Apostel for Interdisciplinary Studies, Free
University of Brussels (VUB), Krijgskundestraat 33, 1160 Brussels, Belgium;
email addresses: diraerts@vub.ac.be,lestercc21<EMAIL_ADDRESS>and Sandro Sozzo222School of Business and Centre IQSCS, University of
Leicester, University Road, LE17RH Leicester, United Kingdom; email address:
<EMAIL_ADDRESS>
###### Abstract

We present the results of two tests where a sample of human participants were
asked to make judgements about the conceptual combinations The Animal Acts and
The Animal eats the Food. Both tests significantly violate the Clauser-Horne-
Shimony-Holt version of Bell inequalities (‘CHSH inequality’), thus exhibiting
manifestly non-classical behaviour due to the meaning connection between the
individual concepts that are combined. We then apply a quantum-theoretic
framework which we developed for any Bell-type situation and represent
empirical data in complex Hilbert space. We show that the observed violations
of the CHSH inequality can be explained as a consequence of a strong form of
‘quantum entanglement’ between the component conceptual entities in which both
the state and measurements are entangled. We finally observe that a quantum
model in Hilbert space can be elaborated in these Bell-type situations even
when the CHSH violation exceeds the known ‘Cirel’son bound’, in contrast to a
widespread belief. These findings confirm and strengthen the results we
recently obtained in a variety of cognitive tests and document and image
retrieval operations on the same conceptual combinations.

Keywords: Cognition; Bell-type tests; CHSH inequality; quantum entanglement;
quantum structures.

##  1 Introduction

In physics, the violation of the so-called ‘Bell inequalities’ is generally
maintained to prove that quantum entities exhibit specific aspects, as
‘contextuality’, ‘entanglement’ and ‘nonseparability’, which cannot be
explained in terms of the mathematical structures that are typically used in
classical physics. This contextuality also persists for far away quantum
entities, an aspect known as ‘nonlocality’ after Bell’s seminal paper [1, 2].
The consistent violation of Bell inequalities in physics tests performed on a
variety of quantum entities, in addition to confirming the predictions of
quantum theory, in particular reveals that the statistics of repeated
experiments shows connections between quantum entities that cannot be
represented within a classical probabilistic model satisfying the axioms of
Kolmogorov (‘Kolmogorovian probability’) (see, e.g., [3]).

The setting of a Bell-type test is relatively simple, if formulated in the so-
called ‘EPR-Bohm form’ [4, 5]. We briefly review it here, as Bell-type tests
are relevant to the purposes of the present paper. One considers a composite
physical entity ${\mathscr{S}}_{12}$, prepared in an initial state $p$ and
such that two individual entities ${\mathscr{S}}_{1}$ and ${\mathscr{S}}_{2}$
that have interacted in the past but are now far away (‘space-like
separation’) can be recognized as parts of ${\mathscr{S}}_{12}$. Then, 4
coincidence experiments $AB$, $AB^{\prime}$, $A^{\prime}B$ and
$A^{\prime}B^{\prime}$ are performed on ${\mathscr{S}}_{12}$ which consist in
performing measurements $e_{A}$, with outcomes $A_{1}$ and $A_{2}$, and
$e_{A^{\prime}}$ with outcomes $A^{\prime}_{1}$ and $A^{\prime}_{2}$, on
entity ${\mathscr{S}}_{1}$, and measurements $e_{B}$, with outcomes $B_{1}$
and $B_{2}$, and $e_{B^{\prime}}$ with outcomes $B^{\prime}_{1}$ and
$B^{\prime}_{2}$, on entity ${\mathscr{S}}_{2}$. If the measurement outcomes
can be only $-1$ or $+1$, then the expected values of $AB$, $AB^{\prime}$,
$A^{\prime}B$ and $A^{\prime}B^{\prime}$ become the correlation functions
$E(A,B)$, $E(A,B^{\prime})$, $E(A^{\prime},B)$ and $E(A^{\prime},B^{\prime})$,
respectively, and a specific inequality, which is the ‘Clauser-Horne-Shimony-
Holt version of Bell inequalities’ (briefly, ‘CHSH inequality’)

$-2\leq E(A^{\prime},B^{\prime})+E(A^{\prime},B)+E(A,B^{\prime})-E(A,B)\leq 2$
(1)

can be formulated [2]. John Bell put forward the type of inequality carrying
his name [1], with the aim of making experimental tests possible with respect
to the strange quantum phenomenon revealed in the Einstein Podolsky Rosen
paradox reasoning [4] called ‘entanglement’, i.e. if this phenomenon would not
be present the inequality would not be violated. The violation of Bell’s
inequality can also be connected with the non-existence of a Kolmogorovian
probability model for the joint entity [3], but in the investigation we put
forward here our interest goes to the original situation for which the
inequalities were formulated, i.e. the connections possible between individual
entities recognizable within a joint entity. It is indeed the case that
quantum theory makes specific predictions about Equation (1). More
specifically, one can choose the individual entities, e.g., pair of spin 1/2
quantum particles, states, e.g., entangled state (typically, the singlet spin
state), and observables, e.g., pairs of spin along different directions, in
such a way that Equation (1) is violated [1, 2]. Moreover, one can show that
the ‘CHSH factor’
$\Delta_{CHSH}=E(A^{\prime},B^{\prime})+E(A^{\prime},B)+E(A,B^{\prime})-E(A,B)$
is bound in quantum theory by the numerical value
$\Delta_{QMC}=2\sqrt{2}\approx 2.83$, known as the ‘Cirel’son bound’ [6, 7].
It is important to observe, at this stage, that the existence of such a
‘quantum bound’ is not trivial because, from a mathematical point of view, the
CHSH factor can take any value between $-4$ and $+4$. In addition, the
derivation of the Cirel’son bound explicitly refers to measurements $e_{XY}$,
corresponding to coincidence experiments $XY$, $X=A,A^{\prime}$,
$Y=B,B^{\prime}$, on the composite entity that are represented by product
self-adjoint operators whenever the states of the composite entity
${\mathscr{S}}_{12}$ are represented by the unit vectors of the tensor product
Hilbert space of two Hilbert spaces of which their unit vectors represent the
states of the possible to be recognized entities ${\mathscr{S}}_{1}$ and
${\mathscr{S}}_{2}$. This remark will play an important role in what follows.

In the last two decades, several theoretical studies, mainly inspired by
quantum computation and quantum information and their flourishing
applications, have deeply analysed and extended Bell inequalities (see, e.g.,
[8, 9]). In addition, numerous empirical tests have followed the seminal
experiments of Aspect and his collaborators [10, 11]. All empirical tests that
have been performed so far strongly confirm the predictions of quantum theory
(see, e.g., [12, 13, 14, 15]). The natural consequence of these results is
that ‘entanglement’, i.e. unavoidable connection between the individual
entities recognizable in a composite quantum entity, give rise to empirically
observable phenomena, so much that, in particular, entanglement is nowadays
considered as one of the fingerprints of quantum theory.

Growing theoretical and empirical research reveals that several quantum
structures, such as ‘contextuality’, ‘entanglement’, ‘indistinguishability’,
‘interference’ and ‘superposition’, also manifest themselves in other domains
than micro-physics, which include, in particular, cognition (e.g., human
probability and similarity judgements, decision-making and visual perception),
socio-economic domains (economics and finance) and information systems
(computer science and artificial intelligence) (see, e.g., [16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and references therein). In
particular, interesting results have been obtained in the identification of
quantum structures and, in particular, quantum entanglement, in the
combination of natural concepts and applied computer science domains, as
information retrieval and natural language processing (see, e.g., [31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] and references
therein). In this regard, our research team has produced both theoretical and
empirical research on how to identify entanglement in conceptual combinations
and document retrieval applications.

At an empirical level, in particular, we performed both cognitive tests on
human participants [38], document retrieval tests on structured corpuses of
documents [46] and image retrieval tests on the web [29]. In these tests, we
considered the composite conceptual entity The Animal Acts as a combination of
the individual conceptual entities Animal and Acts. We performed an additional
document retrieval test on the web in which we considered the composite
conceptual entity The Animal eats the Food as a combination of the individual
conceptual entities Animal and Food [32]. In all these tests, we observed a
systematic violation of the CHSH inequality, accompanied by a systematic
violation of a property called the ‘marginal law’.

In particular, the violation of the marginal law, which is believed not to
occur in principle in Bell-type tests in quantum physics – we come back to
this later, led us to study in detail a theoretical problem which is usually
overlooked in physics, the ‘identification problem’, i.e. the problem of
recognising individual entities of a composite entity by performing on the
latter measurements that resemble the typical coincidence experiments of Bell-
type tests [41, 42]. We constructed a general theoretical framework to model
any Bell-type situation in the Hilbert space formalism of quantum theory [48].
In this framework, one explicitly applies the prescription of the standard
formalism of quantum theory that the joint entity needs to be modelled in a
complex Hilbert space of which the dimension is given by the number of
outcomes of the defining measurements, which is hence $\mathbb{C}^{4}$. Only
then as a secondary step, with respect to the attempt to ‘recognize’ entities
in the joint entity, one considers possible isomorphisms with another Hilbert
space, namely, the one built as the tensor product
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ of two-dimensional complex Hilbert
spaces which need to describe the two possible entities, since the defining
measurements to recognize these entities have two outcomes. Like we analyzed
in detail in [41, 42], there is no unique isomorphism between $\mathbb{C}^{4}$
and $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ and this is the reason that from a
mathematical point of view there are different ways to account for
entanglement being present within the joint entity with respect to the
entities to be recognized individual entities. Essentially, entanglement
manifests on the level of the probabilities of a joint measurements not being
able to be written as products of probabilities on the level of the component
measurements of this joint measurement and hence it is a property of the
relation between the joint measurements and their components. Only when an
extra symmetry is present connecting different joint measurements the
entanglement of these different joint measurements can be captured in a state
of the joint entity. This symmetry can be detected by verifying whether the
marginal law corresponding to the considered joint measurements is satisfied,
in that case, the entanglement of these joint measurements can together be
captured in a state of the joint entity. If this is true for all joint
measurements one can prove that there is only one isomorphism connecting
$\mathbb{C}^{4}$ with $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ and in that case
$\mathbb{C}^{4}$ can be substituted by $\mathbb{C}^{4}$ with
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ and the joint entity can be directly
modeled in $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. With respect to
‘entanglement’, a property of the structure of the probabilities of a joint
measurement with respect to the probabilities of the component measurements,
this situation is exceptional and definitely not the general one. In [42] we
work out in detail several experimental examples and how how entanglement can
generally only be located in the joint measurements and only exceptionally in
the state of the joint entity. It is widely accepted that in the typically
considered quantum physics situations where entanglement is being measured and
studied this exceptional symmetry is always present and hence apriori the
tensor product Hilbert space is taken to be the Hilbert space to model the
joint entity, although already the first experiments of Alain Aspect [10]
showed violations of the marginal law, confirmed manifestly by later
experiments [13]. For the cognition experiments that we consider in the
present article where joint measurements are performed on joint concepts The
Animal Acts and The Animal eats the Food the symmetry is not fulfilled and
hence no unique isomorphism exists between $\mathbb{C}^{4}$ and
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. The new quantum-theoretic perspective
enabled modelling of both cognitive and web tests on The Animal Acts and
introduced a novel mathematical ingredient in the modelling of Bell-type
situations, namely, ‘entangled measurements’ [30, 41, 42, 47].

Entangled measurements have recently been applied with success to perform
quantum computation and information tasks (see, e.g., [9]). We show in the
present paper that ‘entanglement measurement also allow to identify the
presence of quantum entanglement also in Bell-type tests where the violation
of the Cirel’son bound occurs in addition to the violation of the CHSH
inequality’. More specifically, we present the results of two cognitive tests,
one on The Animal Acts and the other on The Animal eats the Food situations,
that we have recently performed on a sample of 81 participants. In both tests,
a significant violation of the CHSH has been observed. Furthermore, while The
Animal Acts test has revealed a result very close to the Cirel’son bound, The
Animal eats the Food test has revealed a significant violation of the latter
bound – a result which we had already been obtained in the web tests above on
The Animal Acts [46, 47]. We successfully apply and refine the quantum-
theoretic framework to The Animal Acts and The Animal eats the Food situations
and get a confirmation that a ‘strong form of quantum entanglement’ exists
between the respective individual concepts which involves both states and
measurements’. Moreover, we get a further confirmation that ‘such a form of
entanglement captures the deeply non-classical meaning connections existing
between those individual concepts’. Finally, we prove that ‘quantum
entanglement also exists beyond Cirel’son bound, but it is a joint effect of
state-entanglement and measurement-entanglement. This final result is
relevant, in our opinion, as it confutes the widespread belief that one cannot
model situations exceeding Cirel’son in the Hilbert space formalism in Hilbert
space (see, e.g., [6, 7, 9]). As such, this final result may also have an
impact on the foundations of quantum theory.

For the sake of completeness, we summarise the content of this paper in the
following.

In Section 2, we briefly review the theoretical and empirical results that our
research team has obtained in the last years on the identification of quantum
entanglement in conceptual combinations. In Section 3, we report and analyse
the data collected in two cognitive tests that we have recently performed on
human participants about the conceptual combinations The Animal Acts and The
Animal eats the Food. In Section 4, we explicate, refine and particularise to
The Animal Acts and The Animal eats the Food situations the general quantum-
theoretic framework that we have developed in [48] to model any Bell-type
situation. In Section 5, we show that the quantum-theoretic framework allows
to faithfully represent empirical data in Section 3. Finally, in Section 6, we
explain how and why a violation of the CHSH inequality which also exceed the
Cirel’son bound may indicate the presence of a stronger form of quantum
entanglement involving both states and measurements.

##  2 A brief history of Bell-type tests in conceptual combinations

We review in this section the results that we have obtained in the last decade
on the identification of entanglement in the combination of natural concepts
and related applied domains, as information retrieval and natural language
processing. For a detailed description of the tests and a complete analysis of
the obtained results, the reader is referred to the papers cited below (see
also Section 3).

In the first empirical test of entanglement involving concepts, we studied the
combination The Animal eats the Food, which we regarded as a combination of
the individual concepts Animal and Food [32]. We considered different items of
Animal, namely, Cat, Cow, Horse and Fish, and different items of Food, namely,
Grass, Meat, Fish and Nuts, and combined them to form all possible
combinations, i.e. The Cow eats the Grass, The Cat eats the Meat, The Cat eats
the Fish, The Squirrel eats the Nuts, and so on, which were considered as
items of the combination The Animal eats the Food. Next, we split the 16
combinations obtained in this way into 4 groups of 4 combinations each in
order to reproduce the 4 coincidence experiments described in Section 1. For
example, coincidence experiment $AB$ had 4 outcomes, The Cat eats the Grass,
The Cat eats the Meat, The Cow eats the Grass and The Cow eats the Meat; the
other experiments were constructed in an analogous way. We used the World Wide
Web as a conceptual space and counted co-occurrence of words, e.g., “cow” and
“grass”, “cat” and “fish”, and so on, in document retrieval operations by
means of the ‘Google’ search engine. We collected relative frequencies of co-
occurrences which we considered, in the large number limit, as joint
probabilities and inserted them into the correlation functions in Equation
(1). With our surprise, we found $\Delta_{CHSH}=2.86$, hence a significant
violation of the CHSH inequality. We put forward the presence of some type of
entanglement between the concepts Animal and Food was responsible of the
observed deviation from the CHSH inequality. At that time, we also noticed
without however deepening it, the simultaneous violation of Cirel’son bound.

This empirical finding was really exciting for us, also because it was
partially unexpected, thus we decided to deepen the investigation on
entanglement in conceptual combinations and performed a cognitive test in
which the direct responses of a sample of participants were collected [38]. In
this case, we analysed the conceptual combination The Animal Acts, which we
regarded as a combination of the individual concepts Animal and Acts – “acts”
refers here to the action of an animal emitting a sound. We again considered
different items of Animal, namely, Horse, Bear, Tiger and Cat, and different
items of Atcs, namely, Growls, Whinnies, Snorts and Meows, and combined them
to form all possible combinations, i.e. The Horse Growls, The Bear Whinnies,
The Tiger Snorts, The Cat Meows, and so on, which were considered as items of
the combination The Animal Acts. Then, we split the 16 combinations obtained
in this way into 4 groups of 4 combinations each in order to reproduce the 4
coincidence experiments in Section 1. For example, coincidence experiment $AB$
had 4 outcomes, The Horse Growls, The Horse Whinnies, The Bear Growls, The
Bear Whinnies; the other experiments were constructed in a similar way. After
reading an introductory text, a sample of 81 respondents had to fill in a
questionnaire where, in each coincidence experiment, they had to choose which
item in a list of 4 items they judged as a ‘good example’ of the conceptual
combination The Animal Acts. Relative frequencies of positive responses were
considered, in the large number limit, as joint probabilities and inserted
into the correlation functions in Equation (1). Also in this case, we found a
significant violation the CHSH inequality, $\Delta_{CHSH}=2.42$, which we
again interpreted as the consequence of a form of entanglement in the
combination The Animal Acts, due to the connections of meaning between the
component concepts Animal and Acts.

The numerical value $2.42$ obtained in the cognitive test on The Animal Acts
resembled the values found in Bell-type tests in quantum physics (see Section
1). To make the analogy with physics more convincing, we then decided to
construct an explicit quantum-theoretic model in Hilbert space of empirical
data [41]. But, we realised at once that we could not reproduce empirical data
on The Animal Acts test by representing the initial state of The Animal Acts
by an entangled state and the 4 measurements by product measurements, as one
would be tempted to do in analogy with a quantum representation in physics. We
discovered that the main reason of this impossibility was that, if one wants
to model The Animal Acts situation starting from the modelling of Animal and
Acts, hence using the usual tensor product representation, then the ‘marginal
law of probability’ has to be satisfied, as it occurs in quantum physics,
whereas it was systematically violated in the cognitive test. This difference
between physical and cognitive realms led us to study in detail a problem that
is generally overlooked in physics, the ‘identification problem’, that is, the
problem of recognising individual entities, e.g., concepts, from measurements
performed on composite entities, e.g., conceptual combinations. The
investigation led us to develop a novel quantum-theoretic framework for any
Bell-type situation, which allowed us to recognise systematic analogies and
differences between violations of Bell inequalities in physics and cognition
[48].333A closer analogy between Bell-type situations in physical and
cognitive realms was identified in a cognitive test we performed on the
conceptual combination Two Different Wind Directions, where the marginal law
was not violated and we could represent data in Hilbert space using entangled
states and product measurements [50, 51]. But, the quantum-theoretic framework
also introduced a new fundamental element in the modelling of such situations,
namely, the empirically justified use of ‘entangled measurements’ in addition
to entangled states. Entangled measurements were indeed successfully used in
the quantum representation of The Animal Acts situation [41].

In the meanwhile, we had also started systematically looking into possible
applications of quantum structures in information retrieval and natural
language processing, where representation of meaning entities, like concepts
and conceptual combinations, play a crucial role (see, e.g., [44, 27]). In
particular, we considered again The Animal Acts situation in a document
retrieval test in which we collected data on co-occurrence of words, as we had
done for The Animal eats the Food situation, but using known corpuses of
documents instead of web search engines – corpuses of documents provide more
reliable word counts than search engines [46]. We used ‘Google Books’, ‘Corpus
of Contemporary American English (COCA)’ and ‘News On Web (NOW)’ as corpuses
of documents and performed the coincidence experiments described in Section 1,
finding 3.41, 3.00 and 3.33, respectively in the factor appearing in Equation
(1). These empirical findings were statistically significant and, more
important, they were completely unexpected. Indeed, not only we had obtained
consistent violations of the CHSH inequality across all corpuses of documents,
but the numerical value of the violation systematically exceeded the Cirel’son
bound. According to a widespread belief, a violation of the CHSH inequality
which also violates Cirel’son bound cannot be modelled in Hilbert space
formalism of quantum theory, hence does not represent a ‘quantum effect’ [9].
On the other side, we noticed that the derivation of this bound does not
consider entangled measurements, but only product measurements [6, 7]. If one
allows entangled measurements, then it is in principle possible to model in
Hilbert space any violation of the CHSH inequality whose numerical value lies
in the mathematically permitted interval from $-4$ to $+4$. Indeed, we applied
in [47] the general quantum-theoretic framework and faithfully represented the
web data presented in [46].

Finally, in an independent but related investigation, we performed a web test
on artefacts of visual perception, such as images, using ‘Google Images’ as a
web search engine to count images representing items of the conceptual
combination The Animal Acts [29]. Also in this case, we obtained the value
$\Delta_{CHSH}=2.41$ for the CHSH factor in Equation (1), which again
indicated the presence of entanglement in visual perception. The corresponding
quantum-theoretic modelling enabled faithful representation of empirical data
and confirmed that a strong form of entanglement, involving both states and
measurements, exists between the component concepts Animal and Acts, again due
to their meaning connections.

We now intend to complete the investigation above and consider two cognitive
tests on the conceptual combinations The Animal Acts and The Animal eats the
Food with the aim of ‘identifying the presence of quantum entanglement beyond
the limits imposed by Cirel’son bound’. We will dedicate the next sections to
this purpose.

##  3 New empirical results

We report in this section the details of the two cognitive tests that we have
performed on the conceptual combinations The Animal Acts and The Animal eats
the Food. As we will see, their results substantially confirm the empirical
patterns highlighted in Section 2.

Let us start by The Animal Acts test. As anticipated in Section 2, we consider
the concept The Animal Acts as a combination of the individual concepts Animal
and Acts, where “acts” refers to the sound, or noise, produced by an animal.
Next, we consider two pairs of items of Animal, namely, (Horse, Bear) and
(Tiger, Cat), and two pairs of items of Acts, namely, (Growls, Whinnies) and
(Snorts, Meows). We are now ready to illustrate the test.

A sample of 81 individuals were presented in a ‘within subjects design’ a
questionnaire which contained 4 coincidence experiments $AB$, $AB^{\prime}$,
$A^{\prime}B$ and $A^{\prime}B^{\prime}$ whose setting was similar to the
typical setting of a Bell-type test sketched in Section 1. More specifically,
participants were preliminarily asked to read an ‘introductory text’ where the
concepts under study were introduced and a description of the tasks involved
in the judgement test was provided. Then, in each coincidence experiment
participants were asked to choose which item in a list of 4 items they judged
as a good example of the conceptual combination The Animal Acts.

In coincidence experiment $AB$, participants had to choose among the 4 items:

($A_{1}B_{1}$) The Horse Growls

($A_{2}B_{2}$) The Bear Whinnies

($A_{1}B_{2}$) The Horse Whinnies

($A_{2}B_{1}$) The Bear Growls

If the response was ($A_{1}B_{1}$) or ($A_{2}B_{2}$), then experiment $AB$ was
attributed outcome $+1$; if the response was ($A_{1}B_{2}$) or ($A_{2}B_{1}$),
then experiment $AB$ was attributed outcome $-1$.

In coincidence experiment $AB^{\prime}$, participants had to choose among the
4 items:

($A_{1}B^{\prime}_{1}$) The Horse Snorts

($A_{1}B^{\prime}_{2}$) The Horse Meows

($A_{2}B^{\prime}_{1}$) The Bear Snorts

($A_{2}B^{\prime}_{2}$) The Bear Meows

If the response was ($A_{1}B^{\prime}_{1}$) or ($A_{2}B^{\prime}_{1}$), then
experiment $AB^{\prime}$ was attributed outcome $+1$; if the response was
($A_{1}B^{\prime}_{2}$) or ($A_{2}B^{\prime}_{1}$), then experiment
$AB^{\prime}$ was attributed outcome $-1$.

In coincidence experiment $A^{\prime}B$, participants had to choose among the
4 items:

($A^{\prime}_{1}B_{1}$) The Tiger Growls

($A^{\prime}_{1}B_{2}$) The Tiger Whinnies

($A^{\prime}_{2}B_{1}$) The Cat Growls

($A^{\prime}_{2}B_{2}$) The Cat Whinnies

If the response was ($A^{\prime}_{1}B_{1}$) or ($A^{\prime}_{2}B_{2}$), then
experiment $A^{\prime}B$ was attributed outcome $+1$; if the response was
($A^{\prime}_{1}B_{2}$) or ($A^{\prime}_{2}B_{1}$), then experiment
$A^{\prime}B$ was attributed outcome $-1$.

Finally, in coincidence experiment $A^{\prime}B^{\prime}$, participants had to
choose among the 4 items:

($A^{\prime}_{1}B^{\prime}_{1}$) The Tiger Snorts

($A^{\prime}_{1}B^{\prime}_{2}$) The Tiger Meows

($A^{\prime}_{2}B^{\prime}_{1}$) The Cat Snorts

($A^{\prime}_{2}B^{\prime}_{2}$) The Cat Meows

If the response was ($A^{\prime}_{1}B^{\prime}_{1}$) or
($A^{\prime}_{2}B^{\prime}_{2}$), then experiment $A^{\prime}B^{\prime}$ was
attributed outcome $+1$; if the response was ($A^{\prime}_{1}B^{\prime}_{2}$)
or ($A^{\prime}_{2}B^{\prime}_{1}$), then experiment $A^{\prime}B^{\prime}$
was attributed outcome $-1$.

For each coincidence experiment $AB$, $AB^{\prime}$, $A^{\prime}B$ and
$A^{\prime}B^{\prime}$, we collected the relative frequencies of the obtained
responses which we considered, in the large number limit, as the probability
$\mu(A_{i}B_{j})$, $\mu(A_{i}B^{\prime}_{j})$, $\mu(A^{\prime}_{i}B_{j})$ and
$\mu(A^{\prime}_{i}B_{j})$ that the outcome $A_{i}B_{j}$,
$A_{i}B^{\prime}_{j}$, $A^{\prime}_{i}B_{j}$ and
$A^{\prime}_{i}B^{\prime}_{j}$, $i,j=1,2$, respectively, is obtained in the
corresponding experiment. Table 1 reports the judgement probabilities computed
in this way. Referring to these probabilities, we can then calculate the
expectation values, or correlation functions, of coincidence experiments $AB$,
$AB^{\prime}$, $A^{\prime}B$ and $A^{\prime}B^{\prime}$, as follows:

$\displaystyle E(AB)$ $\displaystyle=$
$\displaystyle\mu(A_{1}B_{1})-\mu(A_{1}B_{2})-\mu(A_{2}B_{1})+\mu(A_{2}B_{2})=-0.8025$
(2) $\displaystyle E(AB^{\prime})$ $\displaystyle=$
$\displaystyle\mu(A_{1}B^{\prime}_{1})-\mu(A_{1}B^{\prime}_{2})-\mu(A_{2}B^{\prime}_{1})+\mu(A_{2}B^{\prime}_{2})=0.4568$
(3) $\displaystyle E(A^{\prime}B)$ $\displaystyle=$
$\displaystyle\mu(A^{\prime}_{1}B_{1})-\mu(A^{\prime}_{1}B_{2})-\mu(A^{\prime}_{2}B_{1})+\mu(A^{\prime}_{2}B_{2})=0.7037$
(4) $\displaystyle E(A^{\prime}B^{\prime})$ $\displaystyle=$
$\displaystyle\mu(A^{\prime}_{1}B^{\prime}_{1})-\mu(A^{\prime}_{1}B^{\prime}_{2})-\mu(A^{\prime}_{2}B^{\prime}_{1})+\mu(A^{\prime}_{2}B^{\prime}_{2})=0.8765$
(5)

Inserting Equations (2)–(5) into Equation (1), we get

$\Delta_{CHSH}=E(A^{\prime},B^{\prime})+E(A^{\prime},B)+E(A,B^{\prime})-E(A,B)=2.7901$
(6)

Experiment $AB$ | _Horse Growls_ | _Horse Whinnies_ | _Bear Growls_ | _Bear Whinnies_
---|---|---|---|---
Probability | $\mu(A_{1}B_{1})=0.0494$ | $\mu(A_{1}B_{2})=0.1235$ | $\mu(A_{2}B_{1})=0.7778$ | $\mu(A_{2}B_{2})=0.0494$
Experiment $AB^{\prime}$ | _Horse Snorts_ | _Horse Meows_ | _Bear Snorts_ | _Bear Meows_
Probability | $\mu(A_{1}B^{\prime}_{1})=0.7160$ | $\mu(A_{1}B^{\prime}_{2})=0.0494$ | $\mu(A_{2}B^{\prime}_{1})=0.2222$ | $\mu(A_{2}B^{\prime}_{2})=0.0123$
Experiment $A^{\prime}B$ | _Tiger Growls_ | _Tiger Whinnies_ | _Cat Growls_ | _Cat Whinnies_
Probability | $\mu(A^{\prime}_{1}B_{1})=0.7778$ | $\mu(A^{\prime}_{1}B_{2})=0.0864$ | $\mu(A^{\prime}_{2}B_{1})=0.0617$ | $\mu(A^{\prime}_{2}B_{2})=0.0741$
Experiment $A^{\prime}B^{\prime}$ | _Tiger Snorts_ | _Tiger Meows_ | _Cat Snorts_ | _Cat Meows_
Probability | $\mu(A^{\prime}_{1}B^{\prime}_{1})=0.0864$ | $\mu(A^{\prime}_{1}B^{\prime}_{2})=0.0617$ | $\mu(A^{\prime}_{2}B^{\prime}_{1})=0.0247$ | $\mu(A^{\prime}_{2}B^{\prime}_{2})=0.8272$
Table 1: The data collected in coincidence experiments on entanglement in the
conceptual combination The Animal Acts.

The numerical value $2.7901\approx 2.79$ exceeds the classical limit imposed
by the CHSH inequality and is slightly below the Cirel’son bound. However,
while the deviation from the value $2$ is statistically significant (p-value
$1.44*10^{-5}$), the deviation from the value $2\sqrt{2}$ is not significant
(p-value $0.4151$), which entails that The Animal Acts data show a non
definite behaviour with respect to the Cirel’son bound. This empirical pattern
confirms and strengthens the results obtained in [38] where a violation of the
CHSH inequality was observed. This non-classical behaviour admits the presence
of entanglement between the concepts Animal and Acts as a natural explanation,
because Animal and Acts are connected by meaning and this connection gives
rise to statistical correlations, those expressed by Equations (2)–(5).

Let us now come to The Animal eats the Food test. As mentioned in Section 2,
we consider the concept The Animal eats the Food as a combination of the
individual concepts Animal and Food. Let us then consider two pairs of items
of Animal, namely, (Cat, Cow) and (Horse, Squirrel), and two pairs of items of
Food, namely, (Grass, Meat) and (Fish, Nuts). The test can be illustrated as
follows.

A sample of 81 individuals were presented in a ‘within subjects design’ a
questionnaire which contained again the 4 coincidence experiments $AB$,
$AB^{\prime}$, $A^{\prime}B$ and $A^{\prime}B^{\prime}$ that are typical of a
Bell-type test. More precisely, participants, after reading an introductory
text on concepts and their combinations as in the first test, were asked to
choose in each coincidence experiment one item in a list of 4 items, namely,
the item that they judged as a good example of the conceptual combination The
Animal eats the Food.

In coincidence experiment $AB$, participants had to choose among the 4 items:

($A_{1}B_{1}$) The Cat eats the Grass

($A_{2}B_{2}$) The Cat eats the Meat

($A_{1}B_{2}$) The Cow eats the Grass

($A_{2}B_{1}$) The Cow eats the Meat

If the response was ($A_{1}B_{1}$) or ($A_{2}B_{2}$), then experiment $AB$ was
attributed outcome $+1$; if the response was ($A_{1}B_{2}$) or ($A_{2}B_{1}$),
then experiment $AB$ was attributed outcome $-1$.

In coincidence experiment $AB^{\prime}$, participants had to choose among the
4 items:

($A_{1}B^{\prime}_{1}$) The Cat eats the Fish

($A_{1}B^{\prime}_{2}$) The Cat eats the Nuts

($A_{2}B^{\prime}_{1}$) The Cow eats the Fish

($A_{2}B^{\prime}_{2}$) The Cow eats the Nuts

If the response was ($A_{1}B^{\prime}_{1}$) or ($A_{2}B^{\prime}_{1}$), then
experiment $AB^{\prime}$ was attributed outcome $+1$; if the response was
($A_{1}B^{\prime}_{2}$) or ($A_{2}B^{\prime}_{1}$), then experiment
$AB^{\prime}$ was attributed outcome $-1$.

In coincidence experiment $A^{\prime}B$, participants had to choose among the
4 items:

($A^{\prime}_{1}B_{1}$) The Horse eats the Grass

($A^{\prime}_{1}B_{2}$) The Squirrel eats the Meat

($A^{\prime}_{2}B_{1}$) The Horse eats the Grass

($A^{\prime}_{2}B_{2}$) The Cow eats the Meat

If the response was ($A^{\prime}_{1}B_{1}$) or ($A^{\prime}_{2}B_{2}$), then
experiment $A^{\prime}B$ was attributed outcome $+1$; if the response was
($A^{\prime}_{1}B_{2}$) or ($A^{\prime}_{2}B_{1}$), then experiment
$A^{\prime}B$ was attributed outcome $-1$.

Finally, in coincidence experiment $A^{\prime}B^{\prime}$, participants had to
choose among the 4 items:

($A^{\prime}_{1}B^{\prime}_{1}$) The Horse eats the Fish

($A^{\prime}_{1}B^{\prime}_{2}$) The Horse eats the Nuts

($A^{\prime}_{2}B^{\prime}_{1}$) The Squirrel eats the Fish

($A^{\prime}_{2}B^{\prime}_{2}$) The Squirrel eats the Nuts

If the response was ($A^{\prime}_{1}B^{\prime}_{1}$) or
($A^{\prime}_{2}B^{\prime}_{2}$), then experiment $A^{\prime}B^{\prime}$ was
attributed outcome $+1$; if the response was ($A^{\prime}_{1}B^{\prime}_{2}$)
or ($A^{\prime}_{2}B^{\prime}_{1}$), then experiment $A^{\prime}B^{\prime}$
was attributed outcome $-1$.

As in the first test, after collecting relative frequencies of responses, we
calculated in the large number limit the probability $\mu(X_{i}Y_{j})$ that
the outcome $X_{i}Y_{j}$ is obtained in the coincidence experiment $XY$,
$i,j=1,2$, $X=A,A^{\prime}$, $Y=B,B^{\prime}$. Table 2 reports the judgement
probabilities computed in this way. The corresponding expectation values, or
correlation functions, are then calculated as follows:

$\displaystyle E(AB)$ $\displaystyle=$
$\displaystyle\mu(A_{1}B_{1})-\mu(A_{1}B_{2})-\mu(A_{2}B_{1})+\mu(A_{2}B_{2})=-0.7284$
(7) $\displaystyle E(AB^{\prime})$ $\displaystyle=$
$\displaystyle\mu(A_{1}B^{\prime}_{1})-\mu(A_{1}B^{\prime}_{2})-\mu(A_{2}B^{\prime}_{1})+\mu(A_{2}B^{\prime}_{2})=0.8025$
(8) $\displaystyle E(A^{\prime}B)$ $\displaystyle=$
$\displaystyle\mu(A^{\prime}_{1}B_{1})-\mu(A^{\prime}_{1}B_{2})-\mu(A^{\prime}_{2}B_{1})+\mu(A^{\prime}_{2}B_{2})=0.9012$
(9) $\displaystyle E(A^{\prime}B^{\prime})$ $\displaystyle=$
$\displaystyle\mu(A^{\prime}_{1}B^{\prime}_{1})-\mu(A^{\prime}_{1}B^{\prime}_{2})-\mu(A^{\prime}_{2}B^{\prime}_{1})+\mu(A^{\prime}_{2}B^{\prime}_{2})=0.8519$
(10)

Inserting Equations (7)–(10) into Equation (1), we get
$\Delta_{CHSH}=3.2840\approx 3.28$.

Experiment $AB$ | _Cat eats Grass_ | _Cat eats Meat_ | _Cow eats Fish_ | _Cow eats Meat_
---|---|---|---|---
Probability | $\mu(A_{1}B_{1})=0.1111$ | $\mu(A_{1}B_{2})=0.3457$ | $\mu(A_{2}B_{1})=0.5185$ | $\mu(A_{2}B_{2})=0.0247$
Experiment $AB^{\prime}$ | _Cat eats Fish_ | _Cat eats Nuts_ | _Cow eats Fish_ | _Cow eats Nuts_
Probability | $\mu(A_{1}B^{\prime}_{1})=0.8765$ | $\mu(A_{1}B^{\prime}_{2})=0.0494$ | $\mu(A_{2}B^{\prime}_{1})=0.0494$ | $\mu(A_{2}B^{\prime}_{2})=0.0247$
Experiment $A^{\prime}B$ | _Horse eats Grass_ | _Horse eats Meat_ | _Squirrel eats Grass_ | _Squirrel eats Meat_
Probability | $\mu(A^{\prime}_{1}B_{1})=0.8889$ | $\mu(A^{\prime}_{1}B_{2})=0.0494$ | $\mu(A^{\prime}_{2}B_{1})=0$ | $\mu(A^{\prime}_{2}B_{2})=0.0617$
Experiment $A^{\prime}B^{\prime}$ | _Horse eats Fish_ | _Horse eats Nuts_ | _Squirrel eats Fish_ | _Squirrel eats Nuts_
Probability | $\mu(A^{\prime}_{1}B^{\prime}_{1})=0.0494$ | $\mu(A^{\prime}_{1}B^{\prime}_{2})=0.1235$ | $\mu(A^{\prime}_{2}B^{\prime}_{1})=0.0617$ | $\mu(A^{\prime}_{2}B^{\prime}_{2})=0.8765$
Table 2: The data collected in coincidence experiments on entanglement in the
conceptual combination The Animal eats the Food.

The numerical value 3.28 violates both the CHSH inequality and the Cirel’son
bound. In both cases, the deviation is statistically significant (p-values
$2.81*10^{-11}$ and $4.40*10^{-2}$, respectively). Hence, this deviation
confirms and strengthens the empirical patterns identified in document
retrieval tests on the web in [32] and [46]. Also in this case, entanglement
between the concepts Animal and Food is a natural candidate to explain this
non-classical behaviour, due to the meaning connections existing between
Animal and Food.

We will show in the next two sections that these empirical findings can be
represented in the Hilbert space formalism of quantum theory, independently of
their behaviour with respect to the Cirel’son bound. However, we firstly need
to present the essentials of a general quantum-theoretic framework that we
have recently elaborated to model Bell-type situations in any empirical
domain.

##  4 Construction of a general Hilbert space model

In this section we study how a theoretical framework can be constructed which
models the two cognitive situations in Section 3 but is also general enough to
cope with any domain in which Bell-type tests are performed. We will show that
the violations of the CHSH inequality and the Cirel’son bound can be
simultaneously explained by assuming that a strong form of ‘quantum
entanglement’ is present in conceptual combinations. We refine and generalise
the modelling scheme that we have elaborated in [41, 42] to identify the
entanglement that occurs in Bell-type settings. The theoretical scheme has
already been applied with success to model the web tests in [47] (see also
Section 2). We refer to [48] for a detailed analysis and comparison of the
entanglement situations occurring in physical and cognitive realms.

The general modelling scheme for Bell-type situations consists in the
implementation of three main steps which we summarise in the following.

(i) One identifies in the empirical situation under investigation the
composite conceptual entity and the individual conceptual entities composing
it.

(ii) One recognises in the composite conceptual entity the states,
measurements and outcome probabilities that are relevant to the phenomenon
under study.

(iii) One represents entities, states, measurements and outcome probabilities
using the Hilbert space representation of entities, states, measurements and
outcome probabilities of quantum theory.

The application of steps (i)–(iii) to The Animal Acts situation has been
presented in various papers [38, 41, 47]. Let us then focus here on how
(i)–(iii) are applied to The Animal eats the Food situation. Let us start by
coincidence experiment $AB$ and its outcomes The Cat eats the Grass, The The
Cat eats the Meat, The Cow eats the Grass and The Cow eats the Meat.

(i) The conceptual combination The Animal eats the Food can be considered as a
composite conceptual entity made up of the individual conceptual entities
Animal and Food.

(ii) Whenever an individual reads the introductory text which explains the
details of the test and nature of the involved concepts, this set of
instructions prepares the composite entity The Animal eats the Food in an
initial state $p$ which describes the general situation of an animal that eats
food. This initial state is the unique conceptual state which all participants
are confronted with in the test. In coincidence experiment $AB$, each
respondent interacts with this uniquely prepared state $p$ and operates as a
measurement context $e_{AB}$ for The Animal eats the Food which changes $p$
into a generally different state. The latter state is not predetermined, as it
depends on the concrete choice being made, which results as a consequence of
this ‘contextual interaction’ between the respondent and the conceptual
entity. More specifically, if the respondent chooses The Cow eats the Grass,
that is, the outcome $A_{2}B_{1}$ is obtained in $AB$ (see Section 3), the
interaction between the entity The Animal eats the Food prepared in the state
$p$ and the (mind of the) respondent will determine a change of state of The
Animal eats the Food from $p$ to the state $p_{A_{2}B_{1}}$ which describes
the more concrete situation of a cow that eats grass. When all responses are
collected, a statistics of outcomes arises from this intrinsically and
genuinely indeterministic contextual process of state change. This outcome
statistics is interpreted, in the large number limit, as the probability
$P_{p}(A_{2}B_{1})$ that the outcome $A_{2}B_{1}$ is obtained when the
measurement $e_{AB}$ is performed on the composite entity The Animal eats the
Food in the initial state $p$.444In [49]), we have called ‘realistic’ and
‘operational’ the above description of a measurement process on a conceptual
entity, where the term ‘realistic’ refers to the preparation of the conceptual
entity in a defined state and the term ‘operational’ refers to the
measurements that are performed on the entity.

(iii) Let us finally come to the quantum mathematical representation. The
entity The Animal eats the Food is associated with a Hilbert space and the
state $p$ is represented by a unit vector $|p\rangle$ of this Hilbert space.
The measurement $e_{AB}$ is instead represented by a self-adjoint operator or,
equivalently, by a spectral family, on the Hilbert space whose eigenvectors
represent the outcome states, or ‘eigenstates’, of $e_{AB}$, while outcome
probabilities are obtained through the Born rule of quantum probability. This
representation will become clear in Section 5.

The description above can be extended in a straightforward way to the other
coincidence experiments of The Animal eats the Food situation. Hence, the
complete identification of states, measurements, outcomes and outcome
probabilities that are relevant to the situations in Section 3 is presented in
the following.

For every $X=A,A^{\prime}$, $Y=B,B^{\prime}$, coincidence experiment $XY$
corresponds to a measurement $e_{XY}$ performed on the composite conceptual
entity The Animal Acts (respectively, The Animal eats the Food) with 4
possible outcomes $X_{i}Y_{j}$, $i,j=1,2$, where we choose $X_{i}Y_{j}=+1$ if
$i=j$ and $X_{i}Y_{j}=-1$ if $i\neq j$, and 4 eigenstates $p_{X_{i}Y_{j}}$
describing the state of The Animal Acts (The Animal eats the Food) after the
outcome $X_{i}Y_{j}$ occurs in $XY$. Let us denote by $P_{p}(X_{i}Y_{j})$ the
probability that the outcome $X_{i}Y_{j}$ is obtained when the measurement
$e_{XY}$ is performed on The Animal Acts (The Animal eats the Food) in the
state $p$.

Once we have completed the identification of entities, initial state,
measurements, eigenstates and outcome probabilities occurring in the The
Animal Acts and The Animal eats the Food situations, we need to construct a
general quantum framework in Hilbert space to model these conceptual
situations. However, before doing so, we need to recall a theoretical analysis
that we have presented in detail in [27] (see also [41, 47]).

We preliminarily observe that in both The Animal Acts and The Animal eats the
Food situations, all measurements $e_{XY}$, $X=A,A^{\prime}$,
$Y=B,B^{\prime}$, have 4 outcomes $X_{i}Y_{j}$, $i,j=1,2$, which entails that
both composite entities should be associated, as overall entities, with the
complex Hilbert space $\mathbb{C}^{4}$ of all ordered 4-tuples of complex
numbers. In addition, each state $p$ of The Animal Acts (respectively, The
Animal eats the Food) should be represented by a unit vector of
$\mathbb{C}^{4}$ and each measurement on The Animal Acts (The Animal eats the
Food) should be represented by a self-adjoint operator or, equivalently, by a
spectral family, on $\mathbb{C}^{4}$. On the other side, for every $i,j=1,2$,
each outcome $X_{i}Y_{j}$ is obtained by juxtaposing the outcomes $X_{i}$ and
$Y_{j}$, e.g., The Bear Growls is obtained by syntactically juxtaposing the
words “bear” and “growls”. This operation defines a 2-outcome measurement
$e_{X}$, $X=A,A^{\prime}$ on the individual entity Animal (Animal) and a
2-outcome measurement $e_{Y}$, $Y=B,B^{\prime}$ on the individual entity Acts
(Food). Hence, each of these individual entities should be associated with the
complex Hilbert space $\mathbb{C}^{2}$ of all ordered couples of complex
numbers. But, then, the standard Hilbert space formalism prescribes that both
composite entities The Animal Acts and The Animal eats the Food should be
associated with the tensor product Hilbert space
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. We stress that we are studying here an
‘identification problem’, that is, the problem of how the composite entity The
Animal Acts (The Animal eats the Food) can be decomposed into the individual
entities Animal (Animal) and Acts (Food)’ in such a way that ‘these individual
entities can be recognised from measurements performed on the respective
composite entities’. As such, we are doing an operation that is opposite to
what one typically does in Bell-type situations in quantum physics, where one
constructs or, better, ‘composes’, the measurements on the composite entity
from measurements performed on individual entities.

From a mathematical point of view, the vector spaces $\mathbb{C}^{4}$ and
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ are isomorphic, and any isomorphism can
be expressed in terms of the relationship between the corresponding
orthonormal (ON) bases. States of The Animal Acts are represented by unit
vectors of $\mathbb{C}^{4}$, hence of $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$,
which contains both vectors representing ‘product states’ and vectors
representing ‘entangled states’. Moreover, the vector space
$L(\mathbb{C}^{4})$ of all linear operators on $\mathbb{C}^{4}$ is isomorphic
to the tensor product $L(\mathbb{C}^{2})\otimes L(\mathbb{C}^{2})$, where
$L(\mathbb{C}^{2})$ of all linear operators on $\mathbb{C}^{2}$. Analogously,
the tensor product $L(\mathbb{C}^{2})\otimes L(\mathbb{C}^{2})$ contains both
self-adjoint operators representing ‘product measurements’ and self-adjoint
operators representing ‘entangled measurements’.

Now, let $I:\mathbb{C}^{4}\longrightarrow\mathbb{C}^{2}\otimes\mathbb{C}^{2}$
be an isomorphism mapping a given ON basis of $\mathbb{C}^{4}$ onto a given ON
basis of $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. We say that a state $p$
represented by the unit vector $|p\rangle\in{\mathbb{C}}^{4}$ is a ‘product
state’ with respect to $I$, if two states $p_{A}$ and $p_{B}$, represented by
the unit vectors $|p_{A}\rangle\in{\mathbb{C}}^{2}$ and
$|p_{B}\rangle\in{\mathbb{C}}^{2}$, respectively, exist such that
$I|p\rangle=|p_{A}\rangle\otimes|p_{B}\rangle$. Otherwise, $p$ is an
‘entangled state’ with respect to $I$. Next, we say that a measurement $e$
represented by the self-adjoint operator ${\mathscr{E}}$ on ${\mathbb{C}}^{4}$
is a ‘product measurement’ with respect to $I$, if two measurements $e_{X}$
and $e_{Y}$, represented by the self-adjoint operators ${\mathscr{E}}_{X}$ and
${\mathscr{E}}_{Y}$, respectively, on ${\mathbb{C}}^{2}$ exist such that
$I{\mathscr{E}}I^{-1}={\mathscr{E}}_{X}\otimes{\mathscr{E}}_{Y}$. Otherwise,
$e$ is an ‘entangled measurement’ with respect to $I$. Hence, the notion of
entanglement crucially depends on the ‘isomorphism that is used to identify
individual entities within a given composite entity’.

With reference to the Bell-type setting presented in this section, one can
then prove that, if the measurements $e_{XY}$ and $e_{XY^{\prime}}$,
$X=A,A^{\prime}$, $Y,Y^{\prime}=B,B^{\prime}$, $Y^{\prime}\neq Y$, are product
measurements with respect to the isomorphism $I$, then, for every state $p$ of
the composed entity, the ‘marginal law of Kolmogorovian probability’ is
satisfied, that is, for every $i=1,2$,
$\sum_{j}P_{p}(X_{i}Y_{j})=\sum_{j}P_{p}(X_{i}Y^{\prime}_{j})$. Analogously,
if the measurements $e_{XY}$ and $e_{X^{\prime}Y}$,
$X,X^{\prime}=A,A^{\prime}$, $Y=B,B^{\prime}$, $X^{\prime}\neq X$, are product
measurements with respect to the isomorphism $I$, then, for every state $p$ of
the composed entity, the marginal law is satisfied, that is, for every
$j=1,2$, $\sum_{i}P_{p}(X_{i}Y_{j})=\sum_{i}P_{p}(X^{\prime}_{i}Y_{j})$ (see
Theorem 2, [41]). In the case the marginal law is satisfied in all
measurements, one can also prove that a unique isomorphism exists, which can
be chosen to be the identify operator (see, Theorem 4, [41]).

It follows from the above that, if the marginal law is violated, as it occurs
in both conceptual tests in Section 3,555The marginal law is systematically
violated in The Animal Acts test. For example,
$\mu(A_{2}B_{1})+\mu(A_{2}B_{2})=0.8272\neq
0.2345=\mu(A_{2}B^{\prime}_{1})+\mu(A_{2}B^{\prime}_{2})$. Analogously, the
marginal law is systematically violated in The Animal eats the Food test. For
example, $\mu(A^{\prime}_{1}B_{1})+\mu(A^{\prime}_{1}B_{2})=0.9383\neq
0.1729=\mu(A^{\prime}_{1}B^{\prime}_{1})+\mu(A^{\prime}_{1}B^{\prime}_{2})$.
then one cannot find a unique isomorphism between $\mathbb{C}^{4}$ and
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ such that all measurements are product
measurements with respect to this isomorphism. In this case, one cannot
explain the violation of the CHSH inequality as due to the usual situation in
quantum physics where all measurements are product measurements and only the
initial, or pre-measurement, state is entangled. Furthermore, if the marginal
law is systematically violated, 4 distinct isomorphisms $I_{XY}$,
$X=A,A^{\prime}$, $Y=B,B^{\prime}$, exist such that the measurement $e_{XY}$
is a product measurement with respect to $I_{XY}$. As a consequence, there is
no unique isomorphism allowing to identify individual entities of a given
composite entity. Finally, if we consider a given isomorphism between
$\mathbb{C}^{4}$ and $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ with respect to
which identifying individual entities of a composite entity in a given test,
then it may happen that both the pre-measurement state and all measurements
are entangled. This final remark suggests the following considerations.

Firstly, the non-classical connections which violate the CHSH inequality in
both The Animal Acts and The Animal eats the Food situations can be reasonably
attributed to the fact that ‘the component individual concepts carry meaning
and further meaning is created in the combination process’. Since the
violation of the CHSH inequality indicates the presence of entanglement
between the individual conceptual entities, then it is reasonable to assume
that ‘it is the quantum structure of entanglement which is able to
theoretically capture the meaning connections that are created in these
cases’. This suggests, in a quantum-theoretic perspective, that the initial
state $p$ of both composite entities The Animal Acts and The Animal eats the
Food should be an entangled state. This entangled state would then capture the
meaning connections that are created between Animal and Acts and also between
Animal and Food when the respondent read the introductory text and start the
questionnaire.

Secondly, in both The Animal Acts and The Animal eats the Food situations, all
measurements $e_{XY}$, $X=A,A^{\prime}$, $Y=B,B^{\prime}$, violate the
marginal law of Kolmogorovian probability. This suggests, again in a quantum-
theoretic perspective, that all these measurements should be entangled
measurements. In addition, in each measurement, all outcomes $X_{i}Y_{j}$,
$i,j=1,2$, correspond to combined concepts, e.g., The Cat Meows, The Cat eats
the Fish, etc., which are thus in turn connected by meaning. This also
suggests that all eigenstates $p_{X_{i}Y_{j}}$ should be entangled states.

The two considerations above applied to the Hilbert space representation of
both The Animal Acts cognitive and web tests [41, 47]. We will see in the next
section that these considerations also apply to the empirical tests in in
Section 3.

##  5 Quantum representation of data

We work out in this section a quantum representation in Hilbert space of the
cognitive tests data in Section 3, following the methodology developed in [48]
and already applied to previous tests [41, 47]. The considerations in Section
4 suggest the following quantum mathematical representation for both The
Animal Acts and The Animal eats the Food situations.

The composite conceptual entity is associated with the Hilbert space
$\mathbb{C}^{4}$ of all ordered 4-tuples of complex numbers. Let $(1,0,0,0)$,
$(0,1,0,0)$, $(0,0,1,0)$ and $(0,0,0,1)\\}$ be the unit vectors of the
canonical ON basis of $\mathbb{C}^{4}$ and let us consider the isomorphism
$I:\mathbb{C}^{4}\longrightarrow\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ where
this ON basis coincides with the ON basis of the tensor product Hilbert space
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ made up of the unit vectors
$(1,0)\otimes(1,0)$, $(1,0)\otimes(0,1)$, $(0,1)\otimes(1,0)$ and
$(0,1)\otimes(0,1)$. In these ON bases, the initial state $p$ of the composite
entity is represented by the unit vector
$|p\rangle=(ae^{i\alpha},be^{i\beta},ce^{i\gamma},de^{i\delta})$, where
$a,b,c,d\geq 0$, $a^{2}+b^{2}+c^{2}+d^{2}=1$, $\alpha$, $\beta$, $\gamma$,
$\delta\in\Re$ and $\Re$ is the real line. One easily proves that $|p\rangle$
represents a product state if and only if the following condition is
satisfied:

$ade^{i(\alpha+\delta)}-bce^{i(\beta+\gamma)}=0$ (11)

Otherwise, $p$ represents an entangled state.

Let us now represent the measurements $e_{AB}$, $e_{AB^{\prime}}$,
$e_{A^{\prime}B}$ and $e_{A^{\prime}B^{\prime}}$ in Section 4. Each
measurement $e_{XY}$, $X=A,A^{\prime}$, $Y=B,B^{\prime}$, has 4 outcomes
$X_{1}Y_{1}$, $X_{1}Y_{2}$, $X_{2}Y_{1}$ and $X_{2}Y_{2}$ and 4 eigenstates
$p_{X_{1}Y_{1}}$, $p_{X_{1}Y_{2}}$, $p_{X_{2}Y_{1}}$ and $p_{X_{2}Y_{2}}$. As
in Sections 3 and 4, we set, for every $X=A,A^{\prime}$, $Y=B,B^{\prime}$,
$X_{1}Y_{1}=X_{2}Y_{2}=+1$ and $X_{1}Y_{2}=X_{2}Y_{1}=-1$. The measurement
$e_{XY}$ is represented by the self-adjoint operator ${\mathscr{E}}_{XY}$ on
${\mathbb{C}}^{4}$ or, equivalently, by the spectral family
$\\{|p_{X_{1}Y_{1}}\rangle\langle
p_{X_{1}Y_{1}}|,|p_{X_{1}Y_{2}}\rangle\langle
p_{X_{1}Y_{2}}|,|p_{X_{2}Y_{1}}\rangle\langle
p_{X_{2}Y_{1}}|,|p_{X_{2}Y_{2}}\rangle\langle p_{X_{2}Y_{2}}|\\}$, such that
the eigenstates $p_{X_{1}Y_{1}}$, $p_{X_{1}Y_{2}}$, $p_{X_{2}Y_{1}}$ and
$p_{X_{2}Y_{2}}$ are represented by the eigenvectors

$\displaystyle|p_{11}\rangle$ $\displaystyle=$
$\displaystyle(a_{11}e^{i\alpha_{11}},b_{11}e^{i\beta_{11}},c_{11}e^{i\gamma_{11}},d_{11}e^{i\delta_{11}})$
(12) $\displaystyle|p_{12}\rangle$ $\displaystyle=$
$\displaystyle(a_{12}e^{i\alpha_{12}},b_{12}e^{i\beta_{12}},c_{12}e^{i\gamma_{12}},d_{12}e^{i\delta_{12}})$
(13) $\displaystyle|p_{21}\rangle$ $\displaystyle=$
$\displaystyle(a_{21}e^{i\alpha_{21}},b_{21}e^{i\beta_{21}},c_{21}e^{i\gamma_{21}},d_{21}e^{i\delta_{21}})$
(14) $\displaystyle|p_{22}\rangle$ $\displaystyle=$
$\displaystyle(a_{22}e^{i\alpha_{22}},b_{22}e^{i\beta_{22}},c_{22}e^{i\gamma_{22}},d_{22}e^{i\delta_{22}})$
(15)

of ${\mathscr{E}}_{XY}$, respectively. In Equations (12)–(15), the
coefficients are such that $a_{ij},b_{ij},c_{ij},d_{ij}\geq 0$ and
$\alpha_{ij},\beta_{ij},\gamma_{ij},\delta_{ij}\in\Re$, $i,j=1,2$. For every
$X=A,A^{\prime}$, $Y=B,B^{\prime}$, the self-adjoint operator
${\mathscr{E}}_{XY}$ can be expressed as a tensor product operator if and only
if all unit vectors in Equations (12)–(15) represent product states.
Otherwise, ${\mathscr{E}}_{XY}$ cannot be expressed as a tensor product
operator, hence $e_{XY}$ is an entangled measurement. For every
$X=A,A^{\prime}$, $Y=B,B^{\prime}$, $i,j=1,2$, the probability
$P_{p}(X_{i}Y_{j})$ of obtaining the outcome $X_{i}Y_{j}$ in a measurement of
$e_{XY}$ on the composite entity in the state $p$ is then given by the Born
rule of quantum probability, that is, $P_{p}(X_{i}Y_{j})=|\langle
p_{ij}|p\rangle|^{2}$.

To find a quantum mathematical representation of the data in Section 3, for
every measurement $e_{XY}$, the unit vectors in (12)–(15) have to satisfy the
following three sets of conditions.

(i) Normalization. The vectors in Equations (12)–(15) are unitary, that is,

$\displaystyle a_{11}^{2}+b_{11}^{2}+c_{11}^{2}+d_{11}^{2}$ $\displaystyle=$
$\displaystyle 1$ (16) $\displaystyle
a_{12}^{2}+b_{12}^{2}+c_{12}^{2}+d_{12}^{2}$ $\displaystyle=$ $\displaystyle
1$ (17) $\displaystyle a_{21}^{2}+b_{21}^{2}+c_{21}^{2}+d_{21}^{2}$
$\displaystyle=$ $\displaystyle 1$ (18) $\displaystyle
a_{22}^{2}+b_{22}^{2}+c_{22}^{2}+d_{22}^{2}$ $\displaystyle=$ $\displaystyle
1$ (19)

(ii) Orthogonality. The vectors in Equations (12)–(15) are mutually
orthogonal, that is,

$\displaystyle 0=\langle
p_{11}|p_{12}\rangle=a_{11}a_{12}e^{i(\alpha_{12}-\alpha_{11})}+b_{11}b_{12}e^{i(\beta_{12}-\beta_{11})}+a_{11}c_{12}c^{i(\gamma_{12}-\gamma_{11})}+d_{11}d_{12}e^{i(\delta_{12}-\delta_{11})}$
(20) $\displaystyle 0=\langle
p_{11}|p_{21}\rangle=a_{11}a_{21}e^{i(\alpha_{21}-\alpha_{11})}+b_{11}b_{21}e^{i(\beta_{21}-\beta_{11})}+a_{11}c_{21}c^{i(\gamma_{21}-\gamma_{11})}+d_{11}d_{21}e^{i(\delta_{21}-\delta_{11})}$
(21) $\displaystyle 0=\langle
p_{11}|p_{22}\rangle=a_{11}a_{22}e^{i(\alpha_{22}-\alpha_{11})}+b_{11}b_{22}e^{i(\beta_{22}-\beta_{11})}+a_{11}c_{22}c^{i(\gamma_{22}-\gamma_{11})}+d_{11}d_{22}e^{i(\delta_{22}-\delta_{11})}$
(22) $\displaystyle 0=\langle
p_{12}|p_{21}\rangle=a_{12}a_{21}e^{i(\alpha_{21}-\alpha_{12})}+b_{12}b_{21}e^{i(\beta_{21}-\beta_{12})}+a_{12}c_{21}c^{i(\gamma_{21}-\gamma_{12})}+d_{12}d_{21}e^{i(\delta_{21}-\delta_{12})}$
(23) $\displaystyle 0=\langle
p_{12}|p_{22}\rangle=a_{12}a_{22}e^{i(\alpha_{22}-\alpha_{12})}+b_{12}b_{22}e^{i(\beta_{22}-\beta_{12})}+a_{12}c_{22}c^{i(\gamma_{22}-\gamma_{12})}+d_{12}d_{22}e^{i(\delta_{22}-\delta_{12})}$
(24) $\displaystyle 0=\langle
p_{21}|p_{22}\rangle=a_{21}a_{22}e^{i(\alpha_{22}-\alpha_{21})}+b_{21}b_{22}e^{i(\beta_{22}-\beta_{21})}+a_{21}c_{22}c^{i(\gamma_{22}-\gamma_{21})}+d_{21}d_{22}e^{i(\delta_{22}-\delta_{21})}$
(25)

(iii) Probabilities. For every $X=A,A^{\prime}$, $Y=B,B^{\prime}$, $i,j=1,2$,
the probability $P_{p}(X_{i}Y_{j})$ coincides with the empirical probability
$\mu(X_{i}Y_{j})$ in Tables 1 and 2, that is,

$\displaystyle\mu(X_{1}Y_{1})=|\langle p_{11}|p\rangle|^{2}$ $\displaystyle=$
$\displaystyle
a^{2}a_{11}^{2}+b^{2}b_{11}^{2}+c^{2}c_{11}^{2}+d^{2}d_{11}^{2}+$ (26)
$\displaystyle+$ $\displaystyle
2aba_{11}b_{11}\cos(\alpha-\alpha_{11}-\beta+\beta_{11})+$ $\displaystyle+$
$\displaystyle 2aca_{11}c_{11}\cos(\alpha-\alpha_{11}-\gamma+\gamma_{11})+$
$\displaystyle+$ $\displaystyle
2ada_{11}d_{11}\cos(\alpha-\alpha_{11}-\delta+\delta_{11})+$ $\displaystyle+$
$\displaystyle 2bcb_{11}c_{11}\cos(\beta-\beta_{11}-\gamma+\gamma_{11})+$
$\displaystyle+$ $\displaystyle
2bdb_{11}d_{11}\cos(\beta-\beta_{11}-\delta+\delta_{11})+$ $\displaystyle+$
$\displaystyle 2cdc_{11}d_{11}\cos(\gamma-\gamma_{11}-\delta+\delta_{11})$
$\displaystyle\mu(X_{1}Y_{2})=|\langle p_{12}|p\rangle|^{2}$ $\displaystyle=$
$\displaystyle
a^{2}a_{12}^{2}+b^{2}b_{12}^{2}+c^{2}c_{12}^{2}+d^{2}d_{12}^{2}+$ (27)
$\displaystyle+$ $\displaystyle
2aba_{12}b_{12}\cos(\alpha-\alpha_{12}-\beta+\beta_{12})+$ $\displaystyle+$
$\displaystyle 2aca_{12}c_{12}\cos(\alpha-\alpha_{12}-\gamma+\gamma_{12})+$
$\displaystyle+$ $\displaystyle
2ada_{12}d_{12}\cos(\alpha-\alpha_{12}-\delta+\delta_{12})+$ $\displaystyle+$
$\displaystyle 2bcb_{12}c_{12}\cos(\beta-\beta_{12}-\gamma+\gamma_{12})+$
$\displaystyle+$ $\displaystyle
2bdb_{12}d_{12}\cos(\beta-\beta_{12}-\delta+\delta_{12})+$ $\displaystyle+$
$\displaystyle 2cdc_{12}d_{12}\cos(\gamma-\gamma_{12}-\delta+\delta_{12})$
$\displaystyle\mu(X_{2}Y_{1})=|\langle p_{21}|p\rangle|^{2}$ $\displaystyle=$
$\displaystyle
a^{2}a_{21}^{2}+b^{2}b_{21}^{2}+c^{2}c_{21}^{2}+d^{2}d_{21}^{2}+$ (28)
$\displaystyle+$ $\displaystyle
2aba_{21}b_{21}\cos(\alpha-\alpha_{21}-\beta+\beta_{21})+$ $\displaystyle+$
$\displaystyle 2aca_{21}c_{21}\cos(\alpha-\alpha_{21}-\gamma+\gamma_{21})+$
$\displaystyle+$ $\displaystyle
2ada_{21}d_{21}\cos(\alpha-\alpha_{21}-\delta+\delta_{21})+$ $\displaystyle+$
$\displaystyle 2bcb_{21}c_{21}\cos(\beta-\beta_{21}-\gamma+\gamma_{21})+$
$\displaystyle+$ $\displaystyle
2bdb_{21}d_{21}\cos(\beta-\beta_{21}-\delta+\delta_{21})+$ $\displaystyle+$
$\displaystyle 2cdc_{21}d_{21}\cos(\gamma-\gamma_{21}-\delta+\delta_{21})$
$\displaystyle\mu(X_{2}Y_{2})=|\langle p_{22}|p\rangle|^{2}$ $\displaystyle=$
$\displaystyle
a^{2}a_{22}^{2}+b^{2}b_{22}^{2}+c^{2}c_{22}^{2}+d^{2}d_{22}^{2}+$ (29)
$\displaystyle+$ $\displaystyle
2aba_{22}b_{22}\cos(\alpha-\alpha_{22}-\beta+\beta_{22})+$ $\displaystyle+$
$\displaystyle 2aca_{22}c_{22}\cos(\alpha-\alpha_{22}-\gamma+\gamma_{22})+$
$\displaystyle+$ $\displaystyle
2ada_{22}d_{22}\cos(\alpha-\alpha_{22}-\delta+\delta_{22})+$ $\displaystyle+$
$\displaystyle 2bcb_{22}c_{22}\cos(\beta-\beta_{22}-\gamma+\gamma_{22})+$
$\displaystyle+$ $\displaystyle
2bdb_{22}d_{22}\cos(\beta-\beta_{22}-\delta+\delta_{22})+$ $\displaystyle+$
$\displaystyle 2cdc_{22}d_{22}\cos(\gamma-\gamma_{22}-\delta+\delta_{22})$

Then, let us represent represent the initial state $p$ of the composite
conceptual entity by the unit vector

$|p\rangle=\frac{1}{\sqrt{2}}(0,1,-1,0)$ (30)

The vector in Equation (30) represents a maximally entangled state and
corresponds to the ‘singlet spin state’ that is used in typical Bell-type
tests in quantum physics (see Section 1). There are different reasons for this
choice. Firstly, our aim is to incorporate all possible entanglement of the
state-measurement situation in the pre-measurement state, so that the ensuing
measurements are as close as possible to product measurements. We are however
aware that some measurements, if not all, are entangled, due to the violation
of the marginal law (see Section 4). Secondly, it is known that the singlet
spin state has specific symmetry properties, namely, it is always represented
by a unit vector of the form in Equation (30) independently of the ON basis in
which the unit vector is expressed. This would intuitively correspond to the
fact both The Animal Acts and The Animal eats the Food express more abstract
concepts than the corresponding outcomes.

Thus, for every $X=A,A^{\prime}$, $Y=B,B^{\prime}$, conditions (i)–(iii)
identify 20 equations which should be satisfied by the 32 real variables
$a_{ij},b_{ij},c_{ij},d_{ij},\alpha_{ij},\beta_{ij},\gamma_{ij},\delta_{ij}$,
$i,j=1,2$. To simplify the calculation, we set, for every $i,j=1,2$,
$\alpha_{ij}=\beta_{ij}=\gamma_{ij}=\delta_{ij}=\theta_{ij}$, where
$\theta_{ij}\in\Re$. Thus, each unit vector in Equations (12)–(15) takes the
form $|p_{ij}\rangle=e^{i\theta_{ij}}(a_{ij},b_{ij},c_{ij},d_{ij})$,
$i,j=1,2$. This reduces the total number of unknown variables to 20.

We are now ready to represent the empirical data in Section 3 in Hilbert
space. We start by The Animal Acts test.

The eigenstates of the measurement $e_{AB}$ are represented by the unit
vectors

$\displaystyle|p_{A_{1}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i174.91^{\circ}}(0.97,-0.10,0.21,0)$ (31)
$\displaystyle|p_{A_{1}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i0.38^{\circ}}(0,0.88,0.38,0.29)$ (32)
$\displaystyle|p_{A_{2}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i32.54^{\circ}}(0.23,0.35,-0.90,0.12)$ (33)
$\displaystyle|p_{A_{2}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i98.90^{\circ}}(0.03,0.32,0,-0.95)$ (34)

By applying the entanglement condition in Equation (11), we can verify that
all unit vectors are entangled, hence $e_{AB}$ is an entangled measurement.
However, one observes that the condition in Equation (11) shows a relatively
larger deviation from zero in the unit vector $|p_{A_{2}B_{1}}\rangle$. We can
then say that the eigenstate $p_{A_{2}B_{1}}$, corresponding to The Bear
Growls, is a ‘relatively more entangled state’.

The eigenstates of the measurement $e_{AB^{\prime}}$ are represented by the
unit vectors

$\displaystyle|p_{A_{1}B^{\prime}_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i0.07^{\circ}}(0.14,0.24,-0.96,0.08)$ (35)
$\displaystyle|p_{A_{1}B^{\prime}_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i18.24^{\circ}}(0.01,0.31,0,-0.95)$ (36)
$\displaystyle|p_{A_{2}B^{\prime}_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i64.68^{\circ}}(-0.02,0.92,0.25,0.30)$ (37)
$\displaystyle|p_{A_{2}B^{\prime}_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i132.68^{\circ}}(0.99,-0.02,0.14,0)$ (38)

Also in this case, all unit vectors are entangled, hence $e_{AB^{\prime}}$ is
an entangled measurement. The entanglement condition in Equation (11) shows a
relatively larger deviation from zero in the unit vector
$|p_{A_{1}B^{\prime}_{1}}\rangle$. We can then say that the eigenstate
$p_{A_{1}B^{\prime}_{1}}$, corresponding to The Horse Snorts, is a ‘relatively
more entangled state’.

The eigenstates of the measurement $e_{A^{\prime}B}$ are represented by the
unit vectors

$\displaystyle|p_{A^{\prime}_{1}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i45.48^{\circ}}(0.25,0.37,-0.88,0.17)$ (39)
$\displaystyle|p_{A^{\prime}_{1}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i2.21^{\circ}}(0.01,0.84,0.42,0.35)$ (40)
$\displaystyle|p_{A^{\prime}_{2}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i308.44^{\circ}}(0.97,-0.13,0.22,0)$ (41)
$\displaystyle|p_{A^{\prime}_{2}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i163.38^{\circ}}(0.05,0.39,0,-0.92)$ (42)

All unit vectors are entangled, hence $e_{A^{\prime}B}$ is an entangled
measurement. The entanglement condition in Equation (11) shows a relatively
larger deviation from zero in the unit vector
$|p_{A^{\prime}_{1}B_{1}}\rangle$. We can then say that the eigenstate
$p_{A^{\prime}_{1}B_{1}}$, corresponding to The Tiger Growls, is a ‘relatively
more entangled state’.

Finally, the eigenstates of the measurement $e_{A^{\prime}B^{\prime}}$ are
represented by the unit vectors

$\displaystyle|p_{A^{\prime}_{1}B^{\prime}_{1}}\rangle$ $\displaystyle=$
$\displaystyle e^{i353.32^{\circ}}(0.96,-0.18,0.23,0)$ (43)
$\displaystyle|p_{A^{\prime}_{1}B^{\prime}_{2}}\rangle$ $\displaystyle=$
$\displaystyle e^{i143.07^{\circ}}(0.07,0.35,0,-0.93)$ (44)
$\displaystyle|p_{A^{\prime}_{2}B^{\prime}_{1}}\rangle$ $\displaystyle=$
$\displaystyle e^{i0.58^{\circ}}(0.01,0.78,0.55,0.30)$ (45)
$\displaystyle|p_{A^{\prime}_{2}B^{\prime}_{2}}\rangle$ $\displaystyle=$
$\displaystyle e^{i41.95^{\circ}}(0.29,0.49,-0.80,0.20)$ (46)

All unit vectors are entangled, hence $e_{A^{\prime}B^{\prime}}$ is an
entangled measurement. The entanglement condition in Equation (11) shows a
relatively larger deviation from zero in the unit vector
$|p_{A^{\prime}_{2}B^{\prime}_{2}}\rangle$. We can then say that the
eigenstate $p_{A^{\prime}_{2}B^{\prime}_{2}}$, corresponding to The Cat Meows
, is a ‘relatively more entangled state’.

Let us now come to the representation of empirical data of The Animal eats the
Food test. The eigenstates of the measurements $e_{AB}$, $e_{AB^{\prime}}$,
$e_{A^{\prime}B}$ and $e_{A^{\prime}B^{\prime}}$ are respectively represented
by the 4 unit vectors

$\displaystyle|p_{A_{1}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i156.99^{\circ}}(0.91,-0.06,0.41,0)$ (47)
$\displaystyle|p_{A_{1}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i0.38^{\circ}}(0.01,0.97,0.13,0.22)$ (48)
$\displaystyle|p_{A_{2}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i117.49^{\circ}}(0.41,0.11,-0.90,0.03)$ (49)
$\displaystyle|p_{A_{2}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i160.39^{\circ}}(0.02,0.22,0,-0.97)$ (50)
$\displaystyle|p_{A_{1}B^{\prime}_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i45.38^{\circ}}(0.22,0.49,-0.84,0.12)$ (51)
$\displaystyle|p_{A_{1}B^{\prime}_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i2.20^{\circ}}(0.01,0.83,0.52,0.19)$ (52)
$\displaystyle|p_{A_{2}B^{\prime}_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i308.12^{\circ}}(0.97,-0.13,0.18,0)$ (53)
$\displaystyle|p_{A_{2}B^{\prime}_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i158.68^{\circ}}(0.03,0.22,0,-0.97)$ (54)
$\displaystyle|p_{A^{\prime}_{1}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i45.43^{\circ}}(0,0.47,-0.87,0.17)$ (55)
$\displaystyle|p_{A^{\prime}_{1}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i2.21^{\circ}}(0.02,0.81,0.50,0.31)$ (56)
$\displaystyle|p_{A^{\prime}_{2}B_{1}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i337.01^{\circ}}(0.99,-0.01,-0.01,0)$ (57)
$\displaystyle|p_{A^{\prime}_{2}B_{2}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i163.38^{\circ}}(0,0.35,0,-0.94)$ (58)
$\displaystyle|p_{A^{\prime}_{1}B^{\prime}_{1}}\rangle$ $\displaystyle=$
$\displaystyle e^{i353.52^{\circ}}(0.97,-0.13,0.19,0)$ (59)
$\displaystyle|p_{A^{\prime}_{1}B^{\prime}_{2}}\rangle$ $\displaystyle=$
$\displaystyle e^{i143.07^{\circ}}(0.02,0.16,0,-0.99)$ (60)
$\displaystyle|p_{A^{\prime}_{2}B^{\prime}_{1}}\rangle$ $\displaystyle=$
$\displaystyle e^{i0.58^{\circ}}(0.02,0.85,0.50,0.14)$ (61)
$\displaystyle|p_{A^{\prime}_{2}B^{\prime}_{2}}\rangle$ $\displaystyle=$
$\displaystyle e^{i42.01^{\circ}}(0.22,0.48,-0.84,0.08)$ (62)

Also in this case, referring to Equation (11), one can show that all unit
vectors in Equations (47)–(62) are entangled, hence ‘all measurements
$e_{AB}$, $e_{AB^{\prime}}$, $e_{A^{\prime}B}$ and $e_{A^{\prime}B^{\prime}}$
are entangled measurements’. In addition, as in The Animal Acts test, in each
measurement, one eigenstate is represented by a unit vectors which exhibits a
larger deviation in the entanglement condition in Equation (11). These
‘relatively more entangled states’ are $p_{A_{2}B_{1}}$, corresponding to The
Cow eats the Grass in $e_{AB}$, $p_{A_{1}B^{\prime}_{1}}$, corresponding to
The Cat eats the Fish in $e_{AB^{\prime}}$, $p_{A^{\prime}_{1}B_{1}}$,
corresponding to The Horse eats the Grass in $e_{A^{\prime}B}$, and
$p_{A^{\prime}_{2}B^{\prime}_{2}}$, corresponding to The Squirrel eats the
Nuts in $e_{A^{\prime}B^{\prime}}$.

We have thus completed the quantum mathematical representation of the data on
The Animal Acts and The Animal eats the Food tests. This representation also
suggests additional considerations, as follows.

We firstly observe that all measurements are entangled in the quantum
representation in both The Animal Acts and The Animal eats the Food
situations. This result is due to the violation of the marginal law of
probability in both tests which forbids concentrating all the entanglement of
the state-measurement situation into the state, as we have seen in Section 4.
Moreover, in each measurement, all eigenstates are entangled. This result
confirms with our suggestion in Section 4 that ‘quantum entanglement
theoretically captures the meaning connections between the individual concepts
that form the composite conceptual entity, and that these meaning connections
are not predetermined, but are created when the test is performed and each
respondent interacts with the composite entity’.

Secondly, in both situations, there are some outcome eigenstates which exhibit
a relatively higher degree of entanglement than others, which can exactly be
explained with the fact that entanglement captures meaning connections, hence
higher degrees of entanglement correspond to higher meaning connections. For
example, in The Animal Acts situation, the eigenstate $p_{A_{2}B_{1}}$ of the
measurement $e_{AB}$, corresponding to The Bear Growls, is relatively more
entangled than the other eigenstates of $e_{AB}$, which can be naturally
explained by the fact that the individual items Bear and Growls, which are
concepts themselves, are relatively more connected by meaning. As a matter of
fact, if we look at empirical probabilities (see Table 1, Section 3), we note
that the outcome $A_{2}B_{1}$, corresponding to The Bear Growls, scores a high
probability to be judged as a good example of the conceptual combination The
Animal Acts. Vice versa, items like Bear Meows and Horse Growls score a low
probability and are less connected by meaning. And, indeed, the corresponding
eigenstates are relatively less entangled. Analogously, in The Animal eats the
Food situation, the eigenstate $p_{A^{\prime}_{2}B^{\prime}_{2}}$ of the
measurement $e_{A^{\prime}B^{\prime}}$, corresponding to The Squirrel eats the
Nuts, is relatively more entangled than the other outcome eigenstates of
$e_{A^{\prime}B^{\prime}}$, which can be again explained by the fact that the
individual items Squirrel and Nuts, which are concepts themselves, are
relatively more connected by meaning. And, indeed, if we look at empirical
probabilities (see Table 2, Section 3), we note that the outcome
$A^{\prime}_{2}B^{\prime}_{2}$, corresponding to The Squirrel eats the Nuts,
scores a high probability to be judged as a good example of the conceptual
combination The Animal eats the Food. On the contrary, items like The Cow eats
the Meat, The Cat eats the Nuts and The Squirrel eats the Grass score a low
probability and are less meaning-connected. And, indeed, the corresponding
eigenstates are relatively less entangled.

Thirdly, we observe that the quantum representation of the cognitive tests
appears to be more complex and less symmetric than the quantum representation
of the information retrieval tests on the web in [46] (see [47]), where all
measurements were entangled but each spectral measure contained two product
eigenstates and two entangled eigenstates. The reason is that many judgement
probabilities were zero in [46], due to the fact that human minds are able to
create additional meaning connections between concepts than corpuses of
documents.

Fourthly, we observe that the violation of Cirel’son bound in The Animal eats
the Food situation can still be explained in terms of quantum entanglement, in
contrast to widespread beliefs. Hence, there is quantum entanglement also
beyond Cirel’son bound, but this type of entanglement involves both states and
measurements. This result is important and we will devote the next section to
deepen it.

##  6 Quantum entanglement beyond Cirel’son bound

The empirical results in the cognitive test on The Animal eats the Food
presented in Section 3 show a significant violation of the Cirel’son bound
and, as such, they confirm the results in the document retrieval tests on the
web in [32] and [46]. As we have noticed throughout the paper, these results
contrast with widespread beliefs in quantum physics. It is thus worth to
carefully explain their meaning and implications.

The original question behind the determination of the Cirel’son bound was
whether there is a fundamental limit to quantum nonlocality, that is, whether
the correlations exhibited by two far away physical entities should satisfy
any condition in order to represent them in the mathematical formalism of
quantum theory – this problem is also connected with the old problem of
exploiting quantum nonlocality to send a faster than light signal between far
away entities (see, e.g., [9]). One then considers a typical Bell-type
situation, as the one presented in Section 1, performs a set of 4 measurements
on a composite entity, made up of two far away individual entities, by
separately performing each time one measurement on one individual entity and
one measurement on the other, and deduces that the CHSH factor in Equation (1)
is bound by the value $\Delta_{QMC}=2\sqrt{2}\approx 2.83$ in quantum theory
[6, 7]. Hence, the Cirel’son bound obtained in this way is usually considered,
especially by quantum physicists, as the limit outside which the correlations
exhibited by any two entities cannot be modelled within the Hilbert space
formalism of quantum theory [6, 7].

On the other side, we have proved, in this and in other papers that it is
possible to represented in Hilbert space empirical data collected in Bell-type
tests which violate the CHSH inequality by an amount which also exceeds
Cirel’son bound (see, e.g., [47, 48]). This means that the statistical
correlations exhibited in these Bell-type tests, which are anyway non-
classical and non-signalling, can be represented within quantum theory.

As we have seen in Sections 4 and 5, the fundamental element which makes it
possible the above mentioned quantum mathematical representation in Hilbert
space is the suggestion that the measurements performed on the composite
entity have outcomes that are products of outcomes obtained in measurements
separately performed on individual entities, but their eigenstates are
generally entangled, that is, they are entangled measurements, hence they
cannot generally be decomposed into product measurements. Cirel’son, and the
scholars who investigated this bound after him, did not envisage such
alternative, hence they implicitly assumed only product measurements in the
derivation of the Cirel’son bound. If one also allows the possibility of using
entangled measurements, then one can in principle violate the CHSH inequality
in Equation (1) by any amount within its mathematical limit, i.e. between $-4$
and $+4$ (see also [48]).

The result above is relevant, in our opinion, for various reasons. Firstly, it
follows from a careful investigation of the identification problem in which
individual entities are identified through measurements that are performed on
composite entities. Secondly, it shows that a violation of Bell inequalities
which also exceeds the Cirel’son bound can still be explained in terms of
quantum entanglement. More important, this ‘quantum entanglement beyond
Cirel’son bound’ is deeper and stronger than the entanglement that is
generally believed to be responsible of the violation of Bell inequalities in
quantum physics., i.e. ‘entanglement in the state’. Indeed, a violation of the
Cirel’son bound, in addition to a violation of Bell inequalities, can be
explained by assuming the presence of entanglement in both the state and the
measurements, i.e. ‘state-measurement entanglement’. Thirdly, together with
the violation of the marginal law, the violation of the Cirel’son bound may
have deep implications on the foundations of quantum physics, as it could shed
new light on the problem of faster than light communication and signalling and
could explain apparent deviations from quantum predictions in some Bell-type
tests in physics. These deviations have been studied putting forward the
hypothesis that they would be due to measurement errors [52, 53], however in
[54] it is shown that they might well be of a fundamental nature and not be
due to measurement errors. If the analysis put forward in [54] turns out to be
true a whole new perspective is needed also on the physical violation of the
Bell inequalities where experimentally results a violation of the marginal law
[10, 13], indicating that the type of signalling believed to be possible as a
consequence of the violation of the marginal law is ‘not’ the one imagined
taking place and hence would not, due to its different nature, be able to be
faster than light even if the coincidence events are space-like separated
events [54].

## Acknowledgements

This work was supported by QUARTZ (Quantum Information Access and Retrieval
Theory), the Marie Skłodowska-Curie Innovative Training Network 721321 of the
European Union’s Horizon 2020 research and innovation programme.

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