Entanglement in Cognition violating Bell Inequalities Beyond Cirel'son's Bound
Diederik Aerts, Jonito Aerts Arguëlles, Lester Beltran, Suzette Geriente, Sandro Sozzo
aa r X i v : . [ q - b i o . N C ] F e b Entanglement in Cognition violating Bell InequalitiesBeyond Cirel’son’s Bound
Diederik Aerts, Jonito Aerts Argu¨elles, Lester BeltranSuzette Geriente ∗ and Sandro Sozzo † Abstract
We present the results of two tests where a sample of human participants were asked to make judge-ments about the conceptual combinations
The Animal Acts and
The Animal eats the Food . Both testssignificantly 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 individ-ual concepts that are combined. We then apply a quantum-theoretic framework which we developed forany Bell-type situation and represent empirical data in complex Hilbert space. We show that the ob-served violations of the CHSH inequality can be explained as a consequence of a strong form of ‘quantumentanglement’ between the component conceptual entities in which both the state and measurementsare entangled. We finally observe that a quantum model in Hilbert space can be elaborated in theseBell-type situations even when the CHSH violation exceeds the known ‘Cirel’son bound’, in contrast toa widespread belief. These findings confirm and strengthen the results we recently obtained in a varietyof cognitive tests and document and image retrieval operations on the same conceptual combinations.
Keywords : Cognition; Bell-type tests; CHSH inequality; quantum entanglement; quantum structures.
In physics, the violation of the so-called ‘Bell inequalities’ is generally maintained to prove that quantumentities exhibit specific aspects, as ‘contextuality’, ‘entanglement’ and ‘nonseparability’, which cannotbe explained in terms of the mathematical structures that are typically used in classical physics. Thiscontextuality also persists for far away quantum entities, an aspect known as ‘nonlocality’ after Bell’sseminal paper [1, 2]. The consistent violation of Bell inequalities in physics tests performed on a varietyof quantum entities, in addition to confirming the predictions of quantum theory, in particular revealsthat the statistics of repeated experiments shows connections between quantum entities that cannot berepresented within a classical probabilistic model satisfying the axioms of Kolmogorov (‘Kolmogorovianprobability’) (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 considersa composite physical entity S , prepared in an initial state p and such that two individual entities S and S that have interacted in the past but are now far away (‘space-like separation’) can be recognizedas parts of S . Then, 4 coincidence experiments AB , AB ′ , A ′ B and A ′ B ′ are performed on S which ∗ Center Leo Apostel for Interdisciplinary Studies, Free University of Brussels (VUB), Krijgskundestraat 33, 1160 Brussels,Belgium; email addresses: [email protected],[email protected],[email protected] † School of Business and Centre IQSCS, University of Leicester, University Road, LE17RH Leicester, United Kingdom;email address: [email protected] e A , with outcomes A and A , and e A ′ with outcomes A ′ and A ′ ,on entity S , and measurements e B , with outcomes B and B , and e B ′ with outcomes B ′ and B ′ , onentity S . If the measurement outcomes can be only − AB , AB ′ , A ′ B and A ′ B ′ become the correlation functions E ( A, B ), E ( A, B ′ ), E ( A ′ , B ) and E ( A ′ , B ′ ), respectively, and aspecific inequality, which is the ‘Clauser-Horne-Shimony-Holt version of Bell inequalities’ (briefly, ‘CHSHinequality’) − ≤ E ( A ′ , B ′ ) + E ( A ′ , B ) + E ( A, B ′ ) − E ( A, B ) ≤ CHSH = E ( A ′ , B ′ ) + E ( A ′ , B ) + E ( A, B ′ ) − E ( A, B ) is bound in quantum theory by the numerical value ∆
QMC = 2 √ ≈ .
83, known asthe ‘Cirel’son bound’ [6, 7]. It is important to observe, at this stage, that the existence of such a ‘quantumbound’ is not trivial because, from a mathematical point of view, the CHSH factor can take any valuebetween − e XY , corresponding to coincidence experiments XY , X = A, A ′ , Y = B, B ′ , on the composite entity thatare represented by product self-adjoint operators whenever the states of the composite entity S arerepresented by the unit vectors of the tensor product Hilbert space of two Hilbert spaces of which theirunit vectors represent the states of the possible to be recognized entities S and S . This remark will playan important role in what follows.In the last two decades, several theoretical studies, mainly inspired by quantum computation andquantum 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 Aspectand his collaborators [10, 11]. All empirical tests that have been performed so far strongly confirm thepredictions 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 compositequantum entity, give rise to empirically observable phenomena, so much that, in particular, entanglementis nowadays considered as one of the fingerprints of quantum theory.Growing theoretical and empirical research reveals that several quantum structures, such as ‘contex-tuality’, ‘entanglement’, ‘indistinguishability’, ‘interference’ and ‘superposition’, also manifest themselvesin other domains than micro-physics, which include, in particular, cognition (e.g., human probability andsimilarity judgements, decision-making and visual perception), socio-economic domains (economics andfinance) 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 havebeen obtained in the identification of quantum structures and, in particular, quantum entanglement, inthe combination of natural concepts and applied computer science domains, as information retrieval andnatural language processing (see, e.g., [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] andreferences therein). In this regard, our research team has produced both theoretical and empirical researchon 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],2ocument 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 combinationof the individual conceptual entities
Animal and
Acts . We performed an additional document retrievaltest on the web in which we considered the composite conceptual entity
The Animal eats the Food as acombination of the individual conceptual entities
Animal and
Food [32]. In all these tests, we observed asystematic violation of the CHSH inequality, accompanied by a systematic violation of a property calledthe ‘marginal law’.In particular, the violation of the marginal law, which is believed not to occur in principle in Bell-typetests in quantum physics – we come back to this later, led us to study in detail a theoretical problem which isusually overlooked in physics, the ‘identification problem’, i.e. the problem of recognising individual entitiesof a composite entity by performing on the latter measurements that resemble the typical coincidenceexperiments 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 explicitlyapplies the prescription of the standard formalism of quantum theory that the joint entity needs to bemodelled in a complex Hilbert space of which the dimension is given by the number of outcomes of thedefining measurements, which is hence C . Only then as a secondary step, with respect to the attemptto ‘recognize’ entities in the joint entity, one considers possible isomorphisms with another Hilbert space,namely, the one built as the tensor product C ⊗ C of two-dimensional complex Hilbert spaces which needto describe the two possible entities, since the defining measurements to recognize these entities have twooutcomes. Like we analyzed in detail in [41, 42], there is no unique isomorphism between C and C ⊗ C and this is the reason that from a mathematical point of view there are different ways to account forentanglement being present within the joint entity with respect to the entities to be recognized individualentities. Essentially, entanglement manifests on the level of the probabilities of a joint measurementsnot being able to be written as products of probabilities on the level of the component measurements ofthis joint measurement and hence it is a property of the relation between the joint measurements andtheir components. Only when an extra symmetry is present connecting different joint measurements theentanglement of these different joint measurements can be captured in a state of the joint entity. Thissymmetry can be detected by verifying whether the marginal law corresponding to the considered jointmeasurements is satisfied, in that case, the entanglement of these joint measurements can together becaptured in a state of the joint entity. If this is true for all joint measurements one can prove that thereis only one isomorphism connecting C with C ⊗ C and in that case C can be substituted by C with C ⊗ C and the joint entity can be directly modeled in C ⊗ C . With respect to ‘entanglement’, aproperty of the structure of the probabilities of a joint measurement with respect to the probabilities ofthe component measurements, this situation is exceptional and definitely not the general one. In [42] wework out in detail several experimental examples and how how entanglement can generally only be locatedin the joint measurements and only exceptionally in the state of the joint entity. It is widely acceptedthat in the typically considered quantum physics situations where entanglement is being measured andstudied this exceptional symmetry is always present and hence apriori the tensor product Hilbert spaceis taken to be the Hilbert space to model the joint entity, although already the first experiments of AlainAspect [10] showed violations of the marginal law, confirmed manifestly by later experiments [13]. For thecognition experiments that we consider in the present article where joint measurements are performed onjoint concepts The Animal Acts and
The Animal eats the Food the symmetry is not fulfilled and henceno unique isomorphism exists between C and C ⊗ C . The new quantum-theoretic perspective enabledmodelling of both cognitive and web tests on The Animal Acts and introduced a novel mathematicalingredient 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 andinformation tasks (see, e.g., [9]). We show in the present paper that ‘entanglement measurement also allow3o identify the presence of quantum entanglement also in Bell-type tests where the violation of the Cirel’sonbound occurs in addition to the violation of the CHSH inequality’. More specifically, we present the resultsof two cognitive tests, one on
The Animal Acts and the other on
The Animal eats the Food situations, thatwe have recently performed on a sample of 81 participants. In both tests, a significant violation of theCHSH has been observed. Furthermore, while
The Animal Acts test has revealed a result very close tothe 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]. Wesuccessfully apply and refine the quantum-theoretic framework to
The Animal Acts and
The Animal eatsthe Food situations and get a confirmation that a ‘strong form of quantum entanglement’ exists between therespective individual concepts which involves both states and measurements’. Moreover, we get a furtherconfirmation that ‘such a form of entanglement captures the deeply non-classical meaning connectionsexisting between those individual concepts’. Finally, we prove that ‘quantum entanglement also existsbeyond 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 modelsituations 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 obtainedin 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 humanparticipants about the conceptual combinations
The Animal Acts and
The Animal eats the Food . In Section4, we explicate, refine and particularise to
The Animal Acts and
The Animal eats the Food situations thegeneral 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 datain Section 3. Finally, in Section 6, we explain how and why a violation of the CHSH inequality whichalso exceed the Cirel’son bound may indicate the presence of a stronger form of quantum entanglementinvolving both states and measurements.
We review in this section the results that we have obtained in the last decade on the identification ofentanglement in the combination of natural concepts and related applied domains, as information retrievaland natural language processing. For a detailed description of the tests and a complete analysis of theobtained 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 Animaleats the Food , which we regarded as a combination of the individual concepts
Animal and
Food [32]. Weconsidered 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.
TheCow eats the Grass , The Cat eats the Meat , The Cat eats the Fish , The Squirrel eats the Nuts , andso on, which were considered as items of the combination
The Animal eats the Food . Next, we splitthe 16 combinations obtained in this way into 4 groups of 4 combinations each in order to reproducethe 4 coincidence experiments described in Section 1. For example, coincidence experiment AB had 4outcomes, The Cat eats the Grass , The Cat eats the Meat , The Cow eats the Grass and
The Cow eats theMeat ; the other experiments were constructed in an analogous way. We used the World Wide Web as aconceptual 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 ofco-occurrences which we considered, in the large number limit, as joint probabilities and inserted them into4he correlation functions in Equation (1). With our surprise, we found ∆
CHSH = 2 .
86, hence a significantviolation of the CHSH inequality. We put forward the presence of some type of entanglement between theconcepts
Animal and
Food was responsible of the observed deviation from the CHSH inequality. At thattime, 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 wedecided to deepen the investigation on entanglement in conceptual combinations and performed a cognitivetest in which the direct responses of a sample of participants were collected [38]. In this case, we analysed theconceptual 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 considereddifferent 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 thecombination
The Animal Acts . Then, we split the 16 combinations obtained in this way into 4 groupsof 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 introductorytext, a sample of 81 respondents had to fill in a questionnaire where, in each coincidence experiment, theyhad 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, asjoint probabilities and inserted into the correlation functions in Equation (1). Also in this case, we found asignificant violation the CHSH inequality, ∆
CHSH = 2 .
42, which we again interpreted as the consequenceof a form of entanglement in the combination
The Animal Acts , due to the connections of meaning betweenthe component concepts
Animal and
Acts .The numerical value 2 .
42 obtained in the cognitive test on
The Animal Acts resembled the values foundin Bell-type tests in quantum physics (see Section 1). To make the analogy with physics more convincing, wethen 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 theinitial 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 discoveredthat the main reason of this impossibility was that, if one wants to model
The Animal Acts situationstarting 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 wassystematically violated in the cognitive test. This difference between physical and cognitive realms led usto 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 compositeentities, e.g., conceptual combinations. The investigation led us to develop a novel quantum-theoreticframework for any Bell-type situation, which allowed us to recognise systematic analogies and differencesbetween violations of Bell inequalities in physics and cognition [48]. But, the quantum-theoretic frameworkalso introduced a new fundamental element in the modelling of such situations, namely, the empiricallyjustified use of ‘entangled measurements’ in addition to entangled states. Entangled measurements wereindeed 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 quantumstructures in information retrieval and natural language processing, where representation of meaning en-tities, like concepts and conceptual combinations, play a crucial role (see, e.g., [44, 27]). In particular, we A closer analogy between Bell-type situations in physical and cognitive realms was identified in a cognitive test weperformed on the conceptual combination
Two Different Wind Directions , where the marginal law was not violated and wecould represent data in Hilbert space using entangled states and product measurements [50, 51].
The Animal Acts situation in a document retrieval test in which we collected data onco-occurrence of words, as we had done for
The Animal eats the Food situation, but using known corpusesof documents instead of web search engines – corpuses of documents provide more reliable word counts thansearch 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 inSection 1, finding 3.41, 3.00 and 3.33, respectively in the factor appearing in Equation (1). These empiricalfindings were statistically significant and, more important, they were completely unexpected. Indeed, notonly we had obtained consistent violations of the CHSH inequality across all corpuses of documents, but thenumerical value of the violation systematically exceeded the Cirel’son bound. According to a widespreadbelief, a violation of the CHSH inequality which also violates Cirel’son bound cannot be modelled in Hilbertspace 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 productmeasurements [6, 7]. If one allows entangled measurements, then it is in principle possible to model inHilbert space any violation of the CHSH inequality whose numerical value lies in the mathematically per-mitted interval from − The Animal Acts [29]. Also in this case, we obtained the value∆
CHSH = 2 .
41 for the CHSH factor in Equation (1), which again indicated the presence of entanglementin visual perception. The corresponding quantum-theoretic modelling enabled faithful representation ofempirical 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 conceptualcombinations
The Animal Acts and
The Animal eats the Food with the aim of ‘identifying the presence ofquantum entanglement beyond the limits imposed by Cirel’son bound’. We will dedicate the next sectionsto this purpose.
We report in this section the details of the two cognitive tests that we have performed on the conceptualcombinations
The Animal Acts and
The Animal eats the Food . As we will see, their results substantiallyconfirm 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 AnimalActs as a combination of the individual concepts
Animal and
Acts , where “acts” refers to the sound, ornoise, 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 arenow ready to illustrate the test.A sample of 81 individuals were presented in a ‘within subjects design’ a questionnaire which contained4 coincidence experiments AB , AB ′ , A ′ B and A ′ B ′ whose setting was similar to the typical setting of aBell-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 involvedin the judgement test was provided. Then, in each coincidence experiment participants were asked tochoose which item in a list of 4 items they judged as a good example of the conceptual combination TheAnimal Acts .In coincidence experiment AB , participants had to choose among the 4 items:( A B ) The Horse Growls A B ) The Bear Whinnies ( A B ) The Horse Whinnies ( A B ) The Bear Growls
If the response was ( A B ) or ( A B ), then experiment AB was attributed outcome +1; if the responsewas ( A B ) or ( A B ), then experiment AB was attributed outcome − AB ′ , participants had to choose among the 4 items:( A B ′ ) The Horse Snorts ( A B ′ ) The Horse Meows ( A B ′ ) The Bear Snorts ( A B ′ ) The Bear Meows
If the response was ( A B ′ ) or ( A B ′ ), then experiment AB ′ was attributed outcome +1; if the responsewas ( A B ′ ) or ( A B ′ ), then experiment AB ′ was attributed outcome − A ′ B , participants had to choose among the 4 items:( A ′ B ) The Tiger Growls ( A ′ B ) The Tiger Whinnies ( A ′ B ) The Cat Growls ( A ′ B ) The Cat Whinnies
If the response was ( A ′ B ) or ( A ′ B ), then experiment A ′ B was attributed outcome +1; if the responsewas ( A ′ B ) or ( A ′ B ), then experiment A ′ B was attributed outcome − A ′ B ′ , participants had to choose among the 4 items:( A ′ B ′ ) The Tiger Snorts ( A ′ B ′ ) The Tiger Meows ( A ′ B ′ ) The Cat Snorts ( A ′ B ′ ) The Cat Meows
If the response was ( A ′ B ′ ) or ( A ′ B ′ ), then experiment A ′ B ′ was attributed outcome +1; if the responsewas ( A ′ B ′ ) or ( A ′ B ′ ), then experiment A ′ B ′ was attributed outcome − AB , AB ′ , A ′ B and A ′ B ′ , we collected the relative frequencies of theobtained responses which we considered, in the large number limit, as the probability µ ( A i B j ), µ ( A i B ′ j ), µ ( A ′ i B j ) and µ ( A ′ i B j ) that the outcome A i B j , A i B ′ j , A ′ i B j and A ′ i B ′ j , i, j = 1 ,
2, respectively, is obtained inthe corresponding experiment. Table 3 reports the judgement probabilities computed in this way. Referringto these probabilities, we can then calculate the expectation values, or correlation functions, of coincidenceexperiments AB , AB ′ , A ′ B and A ′ B ′ , as follows: E ( AB ) = µ ( A B ) − µ ( A B ) − µ ( A B ) + µ ( A B ) = − . E ( AB ′ ) = µ ( A B ′ ) − µ ( A B ′ ) − µ ( A B ′ ) + µ ( A B ′ ) = 0 . E ( A ′ B ) = µ ( A ′ B ) − µ ( A ′ B ) − µ ( A ′ B ) + µ ( A ′ B ) = 0 . E ( A ′ B ′ ) = µ ( A ′ B ′ ) − µ ( A ′ B ′ ) − µ ( A ′ B ′ ) + µ ( A ′ B ′ ) = 0 . CHSH = E ( A ′ , B ′ ) + E ( A ′ , B ) + E ( A, B ′ ) − E ( A, B ) = 2 . . ≈ .
79 exceeds the classical limit imposed by the CHSH inequality and isslightly below the Cirel’son bound. However, while the deviation from the value 2 is statistically significant(p-value 1 . ∗ − ), the deviation from the value 2 √ . The Animal Acts data show a non definite behaviour with respect to the Cirel’son bound. This empiricalpattern confirms and strengthens the results obtained in [38] where a violation of the CHSH inequality wasobserved. This non-classical behaviour admits the presence of entanglement between the concepts
Animal AB Horse Growls Horse Whinnies Bear Growls Bear Whinnies
Probability µ ( A B ) = 0 . µ ( A B ) = 0 . µ ( A B ) = 0 . µ ( A B ) = 0 . AB ′ Horse Snorts Horse Meows Bear Snorts Bear Meows
Probability µ ( A B ′ ) = 0 . µ ( A B ′ ) = 0 . µ ( A B ′ ) = 0 . µ ( A B ′ ) = 0 . A ′ B Tiger Growls Tiger Whinnies Cat Growls Cat Whinnies
Probability µ ( A ′ B ) = 0 . µ ( A ′ B ) = 0 . µ ( A ′ B ) = 0 . µ ( A ′ B ) = 0 . A ′ B ′ Tiger Snorts Tiger Meows Cat Snorts Cat Meows
Probability µ ( A ′ B ′ ) = 0 . µ ( A ′ B ′ ) = 0 . µ ( A ′ B ′ ) = 0 . µ ( A ′ B ′ ) = 0 . Table 1: The data collected in coincidence experiments on entanglement in the conceptual combination
The Animal Acts . and Acts as a natural explanation, because
Animal and
Acts are connected by meaning and this connectiongives 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 thenconsider 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 containedagain the 4 coincidence experiments AB , AB ′ , A ′ B and A ′ B ′ that are typical of a Bell-type test. Moreprecisely, participants, after reading an introductory text on concepts and their combinations as in the firsttest, were asked to choose in each coincidence experiment one item in a list of 4 items, namely, the itemthat 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 B ) The Cat eats the Grass ( A B ) The Cat eats the Meat ( A B ) The Cow eats the Grass ( A B ) The Cow eats the Meat
If the response was ( A B ) or ( A B ), then experiment AB was attributed outcome +1; if the responsewas ( A B ) or ( A B ), then experiment AB was attributed outcome − AB ′ , participants had to choose among the 4 items:( A B ′ ) The Cat eats the Fish ( A B ′ ) The Cat eats the Nuts ( A B ′ ) The Cow eats the Fish ( A B ′ ) The Cow eats the Nuts
If the response was ( A B ′ ) or ( A B ′ ), then experiment AB ′ was attributed outcome +1; if the responsewas ( A B ′ ) or ( A B ′ ), then experiment AB ′ was attributed outcome − A ′ B , participants had to choose among the 4 items:( A ′ B ) The Horse eats the Grass ( A ′ B ) The Squirrel eats the Meat ( A ′ B ) The Horse eats the Grass ( A ′ B ) The Cow eats the Meat
If the response was ( A ′ B ) or ( A ′ B ), then experiment A ′ B was attributed outcome +1; if the responsewas ( A ′ B ) or ( A ′ B ), then experiment A ′ B was attributed outcome − A ′ B ′ , participants had to choose among the 4 items:( A ′ B ′ ) The Horse eats the Fish ( A ′ B ′ ) The Horse eats the Nuts ( A ′ B ′ ) The Squirrel eats the Fish AB Cat eats Grass Cat eats Meat Cow eats Fish Cow eats Meat
Probability µ ( A B ) = 0 . µ ( A B ) = 0 . µ ( A B ) = 0 . µ ( A B ) = 0 . AB ′ Cat eats Fish Cat eats Nuts Cow eats Fish Cow eats Nuts
Probability µ ( A B ′ ) = 0 . µ ( A B ′ ) = 0 . µ ( A B ′ ) = 0 . µ ( A B ′ ) = 0 . A ′ B Horse eats Grass Horse eats Meat Squirrel eats Grass Squirrel eats Meat
Probability µ ( A ′ B ) = 0 . µ ( A ′ B ) = 0 . µ ( A ′ B ) = 0 µ ( A ′ B ) = 0 . A ′ B ′ Horse eats Fish Horse eats Nuts Squirrel eats Fish Squirrel eats Nuts
Probability µ ( A ′ B ′ ) = 0 . µ ( A ′ B ′ ) = 0 . µ ( A ′ B ′ ) = 0 . µ ( A ′ B ′ ) = 0 . Table 2: The data collected in coincidence experiments on entanglement in the conceptual combination
The Animal eats theFood . ( A ′ B ′ ) The Squirrel eats the Nuts
If the response was ( A ′ B ′ ) or ( A ′ B ′ ), then experiment A ′ B ′ was attributed outcome +1; if the responsewas ( A ′ B ′ ) or ( A ′ B ′ ), then experiment A ′ B ′ was attributed outcome − µ ( X i Y j ) that the outcome X i Y j is obtained in the coincidence experiment XY , i, j =1 , X = A, A ′ , Y = B, B ′ . Table 3 reports the judgement probabilities computed in this way. Thecorresponding expectation values, or correlation functions, are then calculated as follows: E ( AB ) = µ ( A B ) − µ ( A B ) − µ ( A B ) + µ ( A B ) = − . E ( AB ′ ) = µ ( A B ′ ) − µ ( A B ′ ) − µ ( A B ′ ) + µ ( A B ′ ) = 0 . E ( A ′ B ) = µ ( A ′ B ) − µ ( A ′ B ) − µ ( A ′ B ) + µ ( A ′ B ) = 0 . E ( A ′ B ′ ) = µ ( A ′ B ′ ) − µ ( A ′ B ′ ) − µ ( A ′ B ′ ) + µ ( A ′ B ′ ) = 0 . CHSH = 3 . ≈ . . ∗ − and 4 . ∗ − , respectively). Hence, thisdeviation confirms and strengthens the empirical patterns identified in document retrieval tests on theweb in [32] and [46]. Also in this case, entanglement between the concepts Animal and
Food is a naturalcandidate 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 Hilbertspace 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 haverecently elaborated to model Bell-type situations in any empirical domain.
In this section we study how a theoretical framework can be constructed which models the two cognitivesituations in Section 3 but is also general enough to cope with any domain in which Bell-type tests areperformed. We will show that the violations of the CHSH inequality and the Cirel’son bound can be simul-taneously explained by assuming that a strong form of ‘quantum entanglement’ is present in conceptualcombinations. We refine and generalise the modelling scheme that we have elaborated in [41, 42] to identifythe entanglement that occurs in Bell-type settings. The theoretical scheme has already been applied withsuccess to model the web tests in [47] (see also Section 2). We refer to [48] for a detailed analysis andcomparison of the entanglement situations occurring in physical and cognitive realms.9he general modelling scheme for Bell-type situations consists in the implementation of three mainsteps which we summarise in the following.(i) One identifies in the empirical situation under investigation the composite conceptual entity and theindividual conceptual entities composing it.(ii) One recognises in the composite conceptual entity the states, measurements and outcome probabil-ities that are relevant to the phenomenon under study.(iii) One represents entities, states, measurements and outcome probabilities using the Hilbert spacerepresentation 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. Letus start by coincidence experiment AB and its outcomes The Cat eats the Grass , The The Cat eats theMeat , 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 conceptualentity 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 natureof the involved concepts, this set of instructions prepares the composite entity
The Animal eats the Food inan initial state p which describes the general situation of an animal that eats food. This initial state is theunique 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 isnot predetermined, as it depends on the concrete choice being made, which results as a consequence ofthis ‘contextual interaction’ between the respondent and the conceptual entity. More specifically, if therespondent chooses The Cow eats the Grass , that is, the outcome A B is obtained in AB (see Section3), the interaction between the entity The Animal eats the Food prepared in the state p and the (mind ofthe) respondent will determine a change of state of The Animal eats the Food from p to the state p A B 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 ofstate change. This outcome statistics is interpreted, in the large number limit, as the probability P p ( A B )that the outcome A B is obtained when the measurement e AB is performed on the composite entity TheAnimal eats the Food in the initial state p . (iii) Let us finally come to the quantum mathematical representation. The entity The Animal eats theFood is associated with a Hilbert space and the state p is represented by a unit vector | p i of this Hilbertspace. The measurement e AB is instead represented by a self-adjoint operator or, equivalently, by a spectralfamily, on the Hilbert space whose eigenvectors represent the outcome states, or ‘eigenstates’, of e AB , whileoutcome probabilities are obtained through the Born rule of quantum probability. This representation willbecome 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, outcomesand outcome probabilities that are relevant to the situations in Section 3 is presented in the following.For every X = A, A ′ , Y = B, B ′ , 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 = − i = j , and4 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 In [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’ refersto the measurements that are performed on the entity. 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 andoutcome probabilities occurring in the The Animal Acts and
The Animal eats the Food situations, we needto 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 ′ , Y = B, B ′ , have 4 outcomes X i Y j , i, j = 1 ,
2, which entails that bothcomposite entities should be associated, as overall entities, with the complex Hilbert space C of all ordered4-tuples of complex numbers. In addition, each state p of The Animal Acts (respectively,
The Animal eatsthe Food ) should be represented by a unit vector of C and each measurement on The Animal Acts ( TheAnimal eats the Food ) should be represented by a self-adjoint operator or, equivalently, by a spectralfamily, on C . On the other side, for every i, j = 1 ,
2, each outcome X i Y j is obtained by juxtaposing theoutcomes 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 ′ on the individual entity Animal ( Animal ) and a 2-outcome measurement e Y , Y = B, B ′ on the individual entity Acts ( Food ). Hence, eachof these individual entities should be associated with the complex Hilbert space C of all ordered couplesof complex numbers. But, then, the standard Hilbert space formalism prescribes that both compositeentities The Animal Acts and
The Animal eats the Food should be associated with the tensor productHilbert space C ⊗ C . We stress that we are studying here an ‘identification problem’, that is, the problemof how the composite entity The Animal Acts ( The Animal eats the Food ) can be decomposed into theindividual entities
Animal ( Animal ) and
Acts ( Food )’ in such a way that ‘these individual entities can berecognised from measurements performed on the respective composite entities’. As such, we are doing anoperation that is opposite to what one typically does in Bell-type situations in quantum physics, where oneconstructs or, better, ‘composes’, the measurements on the composite entity from measurements performedon individual entities.From a mathematical point of view, the vector spaces C and C ⊗ C are isomorphic, and any isomor-phism 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 C , hence of C ⊗ C , which contains both vec-tors representing ‘product states’ and vectors representing ‘entangled states’. Moreover, the vector space L ( C ) of all linear operators on C is isomorphic to the tensor product L ( C ) ⊗ L ( C ), where L ( C ) of alllinear operators on C . Analogously, the tensor product L ( C ) ⊗ L ( C ) contains both self-adjoint operatorsrepresenting ‘product measurements’ and self-adjoint operators representing ‘entangled measurements’.Now, let I : C −→ C ⊗ C be an isomorphism mapping a given ON basis of C onto a given ON basisof C ⊗ C . We say that a state p represented by the unit vector | p i ∈ C is a ‘product state’ with respectto I , if two states p A and p B , represented by the unit vectors | p A i ∈ C and | p B i ∈ C , respectively, existsuch that I | p i = | p A i ⊗ | p B i . Otherwise, p is an ‘entangled state’ with respect to I . Next, we say that ameasurement e represented by the self-adjoint operator E on C is a ‘product measurement’ with respectto I , if two measurements e X and e Y , represented by the self-adjoint operators E X and E Y , respectively,on C exist such that I E I − = E X ⊗ 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 individualentities within a given composite entity’.With reference to the Bell-type setting presented in this section, one can then prove that, if the mea-surements e XY and e XY ′ , X = A, A ′ , Y, Y ′ = B, B ′ , Y ′ = Y , are product measurements with respect tothe isomorphism I , then, for every state p of the composed entity, the ‘marginal law of Kolmogorovianprobability’ is satisfied, that is, for every i = 1 , P j P p ( X i Y j ) = P j P p ( X i Y ′ j ). Analogously, if the mea-surements e XY and e X ′ Y , X, X ′ = A, A ′ , Y = B, B ′ , X ′ = X , are product measurements with respect to11he isomorphism I , then, for every state p of the composed entity, the marginal law is satisfied, that is, forevery j = 1 , P i P p ( X i Y j ) = P i P p ( X ′ i Y j ) (see Theorem 2, [41]). In the case the marginal law is satisfiedin all measurements, one can also prove that a unique isomorphism exists, which can be chosen to be theidentify operator (see, Theorem 4, [41]).It follows from the above that, if the marginal law is violated, as it occurs in both conceptual tests inSection 3, then one cannot find a unique isomorphism between C and C ⊗ C such that all measurementsare product measurements with respect to this isomorphism. In this case, one cannot explain the violationof the CHSH inequality as due to the usual situation in quantum physics where all measurements areproduct measurements and only the initial, or pre-measurement, state is entangled. Furthermore, if themarginal law is systematically violated, 4 distinct isomorphisms I XY , X = A, A ′ , Y = B, B ′ , exist suchthat the measurement e XY is a product measurement with respect to I XY . As a consequence, there isno unique isomorphism allowing to identify individual entities of a given composite entity. Finally, if weconsider a given isomorphism between C and C ⊗ C with respect to which identifying individual entitiesof a composite entity in a given test, then it may happen that both the pre-measurement state and allmeasurements 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 componentindividual concepts carry meaning and further meaning is created in the combination process’. Since theviolation of the CHSH inequality indicates the presence of entanglement between the individual conceptualentities, then it is reasonable to assume that ‘it is the quantum structure of entanglement which is ableto theoretically capture the meaning connections that are created in these cases’. This suggests, in aquantum-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 themeaning connections that are created between
Animal and
Acts and also between
Animal and
Food whenthe 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 ′ , Y = B, B ′ , violate the marginal law of Kolmogorovian probability. This suggests, againin a quantum-theoretic perspective, that all these measurements should be entangled measurements. Inaddition, in each measurement, all outcomes X i Y j , i, j = 1 ,
2, correspond to combined concepts, e.g.,
TheCat Meows , The Cat eats the Fish , etc., which are thus in turn connected by meaning. This also suggeststhat 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 theempirical tests in in Section 3.
We work out in this section a quantum representation in Hilbert space of the cognitive tests data inSection 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
TheAnimal Acts and
The Animal eats the Food situations.The composite conceptual entity is associated with the Hilbert space C of all ordered 4-tuples ofcomplex numbers. Let (1 , , , , , , , , ,
0) and (0 , , , } be the unit vectors of the canonicalON basis of C and let us consider the isomorphism I : C −→ C ⊗ C where this ON basis coincides The marginal law is systematically violated in
The Animal Acts test. For example, µ ( A B ) + µ ( A B ) = 0 . =0 . µ ( A B ′ ) + µ ( A B ′ ). Analogously, the marginal law is systematically violated in The Animal eats the Food test. Forexample, µ ( A ′ B ) + µ ( A ′ B ) = 0 . = 0 . µ ( A ′ B ′ ) + µ ( A ′ B ′ ). C ⊗ C made up of the unit vectors (1 , ⊗ (1 , , ⊗ (0 , , ⊗ (1 ,
0) and (0 , ⊗ (0 , p of the composite entityis represented by the unit vector | p i = ( ae iα , be iβ , ce iγ , de iδ ), where a, b, c, d ≥ a + b + c + d = 1, α , β , γ , δ ∈ ℜ and ℜ is the real line. One easily proves that | p i represents a product state if and only if thefollowing condition is satisfied: ade i ( α + δ ) − bce i ( β + γ ) = 0 (11)Otherwise, p represents an entangled state.Let us now represent the measurements e AB , e AB ′ , e A ′ B and e A ′ B ′ in Section 4. Each measurement e XY , X = A, A ′ , Y = B, B ′ , has 4 outcomes X Y , X Y , X Y and X Y and 4 eigenstates p X Y , p X Y , p X Y and p X Y . As in Sections 3 and 4, we set, for every X = A, A ′ , Y = B, B ′ , X Y = X Y = +1and X Y = X Y = −
1. The measurement e XY is represented by the self-adjoint operator E XY on C or, equivalently, by the spectral family {| p X Y ih p X Y | , | p X Y ih p X Y | , | p X Y ih p X Y | , | p X Y ih p X Y |} , suchthat the eigenstates p X Y , p X Y , p X Y and p X Y are represented by the eigenvectors | p i = ( a e iα , b e iβ , c e iγ , d e iδ ) (12) | p i = ( a e iα , b e iβ , c e iγ , d e iδ ) (13) | p i = ( a e iα , b e iβ , c e iγ , d e iδ ) (14) | p i = ( a e iα , b e iβ , c e iγ , d e iδ ) (15)of E XY , respectively. In Equations (12)–(15), the coefficients are such that a ij , b ij , c ij , d ij ≥ α ij , β ij , γ ij , δ ij ∈ ℜ , i, j = 1 ,
2. For every X = A, A ′ , Y = B, B ′ , the self-adjoint operator E XY canbe expressed as a tensor product operator if and only if all unit vectors in Equations (12)–(15) representproduct states. Otherwise, E XY cannot be expressed as a tensor product operator, hence e XY is an entan-gled measurement. For every X = A, A ′ , Y = B, B ′ , i, j = 1 ,
2, the probability P p ( X i Y j ) of obtaining theoutcome X i Y j in a measurement of e XY on the composite entity in the state p is then given by the Bornrule of quantum probability, that is, P p ( X i Y j ) = |h p ij | p i| .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, a + b + c + d = 1 (16) a + b + c + d = 1 (17) a + b + c + d = 1 (18) a + b + c + d = 1 (19)(ii) Orthogonality. The vectors in Equations (12)–(15) are mutually orthogonal, that is,0 = h p | p i = a a e i ( α − α ) + b b e i ( β − β ) + a c c i ( γ − γ ) + d d e i ( δ − δ ) (20)0 = h p | p i = a a e i ( α − α ) + b b e i ( β − β ) + a c c i ( γ − γ ) + d d e i ( δ − δ ) (21)0 = h p | p i = a a e i ( α − α ) + b b e i ( β − β ) + a c c i ( γ − γ ) + d d e i ( δ − δ ) (22)0 = h p | p i = a a e i ( α − α ) + b b e i ( β − β ) + a c c i ( γ − γ ) + d d e i ( δ − δ ) (23)0 = h p | p i = a a e i ( α − α ) + b b e i ( β − β ) + a c c i ( γ − γ ) + d d e i ( δ − δ ) (24)0 = h p | p i = a a e i ( α − α ) + b b e i ( β − β ) + a c c i ( γ − γ ) + d d e i ( δ − δ ) (25)(iii) Probabilities. For every X = A, A ′ , Y = B, B ′ , i, j = 1 ,
2, the probability P p ( X i Y j ) coincides with13he empirical probability µ ( X i Y j ) in Tables 1 and 2, that is, µ ( X Y ) = |h p | p i| = a a + b b + c c + d d ++ 2 aba b cos( α − α − β + β ) ++ 2 aca c cos( α − α − γ + γ ) ++ 2 ada d cos( α − α − δ + δ ) ++ 2 bcb c cos( β − β − γ + γ ) ++ 2 bdb d cos( β − β − δ + δ ) ++ 2 cdc d cos( γ − γ − δ + δ ) (26) µ ( X Y ) = |h p | p i| = a a + b b + c c + d d ++ 2 aba b cos( α − α − β + β ) ++ 2 aca c cos( α − α − γ + γ ) ++ 2 ada d cos( α − α − δ + δ ) ++ 2 bcb c cos( β − β − γ + γ ) ++ 2 bdb d cos( β − β − δ + δ ) ++ 2 cdc d cos( γ − γ − δ + δ ) (27) µ ( X Y ) = |h p | p i| = a a + b b + c c + d d ++ 2 aba b cos( α − α − β + β ) ++ 2 aca c cos( α − α − γ + γ ) ++ 2 ada d cos( α − α − δ + δ ) ++ 2 bcb c cos( β − β − γ + γ ) ++ 2 bdb d cos( β − β − δ + δ ) ++ 2 cdc d cos( γ − γ − δ + δ ) (28) µ ( X Y ) = |h p | p i| = a a + b b + c c + d d ++ 2 aba b cos( α − α − β + β ) ++ 2 aca c cos( α − α − γ + γ ) ++ 2 ada d cos( α − α − δ + δ ) ++ 2 bcb c cos( β − β − γ + γ ) ++ 2 bdb d cos( β − β − δ + δ ) ++ 2 cdc d cos( γ − γ − δ + δ ) (29)Then, let us represent represent the initial state p of the composite conceptual entity by the unit vector | p i = 1 √ , , − ,
0) (30)The vector in Equation (30) represents a maximally entangled state and corresponds to the ‘singlet spinstate’ that is used in typical Bell-type tests in quantum physics (see Section 1). There are different reasonsfor this choice. Firstly, our aim is to incorporate all possible entanglement of the state-measurement14ituation in the pre-measurement state, so that the ensuing measurements are as close as possible toproduct measurements. We are however aware that some measurements, if not all, are entangled, due tothe violation of the marginal law (see Section 4). Secondly, it is known that the singlet spin state hasspecific 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 correspondto the fact both
The Animal Acts and
The Animal eats the Food express more abstract concepts than thecorresponding outcomes.Thus, for every X = A, A ′ , Y = B, B ′ , conditions (i)–(iii) identify 20 equations which should be satisfiedby the 32 real variables a ij , b ij , c ij , d ij , α ij , β ij , γ ij , δ ij , i, j = 1 ,
2. To simplify the calculation, we set, forevery i, j = 1 , α ij = β ij = γ ij = δ ij = θ ij , where θ ij ∈ ℜ . Thus, each unit vector in Equations (12)–(15)takes the form | p ij i = e iθ ij ( a ij , b ij , c ij , d ij ), i, j = 1 ,
2. This reduces the total number of unknown variablesto 20.We are now ready to represent the empirical data in Section 3 in Hilbert space. We start by
The AnimalActs test.The eigenstates of the measurement e AB are represented by the unit vectors | p A B i = e i . ◦ (0 . , − . , . ,
0) (31) | p A B i = e i . ◦ (0 , . , . , .
29) (32) | p A B i = e i . ◦ (0 . , . , − . , .
12) (33) | p A B i = e i . ◦ (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) showsa relatively larger deviation from zero in the unit vector | p A B i . We can then say that the eigenstate p A B , corresponding to The Bear Growls , is a ‘relatively more entangled state’.The eigenstates of the measurement e AB ′ are represented by the unit vectors | p A B ′ i = e i . ◦ (0 . , . , − . , .
08) (35) | p A B ′ i = e i . ◦ (0 . , . , , − .
95) (36) | p A B ′ i = e i . ◦ ( − . , . , . , .
30) (37) | p A B ′ i = e i . ◦ (0 . , − . , . ,
0) (38)Also in this case, all unit vectors are entangled, hence e AB ′ is an entangled measurement. The entanglementcondition in Equation (11) shows a relatively larger deviation from zero in the unit vector | p A B ′ i . Wecan then say that the eigenstate p A B ′ , corresponding to The Horse Snorts , is a ‘relatively more entangledstate’.The eigenstates of the measurement e A ′ B are represented by the unit vectors | p A ′ B i = e i . ◦ (0 . , . , − . , .
17) (39) | p A ′ B i = e i . ◦ (0 . , . , . , .
35) (40) | p A ′ B i = e i . ◦ (0 . , − . , . ,
0) (41) | p A ′ B i = e i . ◦ (0 . , . , , − .
92) (42)All unit vectors are entangled, hence e A ′ B is an entangled measurement. The entanglement condition inEquation (11) shows a relatively larger deviation from zero in the unit vector | p A ′ B i . We can then saythat the eigenstate p A ′ B , corresponding to The Tiger Growls , is a ‘relatively more entangled state’.15inally, the eigenstates of the measurement e A ′ B ′ are represented by the unit vectors | p A ′ B ′ i = e i . ◦ (0 . , − . , . ,
0) (43) | p A ′ B ′ i = e i . ◦ (0 . , . , , − .
93) (44) | p A ′ B ′ i = e i . ◦ (0 . , . , . , .
30) (45) | p A ′ B ′ i = e i . ◦ (0 . , . , − . , .
20) (46)All unit vectors are entangled, hence e A ′ B ′ is an entangled measurement. The entanglement condition inEquation (11) shows a relatively larger deviation from zero in the unit vector | p A ′ B ′ i . We can then saythat the eigenstate p A ′ B ′ , 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. Theeigenstates of the measurements e AB , e AB ′ , e A ′ B and e A ′ B ′ are respectively represented by the 4 unitvectors | p A B i = e i . ◦ (0 . , − . , . ,
0) (47) | p A B i = e i . ◦ (0 . , . , . , .
22) (48) | p A B i = e i . ◦ (0 . , . , − . , .
03) (49) | p A B i = e i . ◦ (0 . , . , , − .
97) (50) | p A B ′ i = e i . ◦ (0 . , . , − . , .
12) (51) | p A B ′ i = e i . ◦ (0 . , . , . , .
19) (52) | p A B ′ i = e i . ◦ (0 . , − . , . ,
0) (53) | p A B ′ i = e i . ◦ (0 . , . , , − .
97) (54) | p A ′ B i = e i . ◦ (0 , . , − . , .
17) (55) | p A ′ B i = e i . ◦ (0 . , . , . , .
31) (56) | p A ′ B i = e i . ◦ (0 . , − . , − . ,
0) (57) | p A ′ B i = e i . ◦ (0 , . , , − .
94) (58) | p A ′ B ′ i = e i . ◦ (0 . , − . , . ,
0) (59) | p A ′ B ′ i = e i . ◦ (0 . , . , , − .
99) (60) | p A ′ B ′ i = e i . ◦ (0 . , . , . , .
14) (61) | p A ′ B ′ i = e i . ◦ (0 . , . , − . , .
08) (62)Also in this case, referring to Equation (11), one can show that all unit vectors in Equations (47)–(62) areentangled, hence ‘all measurements e AB , e AB ′ , e A ′ B and e A ′ B ′ are entangled measurements’. In addition,as in The Animal Acts test, in each measurement, one eigenstate is represented by a unit vectors whichexhibits a larger deviation in the entanglement condition in Equation (11). These ‘relatively more entangledstates’ are p A B , corresponding to The Cow eats the Grass in e AB , p A B ′ , corresponding to The Cat eatsthe Fish in e AB ′ , p A ′ B , corresponding to The Horse eats the Grass in e A ′ B , and p A ′ B ′ , corresponding to The Squirrel eats the Nuts in e A ′ B ′ . 16e 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
TheAnimal Acts and
The Animal eats the Food situations. This result is due to the violation of the marginallaw of probability in both tests which forbids concentrating all the entanglement of the state-measurementsituation into the state, as we have seen in Section 4. Moreover, in each measurement, all eigenstates areentangled. This result confirms with our suggestion in Section 4 that ‘quantum entanglement theoreticallycaptures the meaning connections between the individual concepts that form the composite conceptual en-tity, and that these meaning connections are not predetermined, but are created when the test is performedand each respondent interacts with the composite entity’.Secondly, in both situations, there are some outcome eigenstates which exhibit a relatively higher degreeof entanglement than others, which can exactly be explained with the fact that entanglement capturesmeaning connections, hence higher degrees of entanglement correspond to higher meaning connections.For example, in
The Animal Acts situation, the eigenstate p A B of the measurement e AB , correspondingto The Bear Growls , is relatively more entangled than the other eigenstates of e AB , which can be naturallyexplained by the fact that the individual items Bear and
Growls , which are concepts themselves, arerelatively more connected by meaning. As a matter of fact, if we look at empirical probabilities (see Table1, Section 3), we note that the outcome A B , corresponding to The Bear Growls , scores a high probabilityto 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, thecorresponding eigenstates are relatively less entangled. Analogously, in
The Animal eats the Food situation,the eigenstate p A ′ B ′ of the measurement e A ′ B ′ , corresponding to The Squirrel eats the Nuts , is relativelymore entangled than the other outcome eigenstates of e A ′ B ′ , which can be again explained by the factthat the individual items Squirrel and
Nuts , which are concepts themselves, are relatively more connectedby meaning. And, indeed, if we look at empirical probabilities (see Table 2, Section 3), we note that theoutcome A ′ B ′ , corresponding to The Squirrel eats the Nuts , scores a high probability to be judged as agood example of the conceptual combination
The Animal eats the Food . On the contrary, items like
TheCow eats the Meat , The Cat eats the Nuts and
The Squirrel eats the Grass score a low probability and areless 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 complexand 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 eigenstatesand two entangled eigenstates. The reason is that many judgement probabilities were zero in [46], dueto the fact that human minds are able to create additional meaning connections between concepts thancorpuses of documents.Fourthly, we observe that the violation of Cirel’son bound in
The Animal eats the Food situation canstill be explained in terms of quantum entanglement, in contrast to widespread beliefs. Hence, there isquantum entanglement also beyond Cirel’son bound, but this type of entanglement involves both statesand measurements. This result is important and we will devote the next section to deepen it.
The empirical results in the cognitive test on
The Animal eats the Food presented in Section 3 show asignificant violation of the Cirel’son bound and, as such, they confirm the results in the document retrievaltests on the web in [32] and [46]. As we have noticed throughout the paper, these results contrast withwidespread 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 funda-17ental limit to quantum nonlocality, that is, whether the correlations exhibited by two far away physicalentities should satisfy any condition in order to represent them in the mathematical formalism of quantumtheory – this problem is also connected with the old problem of exploiting quantum nonlocality to senda faster than light signal between far away entities (see, e.g., [9]). One then considers a typical Bell-typesituation, 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 oneindividual entity and one measurement on the other, and deduces that the CHSH factor in Equation (1) isbound by the value ∆
QMC = 2 √ ≈ .
83 in quantum theory [6, 7]. Hence, the Cirel’son bound obtained inthis way is usually considered, especially by quantum physicists, as the limit outside which the correlationsexhibited 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 Hilbertspace empirical data collected in Bell-type tests which violate the CHSH inequality by an amount whichalso exceeds Cirel’son bound (see, e.g., [47, 48]). This means that the statistical correlations exhibitedin these Bell-type tests, which are anyway non-classical and non-signalling, can be represented withinquantum theory.As we have seen in Sections 4 and 5, the fundamental element which makes it possible the abovementioned quantum mathematical representation in Hilbert space is the suggestion that the measurementsperformed on the composite entity have outcomes that are products of outcomes obtained in measurementsseparately performed on individual entities, but their eigenstates are generally entangled, that is, theyare 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, hencethey implicitly assumed only product measurements in the derivation of the Cirel’son bound. If onealso allows the possibility of using entangled measurements, then one can in principle violate the CHSHinequality in Equation (1) by any amount within its mathematical limit, i.e. between − cknowledgements This work was supported by QUARTZ (Quantum Information Access and Retrieval Theory), the MarieSk lodowska-Curie Innovative Training Network 721321 of the European Union’s Horizon 2020 research andinnovation programme.
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