Beyond-Quantum Modeling of Question Order Effects and Response Replicability in Psychological Measurements
BBeyond-Quantum Modeling of Question Order Effects and ResponseReplicability in Psychological Measurements
Diederik Aerts and Massimiliano Sassoli de Bianchi Center Leo Apostel for Interdisciplinary StudiesBrussels Free University, 1050 Brussels, Belgium
E-Mail: [email protected] Laboratorio di Autoricerca di Base6914 Lugano, Switzerland
E-Mail: [email protected]
Abstract
A general tension-reduction (GTR) model was recently considered (Aerts & Sassoli de Bianchi, 2015a,b)to derive quantum probabilities as (universal) averages over all possible forms of non-uniform fluctu-ations, and explain their considerable success in describing experimental situations also outside of thedomain of physics, for instance in the ambit of quantum models of cognition and decision. Yet, this resultalso highlighted the possibility of observing violations of the predictions of the Born rule, in those situa-tions where the averaging would not be large enough, or would be altered because of the combination ofmultiple measurements. In this article we show that his is indeed the case in typical psychological mea-surements exhibiting question order effects, by showing that their statistics of outcomes are inherentlynon-Hilbertian, and require the larger framework of the GTR-model to receive an exact mathematicaldescription. We also consider another unsolved problem of quantum cognition: response replicability.It is has been observed that when question order effects and response replicability occur together, thesituation cannot be handled anymore by quantum theory. However, we show that it can be easily andnaturally described in the GTR-model. Based on these findings, we motivate the adoption in cognitivescience of a hidden-measurements interpretation of the quantum formalism, and of its GTR-model gen-eralization, as the natural interpretational framework explaining the data of psychological measurementson conceptual entities.
Keywords : Question order effects, Response replicability, Quantum cognition, Quantum probability,Hidden-measurements, Born rule, Bloch sphere.
Quoting from Wikipedia: “Quantum cognition is an emerging field which applies the mathematical formal-ism of quantum theory to model cognitive phenomena such as information processing by the human brain,decision making, human memory, concepts and conceptual reasoning, human judgment, and perception.”(Kitto, 2008; Khrennikov, 2010; Busemeyer & Bruza, 2012; Aerts et al., 2013a,b; Pothos & Busemeyer,2013; Wang et al., 2013; Blutner & beim Graben, 2014). Quantum cognition (n.d.). In Wikipedia. Retrieved August 10, 2015, from https://en.wikipedia.org/wiki/Quantum_cognition . a r X i v : . [ c s . A I] A ug o explain why quantum physics is so adequate in the modeling of the human cognitive processes,many reasons are today available that are beginning to be fairly well understood. An important one isthat quantum mechanics was created to model and explain physical phenomena exhibiting a high degreeof contextuality, and contextuality is also the light motive in human cognition and decision processes. Forinstance, in the same way quantum superposition states can describe physical properties only existing inpotential terms, that is, available to be actualized during certain measurements in a genuinely unpredictableway, conceptual situations that lend themselves to judgments and decisions are also generally describablein terms of indeterministic processes of actualization of potential properties, when subjects are somehowforced to produce an answer in situations of ambiguity, uncertainty, confusion, etc.So, the quantum formalism can be very relevant in the description of the human cognitive processes,and it was quite amazing to observe, over the last years, how many aspects that were used to explain thestrange behavior of the microscopic physical entities have proven to be also pertinent in the descriptionof the human conceptual entities, like superposition, interference, entanglement, emergence, and evenquantum fields effects. The amazement was not only in the observation that the human conceptual entitiesand the microscopic physical entities were evidently sharing a common “conceptual nature,” but thatthe very specific Hilbertian structure of quantum mechanics appeared to be sufficient to model a largespectrum of experimental situations.In other terms, it wasn’t at all expected in the beginning of the investigations in quantum cognition,that the Born rule of assignment of the probabilities would prove to be so effective in the modeling ofmany quantum effects identified in the cognitive domain (Aerts & Sozzo, 2012a,b). To put it anotherway: Why the Born rule and not other “rules,” associated with different and/or more general quantum-like structures? This fundamental question was the object of a recent examination by us: starting froma beyond-quantum modeling of psychological measurements, which we have called the general tension-reduction model (GTR-model, in brief), we were able to show that the Born rule is characterizable interms of uniform fluctuations in the experimental context, and that these uniform fluctuations naturallyemerge when a universal average over all possible forms of non-uniform fluctuations is considered (Aerts& Sassoli de Bianchi, 2015a,b). In other terms, in our analysis we were able to show that Born’s quantumprobabilities correspond to a first-order approximation of more general probability models, describing abeyond-quantum class of measurements characterized by non-uniform fluctuations.Let us explain a little more specifically what we exactly mean by this, as this will be important toproperly contextualize the results presented in this article, which can be considered to be the logicalcontinuation of the analysis in Aerts & Sassoli de Bianchi (2015a,b). Consider an opinion poll conductedon a large number of respondents. Each of them will bring into it the uniqueness of their minds, eachone with a specific conceptual network forming its inner memory structure. This means that participantsin the poll will have a different way of choosing an answer among those that are available to them to beselected. And this also means that a cognitive experiment, involving a number of different subjects, isgenerally to be considered not as a single measurement, but as a collection of different measurements, onefor each participant. Each of these different individual measurements, however, is in principle associatedwith different outcome probabilities, so that the overall probabilities deduced from the individual resultsproduced by all the participants are in fact averages over the individual probabilities.What we are here affirming is that in a typical cognitive experiment, like an opinion poll, we are actuallyin the presence of a sort of abstract “collective mind,” producing a collection of different quantum-likemeasurements that remain undistinguished, being simply averaged out in the final statistics. This means This observation also brought one of us to boldly propose a new conceptuality interpretation of quantum mechanics (Aerts,2009, 2010a,b). We use the term “quantum-like” in the sense of “non-classical & non-quantum,” i.e., to describe processes that can bedescribed by probability models which are simultaneously non-Kolmogorovian and non-Hilbertian. universal measurement , describing the most general possiblecondition of lack of knowledge in a measurement, expressed as an average over all imaginable types ofindividual measurements. We have then proved that such huge average exactly yields the Born rule ofquantum mechanics, if the structure of the state space is Hilbertian, providing in this way a convincingexplanation of why the quantum statistics performs so well, in so many experimental ambits. It does sobecause it is a first order non-classical theory in the modeling of measurement data.However, if on one hand the possibility to describe the Born rule as a universal measurement can explainits “unreasonable” success in modeling so many experiments in cognitive science, this also points to thepossibility of observing violations of it, in specific experimental situations. Indeed, there are no a priori reasons to believe that in all experiments a universal average, or a good approximation of it, would begenerated by the participants’ different ‘ways of choosing’ an outcome. This can be the case because thestatistical sample is too small, because at the inter-subjective level certain ‘meaning connections’ wouldbe absent, thus filtering out certain “ways of choosing,” or because measurements would be combinedtogether in a way that would prevent the average to be effective in producing the uniform fluctuations thatcharacterize the Born rule.It is important to observe that possible violations of the Born rule cannot be observed in single measure-ment situations, as is clear that the Hilbert model, with its scalar product, when equipped with the Bornrule becomes a “universal probabilistic machine,” capable of representing all possible probabilities appear-ing in nature, in a given (single) measurement context (Aerts & Sozzo, 2012a; Aerts & Sassoli de Bianchi,2015a). But when for instance sequential measurements are considered, the Hilbertian model is not anymore a universal model, and violations can in principle be observed (Aerts & Sassoli de Bianchi, 2015a,b).So, it is in the ambit of experiments constructed as multiple processes, where aspects of non-commutativityare at stake, that one has to look to find quantum-like structures which are possibly non-Hilbertian, i.e.,non-Bornian.Typical experiments of this kind are those exhibiting question order effects , commonly observed insocial and behavioral research; see for instance Sudman & Bradburn (1974); Schuman & Presser (1981);Tourangeauet al. (2000). Tentative quantum accounts of these and other effects were given by many authorsin recent years (Conte et al., 2009; Busemeyer et al., 2011; Trueblood & Busemeyer, 2011; Atmanspacher& Roemer, 2012; Wang & Busemeyer, 2013; Aerts et al., 2013a,b; Wang et al., 2014; Khrennikov et al.,2014; Boyer-Kassem et al., 2015) and it is generally considered that quantum mechanics can provide anefficacious and consistent modeling of question order effects data. This was recently emphasized by Wang &Busemeyer (2013), showing that different data sets, like those reported by Moore (2002), obey a parameter-free constraint, called the
QQ-equality , which according to these authors is able to test the pure quantumnature of the experimental probabilities.This conclusion, however, was recently called into question by Boyer-Kassem et al. (2015), showingthat the QQ-equality is actually insufficient to test the validity of the Hilbertian model, as other ‘quantumequalities’ can be shown not to be satisfied by the same data, thus demonstrating a violation of theHilbertian structure. But then, how to properly account for the question order effects, if the Hilbertianmodel is unable to do so? Because one thing is to show that the Born rule is violated, and another thing is3o provide a non ad hoc model able to exactly fit the experimental data and, possibly, to shed some lightinto the nature of the cognitive processes that have generated the data.This is precisely what we are going to do in the present article, by means of the GTR-model. Moreprecisely, in Section 2, we show how two sequential 2-outcome measurements can be generally modeled byusing the hidden-measurements representation that is built in the GTR-model. In Section 3, we add morestructure to the model by introducing three simplifying hypothesis (that we call weak compatibility, localuniformity and sensitivity to pre-measurement state) thanks to which, in Section 4, we can present anexplicit exact solution to the model. This solution depends on a certain number of parameters (only one ofwhich is free), and standard quantum mechanics is recovered in the limit where some of these parameterstend to certain specific values.In Section 5, we use this explicit solution to obtain an exact modeling of some typical data from theGallup survey experiments reported in a seminal article on question order effects by Moore (2002), andmore particularly the ‘Clinton/Gore’ and ‘Rose/Jackson’ data. Thanks to our exact modeling, we can thenshow that the experimental probabilities are irreducibly non-Hilbertian, as they cannot be obtained by achoice of parameters defining a pure quantum situation. In Section 6, we also investigate the content ofWang & Busemeyers QQ-equality, showing how and why it can also be obeyed by non-Hilbertian models.We also use our explicit solution to derive additional parameter-free ‘quantum equalities,’ which we showare strongly violated by the Clinton/Gore experimental data.In Section 7, we address the issue of response replicability , which remains an unsolved problem inquantum cognition. Indeed, the quantum formalism does not allow the joint description of question ordereffects and response replicability (Khrennikov et al., 2014). This is so because the quantum model, contraryto the GTR-model, does not allow for the description of different typologies of measurements, associatedwith a same set of outcomes. On the other hand, by allowing the measurements to “evolve,” taking intoaccount the obtained outcomes, the GTR-model can not only describe response replicability, compatiblywith question order effects, but also more complex situations of ‘partial replicability,’ in accordance withour intuitive idea of what it means to have an opinion, or if we had not to have formed one, and to possiblychange it again, under the right circumstances.In Section 8, we then provide what we think is the appropriate interpretation of the different math-ematical objects of the GTR-model, with the states not describing beliefs, but objective elements of aconceptual reality, independent from the minds that can interact with it. And considering that the GTR-model is a natural generalization of the standard quantum formalism, we encourage the adoption of thisinterpretation as the standard one in quantum cognition. In Section 9, we briefly consider what are thepossible effects of the averaging over the different respondents, for what concerns the effective probabilitymodel describing the overall statistics of outcomes. In particular, we show that even when the individualrespondents choose their outcomes according not only to a symmetrical probability distribution, but alsoin the same way in the two sequential measurements, when their ‘ways of choosing’ are averaged out, inthe final statistics, this symmetry will be broken, in the sense that their collective ‘way of choosing’ willbe non-symmetric and sequentially non-uniform. Finally, in Section 10, we recap the obtained results, andoffer some final thoughts regarding the possibility of modeling question order effects by means of the Bornrule, in higher dimensional Hilbert spaces. -outcome measurements Let us start by considering two general measurements, F and G , performed on an entity S , prepared inthe initial state ψ . We assume that each measurement admits 4 different possible outcomes: F F F F
4, and G G G G
4, respectively. We denote P ( F i | ψ ) the probability of obtaining outcome F i , i = 1 , . . . ,
4, when measurement F is performed, conditional to the fact that the pre-measurement4tate is ψ , and similarly, we denote P ( Gi | ψ ) the probability of obtaining outcome Gi , i = 1 , . . . ,
4, whenmeasurement G is performed, with the entity in the initial state ψ . Since for each measurement the 4outcomes describe mutually exclusive events, which are also assumed to be the only possible events, thenaccording to the properties of unitarity and additivity of a probability measure, we have: (cid:88) i =1 P ( F i | ψ ) = 1 , (cid:88) i =1 P ( Gi | ψ ) = 1 . (1)Apart from the above two constraints, and the fact that probabilities must be positive real numbers,no other constraints need to be imposed on the specific values taken by the above eight probabilities,considering that F and G are two different arbitrary measurements.Let us now assume that F and G are two composite measurements, corresponding to the sequentialexecution of two different measurements, A and B , each one admitting two possible outcomes: Ay , An , and By , Bn , respectively (a 2-outcome measurement is sometimes called a “yes-no” measurement, hence theletters “y” and “n” to designate the outcomes). More precisely, F = AB is the measurement obtained byfirst performing measurement A and then, on the resulting outcome state, measurement B (note that thenotation “ F = AB ” is here just a mnemonic device and is not meant to designate any specific operatorialproduct). Therefore, the 4 different possible outcomes of F are: F AyBy , F AyBn , F AnBy and F AnBn . On the other hand, G = BA is the measurement obtained by first performing themeasurement B and then, on the resulting outcome state, measurement A . Therefore, the four possibleoutcomes are: G ByAy , G ByAn , G BnAy and G BnAn . The unitarity relations (1) thenread: P ( AyBy | ψ ) + P ( AyBn | ψ ) + P ( AnBy | ψ ) + P ( AnBn | ψ ) = 1 ,P ( ByAy | ψ ) + P ( ByAn | ψ ) + P ( BnAy | ψ ) + P ( BnAn | ψ ) = 1 . (2)We can write the above 8 probabilities in more explicit terms by introducing the conditional probabilitiesrelative to the second measurement. More precisely, denoting P ( By | ψAy ) the probability that outcome By is obtained in the second B -measurement, when outcome Ay was obtained in the first A -measurement,which in turn was performed on the initial state ψ (hence the sequential notation “ ψAy ” for the conditionalstatement, to be read from the left to the right), and so on, we can write: P ( AyBy | ψ ) = P ( By | ψAy ) P ( Ay | ψ ) , P ( AyBn | ψ ) = P ( Bn | ψAy ) P ( Ay | ψ ) ,P ( AnBy | ψ ) = P ( By | ψAn ) P ( An | ψ ) , P ( AnBn | ψ ) = P ( Bn | ψAn ) P ( An | ψ ) . (3)In the same way, the probabilities for the outcomes of the G = BA sequential measurement can be writtenas: P ( ByAy | ψ ) = P ( Ay | ψBy ) P ( By | ψ ) , P ( ByAn | ψ ) = P ( An | ψBy ) P ( By | ψ ) ,P ( BnAy | ψ ) = P ( Ay | ψBn ) P ( Bn | ψ ) , P ( BnAn | ψ ) = P ( An | ψBn ) P ( Bn | ψ ) , (4)and we now have the four unitarity relations: P ( By | ψAy ) + P ( Bn | ψAy ) = 1 , P ( By | ψAn ) + P ( Bn | ψAn ) = 1 ,P ( Ay | ψBy ) + P ( An | ψBy ) = 1 , P ( Ay | ψBn ) + P ( An | ψBn ) = 1 . (5)Clearly, the above remains a very general representation, not introducing any specific constraint onthe structure of the outcome probabilities of F and G . In other terms, until something more specificis said about the nature of the measured entity and the characteristics of the two measurements A and B , it is always possible to describe two arbitrary 4-outcome measurements, F and G , as two sequentialmeasurements formed by two 2-outcome measurements, performed in different order, i.e. as: F = AB and G = BA . 5 .1 The hidden-measurements representation We now introduce a very general geometrico-dynamical representation of the two sequential measurements F = AB and G = BA . We emphasize from the beginning, to avoid possible misunderstandings, that thisis a general quantum-like representation, generalizing both the classical and quantum probability models,which can be recovered in the appropriate limits.It is worth observing that variants of this geometrico-dynamical representation have received differentnames, in different research contexts. When describing 2-outcome processes, as it will be the case in thepresent article, it was named the sphere-model (Aerts et al., 1997) and also the (cid:15) -model (Aerts, 1998; Sassolide Bianchi, 2013). When describing more general N -outcome measurements with uniform fluctuations,it was called the quantum model theory (Aerts & Sozzo, 2012a,b), and when more general non-uniformfluctuations were also considered, in a recent update, also including degenerate measurements, it was namedthe general tension-reduction model (Aerts & Sassoli de Bianchi, 2015a,b). Also, when the structure of thestate space is purely Hilbertian, as it is the case in microphysics, the model was called the extended Blochrepresentation of quantum mechanics (Aerts & Sassoli de Bianchi, 2014, 2015c), and offers a challengingsolution to the measurement problem for finite-dimensional quantum systems of arbitrary dimension.We mathematically describe the pre-measurement state ψ of the entity subjected to the measurementsby a 3-dimensional real unit vector x ψ , i.e., by a vector at the surface of a 3-dimensional unit sphere,called the Bloch sphere . Measurements will then be represented in the sphere by abstract 1-dimensionalbreakable and elastic structures, anchored at two antipodal points, corresponding to the two possibleoutcomes. More precisely, measurement A will be represented by a breakable “elastic band” stretchedbetween the two points a y and a n = − a y , (cid:107) a y (cid:107) = (cid:107) a n (cid:107) = 1, corresponding to the two outcomes Ay and An , respectively. Similarly, measurement B will be represented by a breakable “elastic band” stretchedbetween the two points b y and b n = − b y , (cid:107) b y (cid:107) = (cid:107) b n (cid:107) = 1, corresponding to the two outcomes By and Bn , respectively (see Fig. 1). Figure 1: The unit vectors a y , a n = − a y , representing the outcomes Ay and An of the A -measurement, the unit vectors b y , b n = − b y , representing the outcomes By and Bn of the B -measurement, and the unit vector x ψ , representing the initial state ψ , in the Bloch sphere. The angles θ , θ A and θ B are given by the scalar products: x ψ · a y = cos θ A , x ψ · b y = cos θ B , and a y · b y = cos θ . We assume that the two breakable elastics are parameterized in such a way that the coordinate x = 1(resp. x = −
1) corresponds to the outcome “yes” (resp. “no”), with x = 0 describing their centers,coinciding also with the center of the Bloch sphere. Each elastic represents a possible measurement, and isdescribed not only by its orientation within the sphere, but also by ‘the way’ it can break. More precisely,6onsidering for instance the elastic associated with measurement A , we can describe its breakability bymeans of a probability distribution ρ A ( x | ψ ), so that (cid:82) x x ρ A ( x | ψ ) dx is the probability that the elastic willbreak in a point in the interval [ x , x ], − ≤ x ≤ x ≤
1, in the course of the measurement, when theinitial state is ψ , and of course we have the unitarity condition (cid:82) − ρ A ( x | ψ ) dx = 1, expressing the fact thatthe elastic will break in one of its points, with certainty.Let us describe more specifically how the quantum-like measurement unfolds (see Fig. 2). To the initial(pre-measurement) vector state x ψ we associate an abstract point particle. When the A -measurement isexecuted, a certain probability distribution ρ A ( x | ψ ) is actualized, describing the way the A -elastic band willbreak, in accordance with the fluctuations that are present in that moment in the experimental context.The abstract point particle then plunges into the Bloch sphere, by orthogonally “falling” from its surfaceonto the elastic band, firmly attaching to it (we can consider that the elastic is sticky, or in some wayattractive). Then, when the elastic breaks in some point, its two broken fragments will contract toward thecorresponding anchor points, bringing with them the point particle. This means that if x A is the positionof the point particle onto the elastic ( x A = x ψ · a y ≡ cos θ A ), then, if the elastic breaks in a point λ , with x A < λ , the particle will be attached to the elastic fragment collapsing toward a y , and will therefore bedrawn to that position, which will correspond to the outcome of the measurement. On the other hand, if x A > λ , the particle will be attached to the elastic fragment collapsing toward a n , and will be drawn tothat position (state) at the end of the process. This means that the conditional (transition) probabilities P ( A y | ψ ) = P ( x ψ → a y ) and P ( A n | ψ ) = P ( x ψ → a n ) will be given by the two integrals: P ( A y | ψ ) = (cid:90) cos θ A − ρ A ( x | ψ ) dx, P ( A n | ψ ) = (cid:90) θ A ρ A ( x | ψ ) dx. (6)When the breaking point of the elastic exactly coincides with the landing point of the particle, i.e., λ = cos θ A , we are in a situation of classical unstable equilibrium, and the outcome is not predetermined.However, these exceptional λ -values are clearly of zero measure, and will not contribute to the determinationof the probabilities (6). We also observe that to each breaking point of the elastic (excluding the abovementioned exceptional points) is associated a deterministic measurement-interaction between the pointparticle (representing the measured entity) and the elastic (representing the measuring system, or better,that part of the measuring systems that is relevant for the description of the process in question). In ageneral measurement context, the deterministic measurement-interactions remain “hidden,” in the sensethat it is impossible to predict in advance which of them will be actualized, at each run of the measurement.This process of actualization of a potential, almost deterministic, measurement-interaction , is what isdescribed in our model by the given of the probability distribution ρ A ( x | ψ ), which we assume can generallydepend also on the pre-measurement state.Before continuing in our analysis of the sequential measurements, it is worth observing that when ρ A ( x | ψ ) = , i.e., when the probability distribution is globally uniform, (6) reduces to the well-knownBorn rule’s probability formulae: P ( A y | ψ ) = 12 (1 + cos θ A ) , P ( A n | ψ ) = 12 (1 − cos θ A ) . (7)In other terms, the model reduces to standard quantum mechanics when the probability distributionsdescribing the elastic bands become globally uniform. This is a general result, not limited to 2-outcomemeasurements, as can be seen from the more general analysis in Aerts & Sassoli de Bianchi (2014, 2015a,b,c).With a similar notation, we can also write the conditional (transition) probabilities P ( B y | ψ ) = P ( x ψ → b y ) and P ( B n | ψ ) = P ( x ψ → b n ), as the integrals: P ( B y | ψ ) = (cid:90) cos θ B − ρ B ( x | ψ ) dx, P ( B n | ψ ) = (cid:90) θ B ρ B ( x | ψ ) dx, (8)7 igure 2: The unfolding of a quantum-like 2-outcome measurement process. Figure (a) shows the abstract point particle,representative of the initial state ψ , at the surface of the sphere, in the process of entering into it, by orthogonally “falling”onto the elastic band, here describing measurement A . In figure (b) the point particle has reached its on-elastic position. Theunpredictable breaking point of the elastic is also shown in the figure, which is here assumed to belong to the elastic’s fragmentgoing from outcome Ay to the particle position. As a consequence, the contraction of the elastic will draw the point particleto the outcome state An , as shown in Figures (c) and (d). where x B = x ψ · b y ≡ cos θ B is the landing point of the point particle onto the B -elastic band, and ρ B ( x | ψ )is the probability distribution associated with the latter. Proceeding with the same logic, defining cos θ ≡ a y · b y , we can also write, for the conditional (transition) probabilites P ( By | ψAy ) = P ( a y → b y | x ψ → a y ), P ( Bn | ψAy ) = P ( a y → b n | x ψ → a y ), P ( By | ψAn ) = P ( a n → b y | x ψ → a n ) and P ( Bn | ψAn ) = P ( a n → b n | x ψ → a n ): P ( By | ψAy ) = (cid:90) cos θ − ρ B ( x | ψAy ) dx, P ( Bn | ψAy ) = (cid:90) θ ρ B ( x | ψAy ) dx,P ( By | ψAn ) = (cid:90) − cos θ − ρ B ( x | ψAn ) dx, P ( Bn | ψAn ) = (cid:90) − cos θ ρ B ( x | ψAn ) dx, (9)where ρ B ( x | ψAy ) (resp., ρ B ( x | ψAn )) is the probability distribution actualized during the second B -measurement, knowing that the first A -measurement produced the transition ψ → Ay (resp., ψ → An ).8nserting (9) and (8) into (3), we thus obtain: P ( AyBy | ψ ) = (cid:90) cos θ − ρ B ( x | ψAy ) dx (cid:90) cos θ A − ρ A ( x | ψ ) dx,P ( AyBn | ψ ) = (cid:90) θ ρ B ( x | ψAy ) dx (cid:90) cos θ A − ρ A ( x | ψ ) dx,P ( AnBy | ψ ) = (cid:90) − cos θ − ρ B ( x | ψAn ) dx (cid:90) θ A ρ A ( x | ψ ) dx,P ( AnBn | ψ ) = (cid:90) − cos θ ρ B ( x | ψAn ) dx (cid:90) θ A ρ A ( x | ψ ) dx, (10)and for the reversed order measurement G = BA , we can also write, mutatis mutandis : P ( ByAy | ψ ) = (cid:90) cos θ − ρ A ( x | ψBy ) dx (cid:90) cos θ B − ρ B ( x | ψ ) dx,P ( ByAn | ψ ) = (cid:90) θ ρ A ( x | ψBy ) dx (cid:90) cos θ B − ρ B ( x | ψ ) dx,P ( BnAy | ψ ) = (cid:90) − cos θ − ρ A ( x | ψBn ) dx (cid:90) θ B ρ B ( x | ψ ) dx,P ( BnAn | ψ ) = (cid:90) − cos θ ρ A ( x | ψBn ) dx (cid:90) θ B ρ B ( x | ψ ) dx. (11) The expressions (10)-(11) that we have just derived remain of course very general and can be used to modelwhatever experimental data coming from two 4-outcome measurements F and G , be them the result of twosequential 2-outcome measurements or not. So, what we need now to do is to add, in a gradual way, morestructure to the model, by enouncing a certain number of simplifying hypothesis that, when verified, im-pose certain constraints on the experimental probabilities. Of course, we need to add sufficient constraintsto make the model solvable, but at the same time we must take care not to add too many constraints, forthe model to remain sufficiently general and be able to describe also beyond-quantum structures. In otherterms, we have to use in combination Chatton’s anti-razor (no less than necessary) and Occam’s razor (nomore than necessary). We denote our first hypothesis weak compatibility . Weak compatibility . Two 2-outcome measurements A and B are said to be weakly compatible , rel-ative to each other and to the initial state ψ , if ρ A ( x | ψ ) = ρ A ( x | ψBy ) = ρ A ( x | ψBn ) ≡ ρ A ( x ), and ρ B ( x | ψ ) = ρ B ( x | ψAy ) = ρ B ( x | ψAn ) ≡ ρ B ( x ).In other terms, weak compatibility tells us that the fluctuations responsible for the actualization of ameasurement-interaction during the execution of the A -measurements (resp., the B -measurement) are thesame if the measurement is performed as a first measurement, on the initial state ψ , or following theexecution of measurement B (resp., A ), the latter being always executed on the initial state ψ .Our second hypothesis is meant to reduce the class of functions that are allowed to describe the prob-ability distributions ρ A ( x ) and ρ B ( x ), to enable us to explicitly perform the integrals in (10)-(11). We call9his hypothesis local uniformity . Local uniformity . A 2-outcome measurement A (resp. B ) is said to be locally uniform if the way itsrepresentative elastic band breaks is described by a probability distribution ρ A ( x ) (resp. ρ B ( x )) of theform: ρ A ( x ) = 12 (cid:15) A , x ∈ [ − , d A − (cid:15) A )1 , x ∈ [ d A − (cid:15) A , d A + (cid:15) A ]0 , x ∈ ( d A + (cid:15) A , , ρ B ( x ) = 12 (cid:15) B , x ∈ [ − , d B − (cid:15) B )1 , x ∈ [ d B − (cid:15) B , d B + (cid:15) B ]0 , x ∈ ( d B + (cid:15) B , , (12)where (cid:15) A , (cid:15) B ∈ [0 , d A ∈ [ − (cid:15) A , − (cid:15) A ], and d B ∈ [ − (cid:15) B , − (cid:15) B ].In other terms, a locally uniform measurement is characterized by an elastic band that can uniformly breakonly inside a connected internal region, not necessarily centered with respect to the origin of the Blochsphere. Eq. (6) can then be explicitly solved, to give: P ( A y | ψ ) = , cos θ A ∈ [ − , d A − (cid:15) A ) (1 + cos θ A − d A (cid:15) A ) , cos θ A ∈ [ d A − (cid:15) A , d A + (cid:15) A ]1 , cos θ A ∈ ( d A + (cid:15) A , , (13) P ( A n | ψ ) = , cos θ A ∈ [ − , d A − (cid:15) A ) (1 − cos θ A − d A (cid:15) A ) , cos θ A ∈ [ d A − (cid:15) A , d A + (cid:15) A ]0 , cos θ A ∈ ( d A + (cid:15) A , , (14)and similarly for (8). And of course, in the (non-singular) limit ( (cid:15) A , d A ) → (1 , sensitivity to pre-measurement state . Sensitivity to pre-measurement state . A 2-outcome measurement A (resp. B ) is said to be sensitiveto the pre-measurement state ψ , if the outcome probability P ( Ay | ϕ ) (resp., P ( By | ϕ )) is not a constantfunction of the initial state ϕ , when the latter is varied in a voisinage of ψ .Clearly, a locally uniform measurement A (resp., B ) will be sensitive to a pre-measurement state ψ only ifthe representative unit vector x ψ is such that: x ψ · a y ≡ cos θ A ∈ [ d A − (cid:15) A , d A + (cid:15) A ] (resp., x ψ · b y ≡ cos θ B ∈ [ d B − (cid:15) B , d B + (cid:15) B ]), i.e., if the point particle representative of the pre-measurement state orthogonally “falls”onto the internal, uniformly breakable part of the elastic, and not on the external unbreakable one (quantummeasurements, characterized by globally uniformly elastics, are of course sensitive to all pre-measurementstates). To draw a parallel with a similar situation in physics, think about when a potential barrier of general shape is approximatedby a square barrier, in order to explicitly solve the stationary Schrœdinger equation. Explicitly solving the model
In the following, we assume that the 2-outcome measurements A and B , forming the two 4-outcomesequential measurements F = AB and G = BA , are weakly compatible, locally uniform and sensitive to allthe pre-measurement states intervening during the execution of the two sequences of measurements. Thiswill be the case if the following 4 constraints are verified:cos θ, cos θ A ∈ [ d A − (cid:15) A , d A + (cid:15) A ] , cos θ, cos θ B ∈ [ d B − (cid:15) B , d B + (cid:15) B ] . (15)Indeed, if this is the case, the point particle will always orthogonally project onto the internal breakablesegment of both the A -elastic and B -elastic, either in the first measurement or in the second one. Then,for the AB sequential measurement, we have: P ( AyBy | ψ ) = 14 (1 + cos θ − d B (cid:15) B )(1 + cos θ A − d A (cid:15) A ) ,P ( AyBn | ψ ) = 14 (1 − cos θ − d B (cid:15) B )(1 + cos θ A − d A (cid:15) A ) ,P ( AnBn | ψ ) = 14 (1 + cos θ + d B (cid:15) B )(1 − cos θ A − d A (cid:15) A ) ,P ( AnBy | ψ ) = 14 (1 − cos θ + d B (cid:15) B )(1 − cos θ A − d A (cid:15) A ) , (16)and similarly, the BA sequential measurement gives: P ( ByAy | ψ ) = 14 (1 + cos θ − d A (cid:15) A )(1 + cos θ B − d B (cid:15) B ) ,P ( ByAn | ψ ) = 14 (1 − cos θ − d A (cid:15) A )(1 + cos θ B − d B (cid:15) B ) ,P ( BnAn | ψ ) = 14 (1 + cos θ + d A (cid:15) A )(1 − cos θ B − d B (cid:15) B ) ,P ( BnAy | ψ ) = 14 (1 − cos θ + d A (cid:15) A )(1 − cos θ B − d B (cid:15) B ) . (17)From the first two equations in (16), we obtain: P ( AyBy | ψ ) P ( AyBn | ψ ) = 1 + cos θ − d B (cid:15) B − cos θ − d B (cid:15) B , (18)from which we deduce that: d B (cid:15) B = cos θ(cid:15) B − P ( AyBy | ψ ) − P ( AyBn | ψ ) P ( AyBy | ψ ) + P ( AyBn | ψ ) . (19)Doing the same with the last two equations in (16), we also find: d B (cid:15) B = − cos θ(cid:15) B + P ( AnBn | ψ ) − P ( AnBy | ψ ) P ( AnBn | ψ ) + P ( AnBy | ψ ) . (20)Combining (19) and (20), we obtain:cos θ(cid:15) B = 12 (cid:20) P ( AnBn | ψ ) − P ( AnBy | ψ ) P ( AnBn | ψ ) + P ( AnBy | ψ ) + P ( AyBy | ψ ) − P ( AyBn | ψ ) P ( AyBy | ψ ) + P ( AyBn | ψ ) (cid:21) ,d B (cid:15) B = 12 (cid:20) P ( AnBn | ψ ) − P ( AnBy | ψ ) P ( AnBn | ψ ) + P ( AnBy | ψ ) − P ( AyBy | ψ ) − P ( AyBn | ψ ) P ( AyBy | ψ ) + P ( AyBn | ψ ) (cid:21) . (21)11onsidering also that P ( Ay | ψ ) = (1 + cos θ A − d A (cid:15) A ), and P ( Ay | ψ ) = P ( AyBy | ψ ) + P ( AyBn | ψ ), we have:cos θ A (cid:15) A − d A (cid:15) A = 2[ P ( AyBy | ψ ) + P ( AyBn | ψ )] − . (22)The same calculation for the BA measurement yields:cos θ(cid:15) A = 12 (cid:20) P ( BnAn | ψ ) − P ( BnAy | ψ ) P ( BnAn | ψ ) + P ( BnAy | ψ ) + P ( ByAy | ψ ) − P ( ByAn | ψ ) P ( ByAy | ψ ) + P ( ByAn | ψ ) (cid:21) ,d A (cid:15) A = 12 (cid:20) P ( BnAn | ψ ) − P ( BnAy | ψ ) P ( BnAn | ψ ) + P ( BnAy | ψ ) − P ( ByAy | ψ ) − P ( ByAn | ψ ) P ( ByAy | ψ ) + P ( ByAn | ψ ) (cid:21) , (23)as well as: cos θ B (cid:15) B − d B (cid:15) B = 2[ P ( ByAy | ψ ) + P ( ByAn | ψ )] − , (24)and inserting the second equation in (23) into (22), and the second equation in (21) into (24), explicitexpressions for cos θ A (cid:15) A and cos θ B (cid:15) B can also be obtained.It is important to observe that the model’s system of equations is underdetermined, in the sense that the8 outcome probabilities can determine all the parameters but one. This means that we are free to chooseone of the parameters, for instance (cid:15) A , and by doing so all the others will be automatically determined.When choosing a value for (cid:15) A , we have however to remember that, in accordance with (12), it needs toobey the constraint: (cid:15) A + d A ≤
1, as is clear that the uniformly breakable region of the elastic cannotextend outside of the Bloch sphere. This means that: (cid:15) A (1 + d A (cid:15) A ) ≤
1, giving: (cid:15) A ≤
11 + d A (cid:15) A . (25)It immediately follows from (25) that when the experimental probabilities are such that the right handside of the second equality in (23) is different from zero, then the data cannot be modeled by quantummechanics, but require a more general class of measurements. The same is true of course when the righthand side of the second equality in (21) is also different from zero, as is clear that a pure quantum modelingrequires both elastic bands to be globally (and not just locally) uniformly breakable. In this section we use the above explicit solution, obtained under the three simplifying hypothesis ofweak compatibility, local uniformity and sensitivity to pre-measurement state, to model some of the dataobtained in a Gallup poll conducted in 1997, as presented in a review of question order effects by Moore(2002). More precisely, we will use the probabilities given by Wang & Busemeyer (2013) (see also (Wanget al., 2014)), which are different from those in (Moore, 2002), as the authors have rightly excluded fromthe statistics those respondents who did not provided a “yes” or “no” answer (in other terms, they did notinclude in the total count the “don’t know” responses).
In one of the experiments, a thousand respondents were subjected to two interrogative contexts, consistingof a pair of questions asked in a sequence. The first question, let us denote it A , is: “Do you generally thinkBill Clinton is honest and trustworthy?” The second question, let us denote it B , is: “Do you generallythink Al Gore is honest and trustworthy?” Half of the subjects were subjected to the two questions in the12rder AB (first “Clinton” then “Gore”) and the other half in the reversed order BA (first “Gore” then“Clinton”). The experimental outcome probabilities are: P C / G ( AyBy ) = 0 . , P C / G ( AyBn ) = 0 . , P C / G ( AnBy ) = 0 . , P C / G ( AnBn ) = 0 . ,P C / G ( ByAy ) = 0 . , P C / G ( ByAn ) = 0 . , P C / G ( BnAy ) = 0 . , P C / G ( BnAn ) = 0 . . (26)The above data clearly show question order effects, considering that the probabilities in each of the 4columns in (26) are sensibly different. To investigate the mathematical structure of these effects, we nowinsert the above values in (21). This gives:cos θ(cid:15) B = 12 (cid:20) . − . . . . − . . . (cid:21) = 11127972073357 ,d B (cid:15) B = 12 (cid:20) . − . . . − . − . . . (cid:21) = − , (27)whereas (22) gives: cos θ A (cid:15) A − d A (cid:15) A = 2(0 . . − . . (28)Doing the same with (23), we obtain:cos θ(cid:15) A = 12 (cid:20) . − . . . . − . . . (cid:21) = 7167451134784 ,d A (cid:15) A = 12 (cid:20) . − . . . − . − . . . (cid:21) = 1752791134784 , (29)and from (24), we have: cos θ B (cid:15) B − d B (cid:15) B = 2[0 . . − . . (30)From (28) and (29), we also find:cos θ A (cid:15) A = 1732500 + 1752791134784 = 158628783709240000 , (31)and (30) and (27) give: cos θ B (cid:15) B = 327625 − . (32)In summary, writing the obtained exact values in approximate form, to facilitate their reading, we have: d A (cid:15) A ≈ .
15 cos θ A (cid:15) A ≈ . , cos θ(cid:15) A ≈ . ,d B (cid:15) B ≈ − . , cos θ B (cid:15) B ≈ . , cos θ(cid:15) B ≈ . . (33)At this point, we need to choose a specific value for (cid:15) A , thus allowing us to determine the values of allthe other parameters and obtain an exact explicit determination of the experimental probabilities. Before Because of a rounding error, the probabilities given in Wang & Busemeyer (2013) do not exactly sum to 1, but to 0 . AB measurement, and to 1 . BA measurement. Since our solution requires the probabilities to exactly sumto 1, we have corrected the value of P C / G ( AnBn ), from 0 . . P C / G ( BnAn ), from 0 . . (cid:15) A = (cid:15) B , d A = d B ) would clearly be unable tomodel the data, and that none of the two elastics can be symmetric ( d A = 0 and/or d B = 0). Also, theirbreakable region cannot be of the same size ( (cid:15) A = (cid:15) B = (cid:15) ). In other terms, the two measurements’ elasticbands have to be very different and non-symmetric, which means that the structure of the probabilisticdata is irreducibly non-Hilbertian (we recall that the Hilbertian structure is recovered when the elasticsare all globally uniform).In view of (25), we have: (cid:15) A ≤
11 + = 11347841310063 ≈ . . (34)Considering that d B (cid:15) B is negative, for (cid:15) B we have to consider the constraint: d B − (cid:15) B ≥ −
1. This meansthat: (cid:15) B (1 − d B (cid:15) B ) ≤
1, i.e., (cid:15) B ≤ − dB(cid:15)B , giving: (cid:15) B ≤
11 + = 20733572687194 ≈ . . (35)These maximum values are clearly very different form the values (cid:15) A = (cid:15) B = 1, characterizing quantummechanics. For simplicity, we choose (cid:15) A = . We then obtain the exact values: (cid:15) A = 12 , (cid:15) B = 14860682629652525568461696 d A = 1752792269568 , d B = − θ = 7167452269568 , cos θ A = 1586287831418480000 , cos θ B = 21096644663643157848028856000 , (36)or in approximate form: (cid:15) A = 0 . , (cid:15) B ≈ . d A ≈ . , d B ≈ − . θ ≈ . , cos θ A ≈ . , cos θ B ≈ . . (37)We can check that (36)-(37) obey (15). This is clearly the case considering that (using the approximatevalues, for clarity): [ d A − (cid:15) A , d A + (cid:15) A ] ≈ [ − . , . .
32 and 0 .
04 of cos θ andcos θ A , respectively. Also, [ d B − (cid:15) B , d B + (cid:15) B ] ≈ [ − . , . .
32 and 0 . θ and cos θ B , respectively. A last point we need to check, to be sure that our solution is consistent, isthat it is always possible to find a 3-dimensional unit vectors x ψ , representing the initial state in the Blochsphere, which can make the angles cos θ A and cos θ B , when projected onto the two unit vectors a y and b y ,corresponding to the yes-outcomes of the A and B measurements, respectively, with a y · b y = cos θ . Thisis indeed the case, as we can always choose: a y = (1 , , b y = (cos θ, sin θ, , x ψ = (cos θ A , cos θ B − cos θ cos θ A sin θ , x ) , (38)where x is such that: (cid:107) x (cid:107) = 1.Figure 3 illustrates our modeling of the Clinton/Gore data, for (cid:15) A = . Of course, by consideringdifferent choices for the free-parameter (cid:15) A , different equivalent representations will be obtained. However,structurally speaking, they will all be very similar. The two black dots in Fig. 3 indicate the two positionsof the particle representative of the initial state ψ , when it lands onto the two elastic bands. We seethat the on-elastic position for the A -elastic is almost in the middle of its breakable region, whereas the14 igure 3: Two non-uniform and non-symmetric elastic bands describing the data of the Clinton/Gore experiment. Theirrelative angle is θ ≈ ◦ . The initial state vector x ψ is not represented in the drawing, being not located on the same plane ofthe two elastic bands (see Fig.1). Only its projection points onto the two elastic bands are represented (the two black dots).We can observe that the end points of the two elastics orthogonally project onto the internal breakable region of the otherelastic, in accordance with the ‘sensitivity to pre-measurement state’ hypothesis. on-elastic position for the B -elastic is much closer to the border of the breakable region, towards the By outcome. This is in accordance with the fact that the probability of answering “yes” to the Clintonquestion (the A -measurement), P C / G ( Ay ) = P C / G ( AyBy )+ P C / G ( AyBn ) ≈ .
53, irrespective of the answergiven to the subsequent Gore question (what is called the non-comparative context by Moore (2002)), isclose to one-half, whereas the probability of answering “yes” to the Gore question (the B -measurement), P C / G ( By ) = P C / G ( ByAy ) + P C / G ( ByAn ) ≈ .
76, irrespective of the answer to the subsequent Clinton’squestion, is much higher (with a significant difference of 0 . P C / G ( Ay ) and P C / G ( By ). On the other hand, the relative orientation of the two elastics (the angle θ ), theextension and displacement of their locally uniform breakable regions (defined by their “ (cid:15) - d ” parameters),is what accounts for the subtle order effects that are observed, which are too complex to be modeled byonly using two globally uniform elastic bands, i.e., two pure quantum measurements. In another experiment reported by Moore, always performed on a thousand respondents, the interrogativecontexts consisted in a pair of questions about the baseball players Pete Rose and Shoeless Joe Jackson.More precisely, the A question was: “Do you think Rose should or should not be eligible for admission tothe Hall of Fame?” Similarly, the B question was: “Do you think Jackson should or should not be eligiblefor admission to the Hall of Fame?” The obtained experimental probabilities are (also in this case, we usethe revisited probability data presented in (Wang & Busemeyer, 2013; Wang et al., 2014)): P R / J ( AyBy ) = 0 . , P R / J ( AyBn ) = 0 . , P R / J ( AnBy ) = 0 . , P R / J ( AnBn ) = 0 . ,P R / J ( ByAy ) = 0 . , P R / J ( ByAn ) = 0 . , P R / J ( BnAy ) = 0 . , P R / J ( BnAn ) = 0 . . (39)15his time we have: cos θ(cid:15) B = 12 (cid:20) . − . . . . − . . . (cid:21) = 5121331118780 ,d B (cid:15) B = 12 (cid:20) . − . . . − . − . . . (cid:21) = 4888111118780 , cos θ(cid:15) A = 12 (cid:20) . − . . . . − . . . (cid:21) = 1554247024970071 ,d A (cid:15) A = 12 (cid:20) . − . . . − . − . . . (cid:21) = − , (40)and from the two equalities:cos θ A (cid:15) A − d A (cid:15) A = 2(0 . . − .
324 = 81250 , cos θ B (cid:15) B − d B (cid:15) B = 2(0 . . − , (41)we obtain: cos θ A (cid:15) A = 14012170016242517750 , cos θ B (cid:15) B = 112525303279695000 . (42)Writing the obtained values in approximate form, we thus have: d A (cid:15) A ≈ − . , cos θ A (cid:15) A ≈ . , cos θ(cid:15) A ≈ . ,d B (cid:15) B ≈ . , cos θ B (cid:15) B ≈ . , cos θ(cid:15) B ≈ . . (43)As we did for the previous data, we choose (cid:15) A = . It is then easy to check that all the constraints areobeyed, and one finds: (cid:15) A = 12 , (cid:15) B = 869430229330012787997371443 ,d A = − , d B = 12662217171954262665790481 , cos θ = 777123524970071 , cos θ A = 140121700112485035500 , cos θ B = 174892114611841639399868572150 , (44)or in approximate form: (cid:15) A = 0 . , (cid:15) B ≈ . d A ≈ − . , d B ≈ . θ ≈ . , cos θ A ≈ . , cos θ B ≈ . . (45) In the previous sections we have provided an exact modeling of some paradigmatic data exhibiting questionorder effects, using a mathematical formalism that is more general than the standard formalism of quantummechanics. If on one hand it was to be expected that the data could not be exactly modeled by the Born rule,16 igure 4: Two non-uniform and non-symmetric elastic bands describing the data of the Rose/Jackson experiment. Theirrelative angle is θ ≈ ◦ . The initial state vector x ψ is not represented in the drawing, being not located on the same plane ofthe two elastic bands (see Fig.1). Only its projection points onto the two elastic bands are represented (the two black dots).We can observe that the end points of the two elastics orthogonally project onto the internal breakable region of the otherelastic, in accordance with the ‘sensitivity to pre-measurement state’ hypothesis. on the other hand it may come as a little surprise that the structure of the Clinton/Gore probabilities are notonly irreducibly non-Hilbertian, but also not at all close to a Hilbertian model. Indeed, the Clinton/Goredata are structurally similar to the Rose/Jackson data, despite of the fact that they are generally consideredto be very different, considering that there is a test, expressed by a so-called QQ-equality , which is passed(although not in an exact way) by the Clinton/Gore data, but not by the Rose/Jackson data (Wang &Busemeyer, 2013; Wang et al., 2014).The explanation that was given for this difference, is that the validity of the QQ-equality in psychologicalorder effect measurements would depend on the critical assumption that, quoting from Wang & Busemeyer(2013): “the only factor that changes the context or the state for answering a question is answering itspreceding question.” In the Rose/Jackson data this assumption is believed to be violated, as during thesurvey the subjects were provided with some background information. For instance, before the Pete Rosequestion, the following information was typically given: “As you may know, former Major League playerPete Rose is ineligible for baseballs Hall of Fame due to charges that he had gambled on baseball games.”Information was also provided before addressing the Joe Jackson question, typically: “As you may know,Shoeless Joe Jackson is ineligible for baseballs Hall of Fame due to charges that he took money fromgamblers in exchange for fixing the 1919 World Series.”This means that, quoting again from Wang & Busemeyer (2013): “the initial belief state when Rose/Jacksonwas asked first was affected by the information provided about the particular player. Furthermore, thecontext for the second question was changed not only by answering the first question but also sequentiallyby the additional background information on the player in the second question. [...] From the perspectiveof the QQ model, actually the two orders in the data set were Rose background-Rose question-Jacksonbackground-Jackson question versus Jackson background-Jackson question-Rose background-Rose ques-tion.”According to Wang & Busemeyer (2013), this would be sufficient to explain why the QQ-equality is(almost) obeyed by the Clinton/Gore data but manifestly disobeyed by the Rose/Jackson data. But thisis quite surprising, considering that it is always possible to incorporate the background information intothe questions themselves, for instance defining the following modified interrogative contexts: “As you may17now, former Major League player Pete Rose is ineligible for baseballs Hall of Fame due to charges that hehad gambled on baseball games; do you think Rose should or should not be eligible for admission to theHall of Fame?” and: “As you may know, Shoeless Joe Jackson is ineligible for baseballs Hall of Fame dueto charges that he took money from gamblers in exchange for fixing the 1919 World Series; do you thinkJackson should or should not be eligible for admission to the Hall of Fame?”So, we now have two rephrased questions, with no background information provided anymore (as itbecame part of the question), and one would expect the QQ-equality to also be (almost) obeyed in this case.There is undoubtedly something that needs to be clarified here, not only as regards the true content of theQQ-equality (is it really a quantum test?), but also the proper way to interpret the quantum formalism,and its GTR-model generalization, when used to model cognitive situations.Let us have a closer look to the QQ-equality, derived by Wang & Busemeyer (2013) as part of their QQ-model, where the subjects’ beliefs are represented by state vectors in a Hilbert space and the interrogativecontexts by generally non commuting orthogonal projection operators. Using the property of the scalarproduct (or of the trace, in the more general setting where density matrices are also allowed to representstates), it is possible to demonstrate the following quantum mechanical equality (Busemeyer & Bruza,2012; Wang & Busemeyer, 2013): q ≡ [ P ( AyBn | ψ ) + P ( AnBy | ψ )] − [ P ( ByAn | ψ ) + P ( BnAy | ψ )] = 0 , (46)which by using (2), can also be written in the form: q ≡ [ P ( ByAy | ψ ) − P ( AyBy | ψ )] + [ P ( BnAn | ψ ) − P ( AnBn | ψ )] = 0 . (47)Inserting the explicit solutions (16)-(17) into (47), one finds, after some simple algebra: q = cos θ (cid:18) (cid:15) A − (cid:15) B (cid:19) − (cid:18) d A (cid:15) A cos θ B (cid:15) B − d B (cid:15) B cos θ A (cid:15) A (cid:19) . (48)As we explained, the pure quantum situation is recovered when the elastic bands become globallyuniform, i.e., in the limit ( (cid:15) A , d A ) , ( (cid:15) B , d B ) → (1 , q →
0. This is so because in thislimit both terms in the difference (48) tend to zero. The first term quantify the difference in terms of theextension of the “potentiality region” characterizing the two measurements, i.e., the difference in terms ofthe number of hidden measurement-interactions that are available to be actualized (characterizing, in asense, the amount of indeterminism expressed by the interrogative context). The second term, instead, ismore articulated, and expresses the relative asymmetry of the two measurements, responsible for part ofthe observed order effects. For simplicity, let us call the first term the “relative indeterminism” and thesecond term the “relative asymmetry” contribution.Inserting into (48) the values (36) of the parameters for the Clinton/Gore data, we find: q = 2232840321174705624699776 (cid:124) (cid:123)(cid:122) (cid:125) ≈ . − (cid:124) (cid:123)(cid:122) (cid:125) ≈ . = − − . . (49)We immediately see in (49) that the reason why the Clinton/Gore probabilities (almost) obey (47) is verydifferent from the reason why the equality is obeyed by quantum probabilities. Here q is almost zerobecause the “relative indeterminism” and the “relative asymmetry” contributions almost perfectly canceleach other, and not because they are both individually close to zero, as it should be the case in a close to Note that in (Wang & Busemeyer, 2013) the slightly different value q = − . q -value for the Rose/Jackson data: q = 460060721515755872032066760 (cid:124) (cid:123)(cid:122) (cid:125) ≈ . − (cid:20) − (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ≈− . = 7575000 = 0 . . (50)We see that the “relative indeterminism” and the “relative asymmetry” contributions are of the same “size”than those of the Clinton/Gore data, the difference being here that they have opposite sign, and thereforecannot compensate each other. We also observe that the maximum absolute value | q max | that can be takenby q in a general sequential measurement is 1, which means that in the Clinton/Gore experiment we havethat | q | is 0 .
32% of | q max | , whereas in the Rose/Jackson experiment | q | is 15% of | q max | .To make more stringent our point that the Clinton/Gore data are decidedly non-Hilbertian, let ususe our solution to derive additional relations that a quantum system must obey, but that are flagrantlydisobeyed by the Clinton/Gore data. When (cid:15) A = 1 and d A = 0, the right hand side of the second equationin (23) has to be zero. This is the case if and only if:[ P ( BnAn | ψ ) − P ( BnAy | ψ )][ P ( ByAy | ψ ) + P ( ByAn | ψ )] (51)= [ P ( BnAn | ψ ) + P ( BnAy | ψ )][ P ( ByAy | ψ ) − P ( ByAn | ψ )] . (52)Simplifying the above expression, we thus obtain: q ≡ P ( ByAn | ψ ) P ( BnAn | ψ ) − P ( BnAy | ψ ) P ( ByAy | ψ ) = 0 . (53)Reasoning in the same way with the B -measurement, (21) yields the equality: q ≡ P ( AyBn | ψ ) P ( AnBn | ψ ) − P ( AnBy | ψ ) P ( AyBy | ψ ) = 0 . (54)Also, when (cid:15) A = (cid:15) B = 1, the right hand sides of the first equations in (21) and (23) have to coincide. Aftera simple calculation, this provides the additional equality: q ≡ P ( AnBy | ψ ) P ( BnAn | ψ ) − P ( AnBn | ψ ) P ( BnAy | ψ ) = 0 , (55)and of course, by combining together the above three equalities, additional relations can be found.The maximum absolute values that can be taken by these three quantities are | q max1 | = | q max2 | = 0 . | q max3 | = 1. If we calculate the values of q , q and q , for the Clinton/Gore data, we find q ≈ . q ≈ − .
073 and q ≈ . | q | , | q | and | q | are now respectively 11 . .
2% and2 .
9% of the maximum values, which clearly represent strong violations of the quantum predictions. So,we can say that it is almost by a fortuitous circumstance that the Clinton/Gore probabilities (almost)obey (46), as the same probabilities violate in a very flagrant way other critical quantum (parameter-free)equalities.A similar conclusion was recently reached by Boyer-Kassem et al. (2015), who by reasoning directly withthe notion of conditional probabilities also pointed out the insufficiency of the QQ-equality in characterizinga (non-degenerate) quantum probability model. Indeed, as we have also seen in the previous sections, in thequantum formalism transition probabilities from one state to another state are expressions of conditionalstatements, and because of the conjugate symmetry of the inner product the probability of a transitionwill not depend on its order (a property requiring the elastics to be globally uniform). This introducesa symmetry also in the conditional probabilities, that Boyer-Kassem et al. (2015) organize in a set ofequations that they call the Grand Reciprocity equations, which need also to be satisfied by all quantum19odels, as the three equations (53)-(55) and the QQ-equation, need to be, but are not satisfied by most ofthe existing data sets.We therefore see that a pure quantum model is not a good one to account for typical probabilistic dataexhibiting question order effects. Of course, one may object that by considering additional dimensions,and maybe degenerate measurements, it could still be possible to model the experimental data by meansof the sole Born rule and the projection postulate. However, as rightly mentioned by Boyer-Kassem et al.(2015): “Introducing dimensions of degeneracy should be justified, so as not to be accused of being just adhoc.” But as we have shown, this is not necessary, as to exactly model the data one only needs to admit amore general class of measurements, also able to describe non-uniform fluctuations, as it can be naturallydone in the GTR-model. The advantage and relevancy of using the latter is that it also allows handlinganother situation that is problematic for the standard quantum formalism: response replicability . This isthe topic we are going to address in the next sections.
Quoting from Khrennikov et al. (2014): “[...] quantum theory (QT) encounters difficulties in accountingfor some very basic empirical properties. In opinion polling [...], there is a class of questions such that arepeated question is answered in the same way as the first time it was asked. This agrees with the L¨udersprojection postulate [...]. In many situations, we also expect that for a certain class of questions the responseto two replications of a given question remains the same even if we insert another question in between andhave it answered. This property can only be handled by QT if the conventional observables representingdifferent questions all pairwise commute, i.e., can be assigned the same set of eigenvectors. This, in turn,leads to a strong prediction: the joint probability of two responses to two successive questions does notdepend on their order. This prediction is known to be violated for some pairs of questions. The explanationof the ‘question order effect’ is in fact one of the most successful applications of QT in psychology (Wang &Busemeyer, 2013), but it requires non-commuting observables, and these, as we have seen, cannot accountfor the repeated answers to repeated questions.”In the previous sections we have slightly “resized” this last statement regarding the success of standardquantum theory in accounting question order effects, as we have seen that the probability models theseeffects are able to generate are generally incompatible with the specific structure of the Born rule andcan only be represented in more general quantum-like frameworks, like the one provided by the GTR-model. As we are now going to explain, different from quantum theory, the GTR-model is also able tojointly handle, in a consistent way, question order effects and response replicability. The reason why thisis possible is that in the GTR-model not only contexts can change states, through the mechanism of thebreaking and collapsing of the elastic bands, but also states can change contexts, by allowing the probabilitydistributions describing the way the elastics break to also depend on the outcomes obtained in the previousmeasurements, exactly in the same way we humans, once we have formed an opinion, we don’t need toform it again, by definition of what an opinion (and consequently an opinion poll) is.So, we start from the observation that there is a nonempty class of measurements such that, whenthey are performed in successive trials, if one of the measurements in the sequence is equal to a previousmeasurement, the same outcome that was previously obtained will be obtained again, but with certainty,i.e. with probability equal to 1. For instance, considering the A measurement consisting of the interrogativecontext “Do you generally think Bill Clinton is honest and trustworthy?” with outcome Ay and An , atypical respondent, actualizing a non pre-existing answer the first time s/he is subjected to A , not havinga predetermined opinion regarding the honesty and trustworthiness of Clinton, when subjected a secondtime to A , s/he will usually respond in the same way. In other terms, the possible outcomes of thesequential measurement AA are only AyAy , and
AnAn , as the outcomes
AyAn and
AnAy are generally20ot observed. By this we are not saying that all measurements are necessarily of this kind, but that, asremarked in Khrennikov et al. (2014), there certainly is a vast class of measurements for which this is thecase.In quantum mechanics, measurements exhibiting this kind of immediate repeatability, i.e., not leadingto a new outcome when they are repeated, are called (according to Pauli’s classification) “of the first kind,”and are precisely those prescribed by the L¨uders-von Neumann projection postulate. The more generalclass of measurements described in the GTR-model are also of the first kind, as is clear that when weperform the same measurement a second time, being the abstract point particle already located in oneof the two end points of the elastic structure, its position cannot be further changed by the collapse ofa new elastic, stretched along the same two end points. So, both the idealized measurements describedin the standard quantum formalism, and those described in the GTR-model, exhibit this nice property ofproducing a stable change of the state of the entity, which can no longer change under the influence of thesame measurement context (clearly an ideal situation to identify an outcome of an experiment by meansof a so-called eigenstate ).The situation is different when the measurement A is repeated not immediately after A , but after havingperformed an inter-measurement B , like the one corresponding to the question: “Do you generally think AlGore is honest and trustworthy?” In other terms, we now consider the sequential measurement ABA , andaccording to our hypothesis, subjects will generally respond to the third A -question in exactly the sameway they have responded to the first A -question, also when in between they have been subjected to the B -question. We are not interested here in explaining what are the psychological mechanisms behind thisrepeatability of the outcome (desire of coherence, learning, fear of being judged when changing opinion,etc.), but in understanding if it can be naturally modeled, in a way that is compatible with the observedquestion order effects. In other terms, compatibly with our previous modeling of the AB and BA measure-ments, we have to investigate if it is possible to model a measurement ABA , only producing as its possibleoutcomes the four outcomes:
AyByAy , AyBnAy , AnByAn and
AnBnAn , or equivalently a measurement
BAB , only producing the four possible outcomes:
ByAyBy , ByAnBy , BnAyBn and
BnAnBn .As demonstrated by Khrennikov et al. (2014), this situation cannot be described by the standard quan-tum formalism, as replicability is only possible if the A and B measurements are described by commutingoperators, which would then be in conflict with the existence of question order effects. Of course, aswe have shown, the experimental incompatibility defined by non-commuting Hermitian operators is perse insufficient to correctly model the data, but certainly no order effects, of whatever kind, are possiblefor compatible observables. On the other hand, and different from quantum mechanics, the GTR-modelallows for the description of ‘cognitive adjustments in accordance with previous experiences,’ by naturallyincorporating them in a process of change of the probability distributions describing the breaking of theelastic structures.In that respect, let us come back to our hypothesis of weak compatibility, according to which theprobability distribution describing the breaking of the A -elastic is the same if the measurement is performedas a first measurement, or following the B -measurement, and similarly that the probability distributiondescribing the breaking of the B -elastic is the same if the measurement is performed as a first measurement,or following the A -measurement. If we assume that the probability distribution ρ A describes the way It is worth mentioning that in quantum physics many bona fide measurements are not of the first kind, i.e., the post-measurement state will be in general substantially different from that predicted by the projection postulate. A simple exampleof a measurement that is not of the first kind is the counting of the photons’ number of an electromagnetic field, by meansof a photo-detector. Here the state after the measurement is always a vacuum state, irrespective of the measurement result.Another example is the measurement of the nuclear spin by free induction decay. In this case the state after the measurement isthe thermal statistical mixture of upper and lower spin states, again irrespective of the measurement result. What is importantto observe is that the validity of the Born rule does not depend on that of the projection postulate, and that the applicabilityof the latter needs to be justified by taking into account the specificities of the measurement protocol. B -question does not alter their way of carrying out the evaluation (and samefor ρ B ). This means that the observed order effects can be understood as resulting only from the “changeof perspective” induced by answering A before or after B , and not by a change in one’s way of evaluatingthe situation.However, our weakly compatibility hypothesis does not say anything about a possible change of theprobability distribution ρ A (resp. ρ B ) following the AB (resp., BA ) measurement. In case measurement A is repeated a second time, in the direct sequence AA , we can assume for simplicity that ρ A doesn’t changeafter the first measurement. In any case, the measurement being of the first kind, the point particle wouldnot be able to further change its position, even if in the second measurement ρ A would have changed. Butwhat about the change of ρ A following the B measurement? In other terms, what should be the changeinduced by the sequential measurement AB , in order to properly account for the replicability effect?Let ρ A ( x ) ≡ ρ A ( x | ψ ) be the initial probability distribution describing the A -measurement. Accordingto weak compatibility, we have ρ A ( x | ψ ) = ρ A ( x | ψBy ) = ρ A ( x | ψBn ), i.e., the probability distribution doesnot change if the B measurement is performed first, whatever its outcome. However, here we are in thesituation where it is the A -measurement that is performed first, in the sequence AB . So, to the fouradmissible outcomes AyBy , AyBn , AnBy and
AnBn , we can let correspond to the following transitionsfor both the states and the probability distributions: ψ → Ay → By, ψ → Ay → Bn, ψ → An → By, ψ → An → Bn,ρ A → ρ A → ρ AyByA , ρ A → ρ A → ρ AyBnA , ρ A → ρ A → ρ AnByA , ρ A → ρ A → ρ AnBnA , (56)where we have defined the truncated and renormalized probability distributions: ρ AyByA ( x ) = ρ A ( x ) (cid:82) cos θ − ρ A ( x ) dx χ [ − , cos θ ) ( x ) , ρ AyBnA ( x ) = ρ A ( x ) (cid:82) − cos θ − ρ A ( x ) dx χ [ − , − cos θ ) ( x ) ,ρ AnByA ( x ) = ρ A ( x ) (cid:82) θ ρ A ( x ) dx χ (cos θ, ( x ) , ρ AnBnA ( x ) = ρ A ( x ) (cid:82) − cos θ ρ A ( x ) dx χ ( − cos θ, ( x ) , (57)with χ I ( x ) the characteristic function of the interval I . Consider for instance the probability P ( AyByAy | ψ ),for the outcome AyByAy , in the sequential measurement
ABA . It is clearly given by the product of thethree conditional probabilities: P ( Ay | ψAyBy ) P ( By | ψAy ) P ( Ay | ψ ), with: P ( Ay | ψAyBy ) = (cid:90) cos θ − ρ AyByA ( x ) dx = (cid:82) cos θ − ρ A ( x ) dx (cid:82) cos θ − ρ A ( x ) dx = 1 , (58)in accordance with our hypothesis of replicability, and same thing for the other admissible outcomes.Figure 5 illustrates this process. Of course, the same holds true, mutatis mutandis for the changesof the probability distribution ρ B in the sequence of measurements BAB . This means that once we haveperformed either AB or BA , the outcomes of additional A or B measurements will become perfectlydeterministic. For instance, if the measurement A gave Ay , and the subsequent measurement of B gave Bn , then, when performing again the A -measurement, the associated elastic band will be characterized by ρ AyBnA ( x ), and if we perform again the B -measurement, the associated elastic band will be characterizedby ρ BnAyB ( x ) (with obvious notation), and from this point forward the probability distributions describingthe two measurements will not change any more, so that subsequent alternations of the measurements A and B will only cause the point particle to transition from (the unit vector representative of) Ay to Bn , to Ay , to Bn , and so on, in a perfectly deterministic way, in accordance with our expectation that answerswill generally be repeated once they have been actualized.22 igure 5: The sequential measurement ABA . Figure (a) represents the situation before the first A -measurement. Therepresentative point particle is located in x ψ , at the surface of the sphere, outside of the plane of the two elastic bandsdescribing the measurements A and B (and therefore cannot be represented in the figure), here described by the probabilitydistributions ρ A and ρ B , given by (12), with the parameters taking the values (36) of the Clinton/Gore data. Following thefirst A -measurement, we have assumed that the indeterministic breaking of the elastic has produced outcome Ay , as indicatedin Figure (b). Then, in the second B -measurement, we have assumed that the transition Ay → By has occurred, to which,according to (56)-(57), also corresponds the transition: ρ A → ρ AyByA , as illustrated in Figure (c). The outcome Ay of the third A -measurement, described in Figure (d), is now certain in advance, in accordance with the hypothesis of replicability, andthis By → Ay deterministic transition is associated with the probability distribution change: ρ B → ρ ByAyB . Then, subsequent B -measurements and A -measurements will not change anymore the nature of the elastic bands and state transitions will beperfectly deterministic. It is worth observing that the prescription (56)-(57), about how the probability distribution ρ A hasto change, in order to assure replicability of the obtained answer when A is repeated, following a B -measurement, is a minimalistic one (no more than necessary), and is somehow reminiscent of the prescrip-tion of the L¨uders-von Neumann projection postulate for the quantum states transitions. Indeed, we justmake unbreakable that specific segment of the A -elastic band that would produce the unwanted transition,but we do not alter the way the elastic can break everywhere else. This means that we also leave open thepossibility that a new measurement C , once performed, say following the sequence AB , could again make asubsequent A -measurement indeterminate. In other terms, we consider the possibility that a measurements C exist such that when A is performed again, following the sequence ABC , the outcome could be differentthan what was obtained in the first A -measurement.As an example, imagine that C corresponds to the following question: “Newspapers report today a new23candal regarding Bill Clinton: does this diminish your trust in him?” Imagine that the answer to the first A -measurement was Ay , that the answer to the subsequent B -measurement was By , and that the answerto this additional C -measurement was also affirmative, i.e. Cy . Here of course it is not important if thestatement that the newspapers reported a new scandal is true or not. What is important is that, at thepragmatic level, this way of formulating the question is able to confuse some of the respondents, so muchso that one can expect that, when asking A again, following the Cy outcome, the two outcomes Ay and An to be again available to be actualized, with a certain probability, thus interrupting the replicability of theoutcomes of the previous A -measurement. Also, one expects that if the answer to the C -question was Cn (suggesting for instance that the respondent did not believe the alleged scandal reported, or supposedlyreported, in the newspapers), the replicability of the A -measurements will be preserved.In the GTR-model a situation of this kind can still be described, by modeling the C -measurement bymeans of an elastic band oriented in such a way that when the point particle orthogonally “falls” ontothe A -elastic from the position c y , describing outcome Cy , it will land inside the interval where ρ AyByA ( x )is breakable, i.e., cos θ (cid:48) ≡ c y · a y ∈ [cos θ, − (cid:15) A + d A ], so producing, again, an indeterministic process (theprobabilities of which will depend on the exact value taken by the angle θ (cid:48) ). In other terms, when theoutcome of the ABC -measurement is
AyByCy , we can consider the transition: ρ AyByA → ρ AyByCyA , where ρ AyByCyA can simply be taken to be identical to ρ AyByA (no change), whereas when the outcome is
AyByCn ,we can consider the transition ρ AyByA → ρ AyByCnA , with: ρ AyByCnA ( x ) = ρ AyByA ( x ) (cid:82) − cos θ (cid:48) − ρ AyByA ( x ) dx χ [ − , − cos θ (cid:48) ) ( x ) , (59)so that in this case an additional A -measurement will give Ay with certainty.The above was just an example meant to illustrate the great versatility of the GTR-model in handlingin a consistent way, compatibly with question order effects, situations not only of repeatability, but alsoof partial (interrupted) repeatability, when additional particular measurements are considered that areable to change the mindset of the respondents in a way that can break the previously obtained responsereplicability.On the other hand, if one is just interested in modeling idealized situations where once a measure-ment has been performed, say the A -measurement, the same answer will always be replicated, when themeasurement is repeated, independently of the number and nature of the additional measurements thatare performed before the repetition, then for the change of the associated ρ A -probability distribution itis sufficient to consider the following simple rule: ρ A ( x ) → δ ( x + 1), in case the answer was Ay , and ρ A ( x ) → δ ( x − An , where δ denotes the Dirac delta function. In other terms,following the A -measurement, with initial probability distribution ρ A , a subsequent A -measurement willsimply be described by an elastic band that is only breakable in its end point opposite to the previouslyobtained outcome, thus guaranteeing that it will be replicated for all pre-measurement states. In the previous sections we have shown that question order effects and response replicability can both becompatibly described in the GTR-model. This is because, different from the standard quantum formalism,the GTR-model provides an explicit representation of the measurements, by means of a mechanism ofactualization of potential measurement-interactions. This mechanism opens to the possibility of associatingdifferent processes of actualization of a potential outcomes to a same set of outcomes, as described by thedifferent possible ρ -probability distributions associated with an elastic band with a given orientation in theBloch sphere, whilst in the quantum formalism only the uniform probability distribution is admitted.24n other terms, the GTR-model, thanks to its greater structural richness, allows for a distinction between‘a question and its possible answers,’ and ‘the way a respondent selects one of these answers.’ Differentrespondents are in principle associated with different set of probabilities for the outcomes, characterizingtheir ‘personal way of choosing an answer,’ whereas in the quantum formalism the different subjects allnecessarily choose in the same way, as if they were perfect “Bornian clones”. This means that when weuse the quantum formalism, order effects can only be described by means of the relative orientations ofthe different states in the Bloch sphere. This however constitutes a severe shortcoming in the descriptionof psychological measurements (and in fact, possibly also in the description of physical measurements, butthis is another story), as the Hilbertian geometry is too specific to account for the experimentally observedprobabilities.As we have explained in Aerts & Sassoli de Bianchi (2015a,b), the breaking of the elastics (and moregenerally the disintegration of the hyper-membranes, when more general N -outcome interrogative contextsare considered) expresses in an intuitive way what we humans can typically feel when confronted withdecisional contexts. Indeed, when a subject is confronted with a question (and more generally with adecision), say the A -question “Do you generally think Bill Clinton is honest and trustworthy?” andthe associated possible answers “Clinton is honest and trustworthy” ( Ay ) or “Clinton is not honest andtrustworthy” ( An ), this will automatically build a mental (neural) state of equilibrium, which results fromthe balancing of the tensions between the initial state (we will make precise in a moment what this initialstate is about) and the two mutually excluding and competing answer-states.The creation of this mental state of equilibrium is described by the abstract point particle plunging intothe sphere and reaching its on-elastic position, giving rise to two “tension lines” going from its positionto the two end points, representative of the two competing outcomes. At some moment, this mentalequilibrium will be disturbed, in a non-predictable way, and the disturbance will cause an irreversibleprocess during which, almost instantaneously, the particle will be drawn to one of the two vertices of theelastic (representing the answers). This “symmetry breaking” process is described in the model by therandom manifestation of a breaking point, which by causing the collapse of the elastic band breaks thetensional equilibrium that had previously been built, hence the name “tension-reduction” given to themodel.If the elastic band is representative of an aspect of the mind of the subject, here generally understood as amemory structure sensitive to meaning, how should we interpret the abstract point particle, which interactswith it? In other terms, how should we interpret the initial state and the final outcome states? From theperspective of the GTR-model, they should not be interpreted as a ‘description of the subject’s beliefs,’on the issue in place, but as a ‘description of an element of conceptual reality which is independent of thehuman mind of the person that can possibly interact with it.’ In the case of the Clinton/Gore experiment,this element of conceptual reality is nothing but the conceptual entity ‘honesty and trustworthiness,’ whichfor simplicity, in the following, we shall simply denote ‘honesty.’Prior to the measurement, this conceptual entity is in its “ground” (most neutral) state ψ , and thiswill be true for all the respondents, as prior to the measurement no specific contextualization of ‘honesty’is given. Then, when performing measurement A , subjects are asked if they think Clinton is honest.This question can be reformulated/reinterpreted in the following equivalent way: “What best represents‘honesty,’ between the two possibilities: ‘Clinton is honest’ ( Ay ) and ‘Clinton is not honest’ ( An ).” Now,‘Clinton is honest’ and ‘Clinton is not honest’ are again to be considered as descriptions of elements ofconceptual reality that are independent of the subjects’ minds, and since they correspond to two differentcontextualizations of the conceptual entity ‘honesty,’ they are to be interpreted as two different “excited”(non-neutral) states of said entity.So, we have distinguished three states of the conceptual entity in question: the ψ -ground state ‘honesty,’the Ay -excited state ‘Clinton is honest’, and the An -excited state ‘Clinton is not honest.’ The reason why25he state ψ can be considered to be a superposition state, relative to the two (mutually exclusive states) Ay and An , is not because, primarily, a human subject would be uncertain about the honesty or dishonestyof Clinton, but because ‘honesty’ is available to be meaningfully contextualized as ‘Clinton is honest’ or as‘Clinton is not honest.’ This is so because ‘honesty’ is an abstract concept admitting a number of possible(less abstract) contextualizations, and the role of a human mind, when subjected to an interrogativecontext, is precisely that of actualizing one of these “more concrete” contexts, thus producing either thetransition ψ → Ay , or ψ → An , generally in an unpredictable way. And this role of the human mindis made manifest in the GTR-model by means of the indeterministic dynamics of an abstract breakableelastic band.The nature of this breakable elastic band will generally depend not only on the two outcome-states thatare available to be actualized, as the elastic will have these two outcome-states as its two end points, butalso on the specific forma mentis of the subject, described by its individual probability distribution ρ ( i ) A ,where the superscript “( i )” identifies the i -th respondent among the n respondents that are contributingto the overall statistics of outcomes.Let us go one step further in our analysis by assuming that the outcome of the A -measurement was Ay , and that right after it was obtained, the i -th respondent is also subjected to the B -measurement.This corresponds to the question “Do you generally think Al Gore is honest?” As we have seen, thissubsequent measurement is now performed not on the pre-measurement state ψ , but on the outcome-state Ay of the previous measurement. To understand why it is correct to proceed in this way, we have toconsider that since the B -measurement is performed immediately after the A -measurement, the answer Ay will still be present in the ‘field of consciousness’ of the respondent, and therefore will also play a role inthe contextualization of the entity subjected to the interrogative process. In other terms, when the second B -measurement is performed, the conceptual entity is not anymore in the ‘honesty’ ground state, but inthe ‘Clinton is honest’ excited state. So, the second B -measurement can be effectively rephrased as thefollowing interrogation: “What best represents ‘Clinton is honest,’ between the two possibilities: ‘Gore ishonest’ ( By ) and ‘Gore is not honest’ ( Bn ).”To answer this second question, the respondent will again use a mental elastic band, which will bedifferent from the previous one because the two outcome-states are of course different, but also because,possibly, the way these two outcome-states are chosen is different, i.e., the associated probability distri-bution ρ ( i ) B is not necessarily equal to ρ ( i ) A . Assume then that the B -measurement has produced the By outcome, and that right after it the i -th respondent is subjected again to the A -measurement. If theanswers to the two previous questions are still present in the respondent’s field of consciousness, we shouldconsider that this third measurement is performed on the conceptual entity in the state “Gore is honest& Clinton is honest.” This state, however, describes a condition that is stronger than that of the state“Clinton is honest,” and therefore cannot anymore be represented in a 3-dimensional Bloch sphere, whereall 2-outcome measurements are necessarily non-degenerate.The reason why in the GTR-model we can keep the description within the 3-dimensional Bloch sphere,is that it is possible to transfer some information from the states to the measurements, by changing themeasurements’ probability distributions, as we have explained in Sec. 7. Intuitively, we can think of thisprocess as a transfer from the conscious (manifest) to the subconscious (hidden) level, in accordance withthe idea that very few cognitive elements can remain in the actual focus of a subject. This means thatthe outcome of the first A -measurement will exit the subject’s field of consciousness, but will possibly beintegrated in its memory structure, still exerting some influence in the form of a new, updated probabilitydistribution. As a consequence, when the i -th respondent is subjected to the third A -measurement, fol-lowing the AB sequential measurement, the state of the conceptual entity will just be “Clinton is honest,”thus still describable by one of the two end points of the B -elastic, and as we have shown in the previoussection the updated probability distribution for the A -measurement will then account for the expected26espone replicability of the Ay outcome of the first measurement. Having explained how the mathematical formalism of the GTR-model can naturally be interpreted, we musttake the final step in our analysis and consider the critical issue of the average over the different respondents.In a typical psychological measurement, data are collected from a certain number n of participants, forinstance about a thousand in the Clinton/Gore and Rose/Jackson experiments. As we mentioned in theIntroduction, in a recent work (Aerts & Sassoli de Bianchi, 2015a,b) we have shown that if a sufficientlylarge number of respondents is considered, who explore a sufficiently wide spectrum of different ‘ways ofchoosing an answer,’ then the average over all their answers will be well approximated by a pure quantummeasurement.Our result, however, only holds for a single measurement context, and not necessarily for a sequentialmeasurement context. In fact, considering the analysis of the previous sections, it is even to be expectedthat the equivalence between a universal average and a pure quantum measurement will generally not holdin a sequential situation, as suggested by the fact that the Clinton/Gore and Rose/Jackson paradigmaticdata are very far from being well approximated by the Born rule (or better, the Born rule in combinationwith the projection postulate).To clarify a bit further this issue, let us consider again the single measurement context of the Clin-ton’s A -measurement. First of all, we have to observe that the fact that a universal average can explainthe emergence of the Born rule does not mean that in practice the different respondents will necessarilychoose their answers by exploring a wide spectrum of probability distributions ρ ( i ) A . Indeed, an effectivequantum measurement can easily be obtained also by considering a collection of individuals all having apredetermined answer regarding Clinton’s A -question. This means that even though each individual mindmay respond in a deterministic way (using for instance elastics that can only break in one of their endpoints), their “collective mind” can nevertheless behave indeterministically, as a “quantum machine.”Of course, we are not saying that this is what happens in reality in the Clinton’s A -measurement: weare just observing that there are a priori two different notions of indeterminism, one possibly manifestingat the level of the individual mind, and another possibly manifesting at the level of the “collective mind,”when the outcomes of the different individuals are considered together, without distinguishing them, bymeans of a uniform average. There are no doubts that A is typically a question where the respondents,apart possible exceptions, will generally do not have an a predetermined opinion, so that their answerswill literally be created during the very survey, in a way that will depend both on the specificities of theirminds (as described by their ρ ( i ) A ) and on the context (as described by the state of the conceptual entity,which according to our interpretation is the same for all the respondents, in the same way as the wordswritten in the present article are the same for all its potential readers).Generally, we don’t have access to the ρ ( i ) A of the different subjects. Indeed, to retrieve part of anindividual’s ρ ( i ) A , we should be able to perform the A -measurement many times on a same person, consideringdifferent initial conditions, also finding a way to avoid response replicability (for instance by waiting enoughtime between two answers, so that the previous outcome has left the field of consciousness of the subject,and also, possibly, its memory). This risks to be an almost impossible measurement to realize. There ishowever an easier way to obtain some information about an individual’s ρ ( i ) A : simply, by asking her/him todirectly evaluate the outcome probabilities, for instance by answering questions like: “between 1 and 100,what is the degree of truthfulness of the statement ‘Clinton is honest’ ?” In other terms, an individual’smind is able not only to select outcomes, but also probabilities of outcomes.We are not saying of course that the probabilities obtained in this way, once averaged, will necessarily27roduce the same probabilities than those obtained when the subjects are just asked to provide a singleoutcome, and the relative frequencies of the different outcomes are then calculated. But considering thestrong meaning connection between the notion of “truthfulness of a statement about honesty” and thatstatement being “a good representative of honesty,” we should expect the final statistics to be quitesimilar. But our point here was just to emphasize that considering the nature of the question addressedto the respondents, when asked to estimate the probabilities they will generally give values different from0 and 1, and this means that when they actualize a single outcome, it is reasonable to expect that theprocess is genuinely indeterministic.In fact, this conclusion also follows from the very observation of the existence of question order effects.Indeed, if the respondents would know in advance the answers to the A and B questions, their orderwould be irrelevant to the answer (assuming here that the respondents do not change their minds duringthe survey). But since we know that the order is not irrelevant, we can deduce that the answers wereactualized in a contextual way during the interrogative process, i.e., that they were created, and not justdiscovered.So, the assumption that each respondent uses a non trivial ρ ( i ) A is a reasonable one. Given this, it isvery important to distinguish the effective ρ A , describing the working of the “collective mind,” and thedifferent ρ ( i ) A , i = 1 , . . . , n , describing the working of the individual minds participating in the survey. Wehave seen that for the Clinton/Gore and Rose/Jackson data the overall probability distribution ρ A has tobe different from ρ B , and also that ρ A and ρ B have to be both non-symmetric with respect to the originof the Bloch sphere. The following question then arises: Does this mean that also for each participant ρ ( i ) A has to be different from ρ ( i ) B , and that they have to be non-symmetric?In other terms: Is it possible that, after all, each participant does choose in the same way, whenconfronted with the A and B measurements, and also in a perfectly symmetrical way with respect tothe two possible outcomes, and that the observed difference and asymmetry of the effective ρ A and ρ B distributions, describing the “collective mind,” is just a combined effect produced by the averaging of theoutcomes and the sequentiality of the measurements?To tentatively answer this question, we consider a very simple situation, consisting in only two respon-dents ( n = 2). We assume that the individual probability distributions are locally uniform, symmetric,and that they are the same for the two measurements: ( (cid:15) ( i ) A , d ( i ) A ) = ( (cid:15) ( i ) B , d ( i ) B ) ≡ ( (cid:15) i , i = 1 ,
2. Accordingto (16), we have for the probability of, say, outcome
AyBy : P ( i ) ( AyBy | ψ ) = 14 (cid:20) θ + cos θ A ) 1 (cid:15) i + cos θ cos θ A (cid:15) i (cid:21) , i = 1 , . (60)For the average probability P ( AyBy | ψ ) ≡ [ P (1) ( AyBy | ψ ) + P (2) ( AyBy | ψ )], we thus have: P ( AyBy | ψ ) = 14 (cid:20) θ + cos θ A ) (cid:15) + (cid:15) (cid:15) (cid:15) + cos θ cos θ A (cid:15) + (cid:15) (cid:15) (cid:15) (cid:21) . (61)Clearly, (61) can only be written in the form (60) if ( (cid:15) + (cid:15) ) = 2( (cid:15) + (cid:15) ), or equivalently, ( (cid:15) − (cid:15) ) = 0,i.e., if (cid:15) = (cid:15) . But since by hypothesis (cid:15) (cid:54) = (cid:15) , we have that the effective elastic band cannot be symmetric.For instance, if we choose: (cid:15) = 1, and (cid:15) = 0 .
4, cos θ = 0 .
3, cos θ A = 0 .
1, and cos θ B = 0 .
2, (16) give thefollowing probabilities for the first respondent (for simplicity, we don’t write the “ ψ -conditional statement): P (1) ( AyBy ) = 0 . P (1) ( AyBn ) = 0 . P (1) ( AnBn ) = 0 . P (1) ( AnBy ) = 0 . P (1) ( ByAy ) =0 . P (1) ( ByAn ) = 0 . P (1) ( BnAn ) = 0 . P (1) ( BnAy ) = 0 .
14. For the second respondent we have: P (2) ( AyBy ) = 0 . P (2) ( AyBn ) = 0 . P (2) ( AnBn ) = 0 . P (2) ( AnBy ) = 0 . P (2) ( ByAy ) = 0 . P (2) ( ByAn ) = 0 . P (2) ( BnAn ) = 0 . P (2) ( BnAy ) = 0 . P ( AyBy ) = 0 . P ( AyBn ) = 0 . P ( AnBn ) = 0 . P ( AnBy ) = 0 . P ( ByAy ) = 0 . P ( ByAn ) = 0 . P ( BnAn ) =0 . P ( BnAy ) = 0 . θ = 0 .
3, one obtains for the other parameters: (cid:15) A ≈ . (cid:15) B ≈ . d A ≈ . d B ≈ . θ B ≈ .
19 and cos θ A ≈ . θ A and cos θ B (0 .
13 instead of 0 . .
19 instead of 0 .
2, respectively).Of course, the above considerations do not prove that at the individual level the different respondentsactually use a single symmetric (cid:15) -elastic band, when answering both the A and B questions. This wedon’t know. The only thing we know is that such hypothesis is not incompatible with the modeling ofthe experimental data that we have presented in this work. Regarding the issue of response replicability,discussed in Sec. 7, it is worth observing that the effective “collective mind” will behave according toresponse replicability only if each one of the different individual minds will do so. This is so becauseif, say, P ( i ) ( AyByAy ) = P ( i ) ( AyBy ), for all i = 1 , . . . , n , then this will also be trivially the case forthe average: P ( AyByAy ) = n (cid:80) ni =1 P ( i ) ( AyByAy ) = n (cid:80) ni =1 P ( i ) ( AyBy ) = P ( AyBy ). In other terms,response replicability is straightforwardly transferred form the individual to the collective level.
10 Concluding remarks
It is time to summarize our findings. We have considered some paradigmatic data exhibiting question ordereffects and we have shown that they are genuinely non-Hilbertian, i.e., that they require a more generalframework than that of quantum mechanics, at least if one wants the modeling to remain within the ambitof 2-outcome, non-degenerate measurements. This more general framework is provided by the GTR-model,where different from quantum mechanics, various typologies of measurements, associated with a same setof outcome-states, can be described.In order to exactly solve the GTR-model, we have considered three simplifying hypothesis: weakcompatibility, local uniformity, and sensitivity to pre-measurement state. The solution obeying these threehypotheses remains sufficiently general to allow for an exact fitting of the experimental data. When this isdone, one finds that the probability distributions describing the two sequential measurements not only needto be different from one another, but also asymmetric. This, however, could very well be a byproduct of theaveraging over the different respondents. Indeed, even if we assume that each respondent selects in a non-contextual way the outcomes (using the same probability distribution in the two sequential measurements),without favoring one of the outcomes (using a parity invariant probability distribution), to the extent thatdifferent respondents are characterized by different probability distributions, contextuality and asymmetrywill necessarily emerge at the level of the more abstract “collective mind.”We have also discussed replicability of the outcomes, showing that it can be naturally accounted inthe GTR-formalism by allowing the measurements’ probability distributions to change at the individuallevel (and consequently also at the collective level), when measurements are repeated, and and that thisprocess of change of the probability distributions, which adds to the process of change of the states, remainscompatible with the observed question order effects.We have also proposed the adoption of a non-subjectivist interpretation of the GTR-model formalism,and a fortiori of the quantum formalism, where states do not describe “states of mind,” or “states of beliefs,”29ut the objective contextual condition of the conceptual entity that is subjected to the measurement, whosereality is independent from the respondents’ belief systems. A mind, instead, is that entity interactingwith the different conceptual entities, which are part of the human culture, by means of a mechanismof actualization of potential measurement interactions, as described in the formalism by the breakingof the elastic bands (or by the disintegration of higher-dimensional hyper-membranes, for more generalmeasurement situations (Aerts & Sassoli de Bianchi, 2014, 2015a,b,c)).Being the explicit description of the measurements absent in the standard quantum formalism, thisis of course the main reason why subjectivist interpretations exist, not only in quantum cognition, butalso in physics (so-called QBism is a typical example (Fuchs et al., 2014)). The GTR-model (Aerts &Sassoli de Bianchi, 2015a,b), and more specifically its Hilbertian extended Bloch representation (Aerts& Sassoli de Bianchi, 2014, 2015c), provides a completed version of quantum mechanics, in which theexplicit fluctuations originating in the interaction between the measured entity and the measuring systemare explicitly represented, thus explaining the possible nature of quantum indeterminism (so providing acompelling solution to the measurement problem). This allows understanding the collapse of the statevector as an objective physical process, and consequently interpreting the state vector as a description ofthe reality of the physical entity under consideration, and not merely of the experimenter’s beliefs aboutit. Similarly, if the probabilities emerging from psychological measurements are the result of objectivefluctuations in the interaction between a human mind and a ‘meaning entity’ prepared in a given context,as described in our beyond-quantum GTR-model, then a state describes the real contextual condition ofthe conceptual entity, and not the beliefs of a human mind about it. Beliefs are stored in the memorystructure of the judging human entity, and when they play a role they do so by altering the dynamics ofthe elastic bands describing the way answers are selected. But there are additional reasons for adoptinga realistic interpretation of the mathematical formalism in quantum cognition. Different from physics,where the hidden-measurements remain an hypothesis awaiting experimental confirmation, in psychologicalmeasurements they are in part manifested by the very existence of the different participants in a survey,and their different way of responding. Last but not least, another important reason for adopting theproposed interpretational framework is that it has demonstrated its relevance in the modeling of conceptcombinations in a comprehensive theory of conceptual representation (Gabora & Aerts, 2002; Aerts &Gabora, 2005a,b).Let us conclude by briefly commenting on the possibility of describing question order effects (but notresponse replicability) by remaining within the strict confines of standard quantum mechanics, by increasingthe number of dimensions. For this, we observe that generally the order of different statements containedin a sentence is relevant for what concerns its perceived meaning. Take the following example (Rampin,2005):A novice asked the prior:“Father, can I smoke when I pray?”And he was severely reprimanded.A second novice asked the prior:“Father, can I pray when I smoke?”And he was praised for his devotion.We see that “pray & smoke” does not elicit the same potential meanings as “smoke & pray.” In thesame way, the potential meanings of the sentence: “Clinton is honest & Gore is honest,” are not the sameas for the sentence: “Gore is honest & Clinton is honest.” The difference is this time subtler, and noteverybody will consciously perceive it, but, nevertheless, it is an objective difference. This means thatwhen we perform the sequential measurement F = AB , and obtain the four outcomes: AyBy , AyBn ,30 nBy and
AnBn , these outcomes will correspond to states that are all different from those associatedwith G = BA , i.e.: ByAy , ByAn , BnAy and
BnAn .From a quantum mechanical point of view, this means that F and G are necessarily described by twodifferent non-commuting Hermitian operators, acting on C , and although we know that, in practice, F and G are executed as two-step processes, i.e., as processes during which an outcome state is createdin a sequential way, we are not allowed to experimentally disentangle such sequence into two separatemeasurements, in what is called a product measurement. This in particular because product measurementsare formed by commuting sub-measurements, and commutation will generally prevent the modeling of ordereffects. Therefore, the only way to go, to possibly obtain a pure quantum modeling of question order effects,is to resort to the notion of ‘entangled measurements,’ as it was considered for instance in Aerts & Sozzo(2014a,b). We plan to come back to this possible alternative modeling strategy in future works. References
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