MODELING COMBINATION OF QUESTION ORDER EFFECT, RESPONSE REPLICABILITY EFFECT, AND QQ-EQUALITY WITH QUANTUM INSTRUMENTS

We continue to analyze basic constraints on human’s decision making from the viewpoint of quantum measurement theory (QMT). As has been found, the conventional QMT based on the projection postulate cannot account for combination of the question order effect (QOE) and the response replicability effect (RRE). This was an alarm signal for quantum-like modeling of decision making. Recently, it was shown that this objection to quantum-like modeling can be removed on the basis of the general QMT based on quantum instruments. In the present paper we analyse the problem of combination of QOE, RRE, and the famous QQ-equality (QQE). This equality was derived by Busemeyer and Wang and it was shown (in the joint paper with Solloway and Shiffrin) that statistical data from many social opinion polls satisfies it. Now, we construct quantum instruments satisfying QOE, RRE, and QQE. The general features of our approach are formalized with postulates which generalize Wang-Busemeyer postulates for quantum-like modeling of decision making. Moreover, we show that our model closely reproduces the statistics of the famous Clinton-Gore Poll data with a prior belief state independent of the question order. This model successfully removes the order effect from the data to determine the genuine distribution of the opinions in the Poll. The paper also provides a psychologist-friendly introduction to the theory of quantum instruments the most general mathematical framework for quantum measurements. We hope that this theory will attract attention of psychologists and will stimulate further applications. keywords: quantum-like models, decision making, social science, quantum instruments, order effect, response replicability effect, QQ-equality


INTRODUCTION
Recently the top level experts in cognitive psychology, behavioural economics, political sciences, and molecular biology begin to show interest in quantum formalism 1 applications. The majority of applications are based on quantum measurement theory (QMT) that is used to model basic psychological effects.
However, as was shown in [5], the conventional QMT based on the projection postulate (representing the mental state update) confronts with combination of some psychological effects. So, QMT may describe each of the effects individually, but not jointly. In [5], it was shown that combination of the question order effect (QOE) 1 and the response replicability effect (RRE) 2 cannot be described by the conventional QMT. This was an alarm signal for quantum-like modeling of decision making.
However, since the 1970s QMT has been further developed towards more flexible approach to treating all the physically realizable quantum measurements by abandoning the projection postulate in the conventional QTM, used during the first years of quantum physics. Nowadays, in quantum physics, especially in quantum information theory, one uses QMT based on theory of quantum instruments [15][16][17][18]. 3 So, say in quantum information theory, nobody would be surprised that in some situations conventional QMT (based on the projection postulate) cannot be applied.
Therefore, the right reply to the challenge presented in [5] should be based on the theory of quantum instruments. The first attempt to proceed in this direction was done in the paper of Basieva and Khrennikov [6]. Still, they used a restricted class of quantum instruments. (At the same time, this class is standard for quantum information theory.) And their paper confirmed the impossibility statement formulated originally in [5]. Only very recently it was shown [8] that by using quantum instruments, it is possible to describe the combination of QOE and RRE. Thus, quantum-like modeling program was secured from the objection of paper [5].
Immediately after publication of article [8], Dzhafarov (the private communication) raised the question whether the model presented in [8] matches the famous QQ-equality (QQE). The latter is one of the most important fruits of the quantum-like approach to decision making. This is a special constraint on probabilities that was derived by Busemeyer and Wang [3] in the quantum probabilistic (QP) framework. So, from the classical probabilistic (CP) viewpoint there is no reasons for QQE to hold, but usage of quantum formalism may require it. In paper [9], it was shown that statistical data from a bunch of social opinion polls satisfies QQE. We stress that this equality was derived on the basis of conventional QMT [3] and that it can be violated in general QMT [15][16][17][18]. Thus, combination of QOE and RRE with QQE is a delicate problem: one has to go beyond conventional QMT in a special way.
The purpose of this paper is two-fold: A). We present the solution of this problem (QOE+RRE+QQE) by constructing the corresponding quantum instruments. This step is important to justify the use of the quantum-like models for decision making. The general features of our approach are formalized with postulates which generalize Wang-Busemeyer postulates for quantumlike modeling of decision making [3]. This is important for continuity of research on quantum-like modeling in psychology.
We emphasize that our model is a complete model for a certain set of data satisfying the QQ-equality including the Clinton-Gore poll in the following sense. Since the Clinton-Gore poll data sets approximately satisfy QQE, there should be a uniform method to make the original data to satisfy QQE with small distortion. Here the main point is that such a method can be defined independent of the model. We show that our model completely reproduces the modified data of the Clinton-Gore poll data and this means our model reproduces the original data as accurately as how accurately the data fits QQE. Moreover, we show that this model closely reproduces the statistics of the famous Clinton-Gore poll data with a prior belief state independent of the question order. The model successfully removes the order effect from the data to determine the genuine distribution of the opinions in the Poll. 4 In this paper we do not discuss alternative theories. We only offer quantum models as viable candidates for the considered experiments, without comparison with possible other state-of the art models conventionally used in psychology. For such discussion we refer the reader to works [21][22][23].
B). Irrespective of the (QOE+RRE+QQE)-problem, the paper presents the most general formalism of QMT based on theory of quantum instruments. We hope that this theory will attract attention of psychologists and stimulate further applications in psychology.
Finally, we remark that the projective type (von Neumann-Lüders [14,30]) instruments, i.e., describing the measurement's feedback onto the system's state by orthogonal projections (section 2.2), are widely used in quantum-like modeling in cognition, psychology, and decision making [2,3], [9]- [12], [28,29]. Although this model is very attractive due to its simplicity, one has to be careful when applying it. In [5], the role of RRE was elevated. In fact, RRE is a special exhibition of long term memory in combination with keeping consistency in decisions (actions). So, RRE is one of the basic characteristics of rational cognition. Quantum instruments constructed in this paper express the long term memory effect and decision consistency (based on this effect) that are sufficient to generate RRE. In contrast, the projective type instruments for observables represented by non-commuting operators do not. (And one has to consider non-commuting operators to generate the QOE [3,5].) In the A − B − A-measurement scheme, this means that generally the B-measurement washes out the memory about the first A-measurement. For observables given by operators with nondegenerate spectra, memory is washed out completely. In the second A-measurement, the system "remembers" only the output of the B-measurement. For observables, with degenerate spectra memory is washed out essentially, but not completely. Such short term memory decision making can be considered as indication of "irrational" behavior. In social opinion polls, people generally behave rationally (by using long term memory), in spite of the order effect. The latter does not contradict rationality

QUANTUM INSTRUMENTS
We briefly present quantum instruments (see [8] for non-physicist friendly presentation, see also basic papers [15][16][17][18][19][20]; see appendix for coupling with generalized observables given by POVMs.) 2.1. Quantum states and observables. In quantum theory, it is postulated that every quantum system S corresponds to a complex Hilbert space H; denote the scalar product of two vectors by the symbol ψ 1 |ψ 2 . Throughout the present paper, we assume H is finite dimensional. States of the quantum system S are represented by density operators acting in H (positive semi-definite operators with unit trace). Denote this state space by the symbol S(H).
In quantum physics (especially quantum information theory), there are widely used notations invented by Dirac: a vector belonging to H is symbolically denoted as |ψ ; orthogonal projector on this vector is denoted as |ψ ψ|, it acts to the vector |ξ as ψ|ξ |ψ .
Observables (or physical quantities) of the quantum system S are represented by selfadjoint operators in H. Each observable A can be represented as where E A (x) is the spectral projection of the observable A corresponding to the eigenvalue x if x is an eigenvalue of A; otherwise, let E A (x) = 0. We note that spectral projectors sum up to the unit operator I : We also remark that each orthogonal projector E is self-adjoint and idempotent, i.e., In what follows, we identify the states of the system with density operators and the observables of the system with self-adjoint operators and make no notational distinctions.
According to the standard interpretation of quantum mechanics, we can measure every observable A in any state ρ in principle, and then we obtain one of its eigenvalues as the outcome of the measurement, while quantum mechanics cannot in general predict the precise outcome, but only predict its probability distribution Pr{A = x ρ} by the Born rule as For a state vector |ψ , this leads to the relation as Thus, quantum mechanics determines the probability distribution of a single observable in a given state. Then, how about the joint probabilities of several observables determined? There are two types of joint probabilities in quantum mechanics: one for simultaneous measurement and the other for successive measurement. We shall discuss them for two observables.
In quantum mechanics, two observables A and B are simultaneously measurable in any state ρ if and only if A and B are commuting, i.e., [A, B] = AB − BA = 0, and in this case, the joint probability distribution of outcomes of simultaneous measurements of A and B in a state ρ is give by For a state vector |ψ , this leads to the relation Since A and B are commuting, in the above expressions we have E A (x)E B (y) = E B (y)E A (x) and the order of A = x and B = y are interchangeable. Note that Eq. (5) and Eq. (6) can be obtained by the Born rule without a new postulate [14]; see Appendix B.
How does quantum mechanics determine the joint probability distribution of successive measurements? What happens if we measure another observable B just after the A measurement? Does the Born rule determine the joint probability of the outcomes of the A measurement and the subsequent B measurement? Suppose that the observable A is measured in a state ρ and the result A = x is obtained and subsequently the observable B is measured and the result B = y is obtained. To determine the joint probability distribution Pr{A = x, B = y ρ} of their outcomes, we consider the conditional probability distribution of B given A = x defined by The definition of conditional probability is based on the following reasoning. The state of the system just after the A measurement is considered to depend on the initial state ρ and the outcome A = x so that we denote it by ρ {A=x} . Since the B measurement is carried out in this state, the Born rule applies to this state to obtain and hence the joint probability distribution is given by The Born rule gives the probability of the output in a given state. But, in order to determine the joint probability distribution Pr{A = x, B = y ρ} we need another postulate to determine the state ρ {A=x} after the measurement.

Von Neumann-Lüders instruments.
In the conventional QMT, the projection postulate 5 is posed, stating: in the measurement of an observable A, the input state ρ is changed to the output state 5 For observables given by self-adjoint operators with non-degenerate spectra, this postulate was suggested by von Neumann [14]. Then Lüders extended it even to observables with degenerate spectra [30]. For the latter, von Neumann used a more general rule (in the spirit of theory of quantum instruments).
provided that the measurement leads to the outcome A = x. 6 If the input state is the state vector |ψ , the output state is also the state vector |ψ {A=x} such that In this case, i.e., for the density operator ρ = |ψ ψ|, we have the relations Note that equality (2) can be rewritten in the form: and the Born rule is written as According to the projection postulate, if observables A and B are successively measured in this order in the initial input state ρ, the joint probability distribution of their outcomes is given by For the state vector |ψ we have Thus, in the conventional QMT the outcome probability distribution and the state change caused by the measurement are uniquely determined by the observable A.
However, the theory should also reflect the evident fact that the same quantum observable, say energy, can be measured with a variety of quantum measuring instruments. In the quantum formalism, these instruments are characterized by back-actions of measurements to system's states. In modern QMT, a more flexible rule is adopted, in which the state change caused by the measurement is not uniquely determined by the observable to be measured. There are many ways to measure the same observable.
2.3. The Davis-Lewis-Ozawa quantum instruments. It has been known that the projection postulate is too restrictive to describe all the physically realizable measurements of the observable A. Davies-Lewis [15] proposed to abandon this postulate and proposed a more flexible approach to QMT.
The space L(H) of linear operators in H is a linear space over the complex numbers. Moreover, L(H) is the complex Hilbert space with the scalar product, A|B = TrA * B. We consider linear operators acting on it; they are called superoperators. The latter terminology is used to distinguish operators acting in the Hilbert spaces H and L(H). Otherwise superoperators are usual linear operators. In particular, for T : L(H) → 6 We recall that this rule was adopted by Wang and Busemeyer [3]. (This notion is rather complicated technically. Therefore we shall not discuss it in more detail.) Consider now a general measurement on the system S. The statistical properties of any measurement are characterized by • (i) the output probability distribution Pr{x = x ρ}, the probability distribution of the output x of the measurement in the input state ρ : • (ii) the quantum state reduction ρ → ρ {x=x} , the state change from the input state ρ to the output state ρ {x=x} conditional upon the outcome x = x of the measurement.
According to Davies-Lewis [15] and Ozawa [17], the modern quantum measurement theory postulates that any measurement of the system S is described by a mathematical structure called a quantum instrument. This is any map x → I(x), where for each real x, the map I(x) is a completely positive superoperator satisfying the normalization condition x Tr[I(x)ρ] = 1 for any state ρ. Davies-Lewis [15] originally postulated that the superoperator I(x) should be positive. However, Yuen [31] pointed out that the Davies-Lewis postulate is too general to exclude physically non-realizable instrument. Ozawa [17] introduced complete positivity to ensure that every quantum instrument is physically realizable. Thus, complete positivity is a sufficient condition for an instrument to be physically realizable. On the other hand, necessity is derived as follows [19, p. 369]. Every observable A of a system S is identified with the observable A ⊗ I of a system S + S with any system S external to S 7 . Then, every physically realizable instrument I A measuring A should be identified with the instrument I A⊗I measuring A ⊗ I such that I A⊗I (x) = I A (x) ⊗ id. This implies that I A (x) ⊗ id is agin a positive superoperator, so that I A (x) is completely positive. Similarly, any physically realizable instrument I(x) measuring system S should have its extended instrument I(x) ⊗ id measuring system S + S for any external system S . This is fulfilled only if I(x) is completely positive. Thus, complete positivity is a necessary condition for I A to describe a physically realizable instrument.
Given a quantum instrument I, the output probability distribution for the input state ρ is defined by the generalized Born rule in the trace-form, 7 For example, the color A of my eyes is not only the property of my eyes S but also the property A ⊗ I of myself S + S . and the quantum state reduction is defined by According to the Kraus theorem [43], for any instrument I, there exists a family {M xj } x,j of operators, called the measurement operators for I, in H such that (19) for any state ρ. In this case, we have Conversely, any family {M xj } x,j of operators in H satisfying (20) defines an instrument I.
The above general formulation of quantum instruments reflects variety of real measuring instruments for the same system S that measure an observable A accurately or with some error. A quantum instrument I A is called an instrument measuring an observable A, or an A-measuring instrument, if the output probability distribution satisfies Born's rule (3) for the A-measurement, i.e, The projective A-measuring instrument is defined by for any state ρ and real number x. Then, this instrument satisfies not only the Born formula (3) or (14), but also the projection postulate (10). Thus, the projection postulate is no longer the requirement for the measurement of the observable A but only one type of the measurement of A.
Conversely, any family {M xj } x,j of operators in H satisfying (23) defines an Ameasuring instrument I A by Eq. (19). For the projective instrument I A measuring A the measurement operators coincide with the spectral projections, i.e., 2.4. Quantum order effect from the quantum instrument viewpoint. Finally, we make the probabilistic comment on quantum instruments. Typically the main attention is given to Born's rule (21) as generating probabilities from quantum states, ρ → Pr{A = x ρ}. We would like to elevate the role of the state transform (18), measurement back-action. It generates quantum probability conditioning. If after measurement of observable A with outcome A = x, one measures observable B, then the probability to get output B = y is given by Born's rule for state ρ {A=x} : Now, by using quantum conditional probability we can define the sequential joint probability distribution of A (first) and B (last), then generally Pr{A = x, B = y ρ} = Pr{B = y, A = x ρ}. This is the mathematical representation of QOE.
In the framework of the quantum instrument theory, one has to distinguish noncommutativity of observables and noncommutativity of instruments. The standard noncommutativity condition describes incompatibility of observables, in the sense that their JPD, the JPD for their simultaneous measurements, is not well defined. So, condition (27) is related to joint measurements, not to sequential measurement and not to QOE. The latter is characterized not via non-existence of JPD, but via noncommutativity of the state updates, formalized via condition (26).
Generally, noncommutativity of instruments (26) does not imply noncommutativity of the observables to be measured (27), whereas the converse statement holds as shown in the following theorem.

Proof. It is well known that
We shall in fact construct instruments I A and I B showing the question order effect with commuting observables A and B, while the commutativity of A and B is necessary for I A and I B to show the response replicability effect.
2.5. Quantum instruments as representation of indirect measurements. The basic model for construction of quantum instruments is based on the scheme of indirect measurements. This scheme formalizes the following situation. As was permanently emphasized by Bohr (one of the founders of quantum mechanics), the results of quantum measurements are generated in the process of interaction of a system S with a measurement apparatus M. This apparatus consists of a complex physical device interacting with S and a pointer that shows the result of measurement, say spin up or spin down. An observer can see only outputs of the pointer and he associates these outputs with the values of the observable A for the system S. So, the observer approaches only the pointer, not the system by it self. Whether the outputs of the pointer can be associated with the "intrinsic properties" of S or not is one of the main problems of quantum foundations, it is still the topic for hot foundational debates [21]- [27]. Thus, the indirect measurement scheme involves: • the states of the systems S and the apparatus M ; • the operator U representing the interaction-dynamics for the system S + M ; • the meter observable M A giving outputs of the pointer of the apparatus M.
We shall make the following remark on the operator U of the interaction-dynamics. As all operations in quantum mechanics, it is a linear operator. In quantum formalism, dynamics of the state of an isolated system is described by the Schrödinger equation and its evolution operator is unitary. In the indirect measurement scheme, it is assume that (approximately) the compound system S + M is isolated. Hence, its evolution operator U is unitary.
Formally, an indirect measurement model, introduced in [17] as a "(general) measuring process", is a quadruple consisting of a Hilbert space K, a density operator σ ∈ S(K), a unitary operator U on the tensor product of the state spaces of S and M, U : H ⊗ K → H ⊗ K, and a selfadjoint operator M A on K. By this measurement model, the Hilbert space K describes the states of the apparatus M , the unitary operator U describes the time-evolution of the composite system S + M , the density operator σ describes the initial state of the apparatus M , and the self-adjoint operator M A describes the meter observable of the apparatus M. Then, the output probability distribution Pr{A = x ρ} in the system state ρ ∈ S(H) is given by where The change of the state ρ of the system S caused by the measurement for the outcome A = x is represented with the aid of the map I A (x) in the space of density operators defined as where Tr K is the partial trace over K. Then, the map x → I A (x) turn out to be a quantum instrument. Thus, the statistical properties of the measurement realized by any indirect measurement model (K, σ, U, M A ) is described by a quantum measurement. We remark that conversely any quantum instrument can be represented via the indirect measurement model [17]. Thus, quantum instruments mathematically characterize the statistical properties of all the physically realizable quantum measurements 8 . Now, we point to a few details which were omitted in the above considerations. The measuring interaction between the system S and the apparatus M turns on at time t 0 , the time of measurement, and turns off at time t = t 0 + ∆t. We assume that the system S and the apparatus M do not interact each other before t 0 nor after t = t 0 + ∆t and that the compound system S + M is isolated in the time interval (t 0 , t). The probe system P is defined to be the minimal part of apparatus M such that the compound system S + P is isolated in the time interval (t 0 , t). Then the above scheme is applied to the probe system P, instead of the whole apparatus M. The rest of the apparatus M performs the pointer measurement on the probe P. In particular, the unitary evolution operator U describing the state-evolution of the system S + P has the form H = e −i∆tH , where H = H S + H P + H SP is Hamiltonian of S + P with the terms H S and H P representing the internal dynamics in the subsystems S and P of the compound system and H SP describing the interaction between the subsystems.
Introduction of probe systems may be seen as unnecessary complication of the scheme of indirect measurements. However, it is useful if the apparatus M is a very complex system that interacts (often in parallel) with many systems S j , j = 1, 2, ...., m. Its different probes are involved solely in interaction with the concrete systems, P j with S j . And the system S j + P j can be considered as an isolated system; in particular, from interactions with other systems S i and probes P i .
The indirect measurement scheme is part of theory of open quantum systems [32]. Instead of a measurement apparatus M, we can consider the surrounding environment E of the system S (see [33]- [35] for applications to psychology). 8 It is an interesting problem whether instruments on a more general state space are realizable by an indirect measurement model (K, σ, U, M A ). This problem was studied extensively by Okamura and Ozawa [20]. Any discrete classical instrument is realizable. There exists a non-realizable continuous classical instrument. There also exists a non-realizable instrument for a local system in algebraic quantum field theory, but all instruments are realizable within arbitrary error. A necessary and sufficient condition for any instrument on a von Neumann algebra (which includes classical and quantum mechanics, and algebraic quantum field theory) to be precisely realizable by an indirect quantum measurement model is known as the normal extension property (NEP) [20].

INDIRECT MEASUREMENTS OF MENTAL OBSERVABLES: UNCONSCIOUS AS A SYSTEM AND CONSCIOUSNESS AS A MEASUREMENT APPARATUS
The scheme of indirect measurements presented in the previous section was created in quantum physics. Our aim is to adapt it to cognitive psychology. The main question is about cognitive analogues of the system S and the measurement apparatus M. We suggest to use the framework developed by Khrennikov [36] for quantum-like modeling of the Helmholtz sensation-perception theory. This scheme can be extended to a general scheme of unconscious-conscious interaction in the process of decision making The measured system S is a sensation (or generally any state of the unconscious mind). Consciousness as the measurement apparatus to the unconscious interacts with sensations to make the decisions (to generate outcomes of measurements). Consciousness, of course, is not concerned just with a single probe. It is a large environment with many probes interacting with the unconscious. There should be a rule of transformation from a sensation to a conscious decision. This is unitary transformation and measurement of the meter in the probe. The operational description neglects neurophysiological and electrochemical structures of interaction, just like the operational description of quantum computers with unitary transformations and measurements [37]. The unconscious is a black box that is mathematically described by the state of sensation space (= the unconscious state), so that the unconscious state probabilistically determines the decision (by interaction with consciousness), then the unconscious state is changed according to the previous unconscious state and the decision made. Thus, each probe is described by a quantum instrument. Instruments are probe dependent.
For the question-measurements, the question A is transferred into the unconscious, where it plays the role of a sensation, so to say a high mental level sensation. Then, interaction described by the unitary operator U generates a new state of the compound system -the unconscious-conscious. And consciousness performs the final "pointer reading", the measurement of the meter observable. Pointer reading can be treated as generation of a perception, a high mental level perception.
We understand that appealing to the unconscious-conscious description of cognitive processes is not so common in the modern psychology. However, this description well matches the indirect measurement scheme, which is broadly used to describe functions of quantum computers and quantum information devices without details of physical constructions. We can appeal to the authority of James [38] (appealing to Freud [40] might generate a negative reaction), see also Jung [39]. We also can mention the series of works on the use of the unconscious-conscious scheme in quantum-like modeling of cognition [36,41,42].

CONSTRUCTIONS OF INSTRUMENTS
4.1. Observables A and B. In modeling successive question-response experiments, such as the Clinton-Gore experiment, we consider two questions A and B for a subject to answer "yes" (y) or "no" (n) and consider the joint probability p(AaBb) for obtaining the answer a (a = y or n) for the question A and the answer b (b = y or n) for the question B, if the question B is asked after the question A, and the analogous joint probability p(BbAa) if the question A is asked after the question B.
We models the above joint probability distributions by the joint probability distributions of outcomes of successive measurements of 2-valued observables A and B in a quantum system in a given state ρ in the A-B order and in the B-A order. According to the general QMT, the joint probability distributions are well defined by two 2- Let A and B be projections on a Hilbert space H. They correspond to questions labeled A and B, respectively. The eigenvalue 1 means the answer "yes" to the questions, and the eigenvalue 0 means the answer "no" to the questions.
Let Denote by I 1 , I 2 , and I 3 the identity operators on the first, second, and the third component of H, respectively. We let A = |1 1| ⊗ I 2 ⊗ I 3 , and B = I 1 ⊗ |1 1| ⊗ I 3 . Thus, we consider the case where projections A and B commute.

4.2.
Instrument I A measuring A. We construct an instrument I A measuring A as follows. The instrument I A carries out a measurement of A by a measuring interaction between the object S described by the state space H and the probe P described by the state space K = C 2 ⊗ C 2 , which is prepared in the state |00 just before the measuring interaction. Denote by I 4 and I 5 the identity operators on the first and second component of K. The time evolution of the composite system S + P during the measuring interaction is described by a unitary operator U A on H ⊗ K satisfying and the outcome of the measurement is obtained by measuring the meter observable of the probe P . Note that both A and M A have the same spectrum {0, 1} and they are projections. We denote by I H and I K the identity operators on H and K, respectively, and denote by M ⊥ A the orthogonal complement of the projection M A , i.e., M ⊥ Equations (31)-(34) describe the change of the mind state + the probe state with the personality state |0 . In this case, the belief state is copied to the probe state and the belief state does not change. Thus, in the decision stage, measuring the first state of the probe state results in the decision equal to the belief for question A just before the question-response process.
Equations (35)- (38) describe the change of the mind state + the probe state with the personality state |1 . In this case, the belief state is copied to the probe state as well but the belief state for question B changes if the belief states for question A and question B are equal. The decision stage results in the preset belief for question A as well.
Equations (39)-(42) describe the change of the mind state + the probe state with the personality state |2 . In this case, the belief state is copied to the probe state as well but the belief state for question B changes if the belief states for question A and question B are not equal. The decision stage results in the preset belief for question A as well.
Suppose that the probe P is prepared in the state |00 just before the measuring interaction, the measuring process described by the indirect measurement model for any density operator ρ on H, where Tr K stands for the partial trace over K and P M A (a) denotes the spectral projection of an observable M A for a, i.e., P M A (0) = M ⊥ A and P M A (1) = M A . Consequently, for any ρ.
The instrument I A determines the probability distribution Pr{a = a ρ} of the outcome a of the measurement, where a = 0, 1, and the state change ρ → ρ {a=a} caused by the measurement is determined by

4.3.
Instrument I B measuring B. The instrument I B is constructed with the same probe system P prepared in the state |00 in an analogous manner with the instrument I A . The unitary operator U B on H ⊗ K, describing the time evolution of the composite system S + P during the measuring interaction, is supposed to satisfy and the meter observable on K is given by Then the instrument I B of this measuring process (K, |00 , U B , M B ) is defined by for any density operator ρ on H and b = 0, 1. Then we have for any density operator ρ on H, and hence the instrument I B measures the observable B.

POSTULATES FOR QUANTUM MODELS
Our model presented above satisfies the following postulates. Postulate 1: A subject's mind state is represented by a state vector in or a density operator on a multidimensional feature space (technically, an N -dimensional Hilbert space).
The above postulate extends Postulate 1 in Wang-Busemeyer [3] to include density operators. State vectors |ψ j can be mixed in two way. For complex numbers α j with j |α j | 2 = 1, we can make a new state vector, called the superposition, j α j |ψ j , and alternatively for positive numbers p j with j p j = 1, we can make a new state, called the mixture, represented by the density operator j p j |ψ j ψ j |. We use density operators if the state is a probability mixture (not a superposition) of state vectors.
Postulate 2: A potential response to a question is represented by a subspace of the multidimensional feature space.
This repeats Postulate 2 in [3]. In our model, we consider two questions A and B corresponding to the two subspaces A and B, which are represented by the two projections A and B that project the whole space H onto A and B, respectively.
Postulate 3: The probability of responding to an opinion question equals the squared length of the projection of the state vector onto the response subspace, or equals the trace of the projection of the density operator onto the response subspace.
This extends Postulate 3 in [3] to include density operators. According to Eq. (48), in our model the probability that a subject in a state vector |ψ responds "yes" or "no" to the opinion question A is given by This postulate is more general than Postulate 4 in [3]. According to Eq. (19) and Eq. (20) for any A-measuring instrument I A , there exists a family {M 1j , M 0j } j of operators such that If the state before answering the question A is ρ = |ψ ψ|, the updated mind state ρ {A=y} after deciding an answer A = y to the question A is determined by the instrument I A as According to Eq. (49), an explicit form of the family {M 1j , M 0j } j is given with parameter j = (β, γ) as follows.

Wang-Busemeyer [3] posed Postulate 4 (Wang-Busemeyer [3]):
The updated belief state after deciding an answer to a question equals the normalized projection on the subspace representing the answer.
This condition corresponds to the projective A-measuring instrument I proj A such that I proj A (1)ρ = AρA and I proj A (0)ρ = A ⊥ ρA ⊥ . This instrument has the measurement operators {A, A ⊥ } so that or states changes as Thus, our model satisfies Postulates 1-3 consistent with those posed by Wang-Busemeyer [3]. Moreover, our model generalizes Postulates 4 posed by Wang-Busemeyer [3] equivalent to the projection postulate, so that we allow any A-measuring instrument for question A, Last but not least, we should pose an additional postulate on the ensemble of subjects, which is not explicit but satisfied by the Wang-Busemeyer model [3].
Postulate 5: If a sequence of question-response experiments are carried out to the same ensemble of individual subjects, the joint probability of the sequence of responses are probability mixture of those joint probabilities with respect to the frequency of individual mind states in the ensemble.
If a sequence of question-response experiments are carried out to the same ensemble of individual subjects, and a randomly chosen individual subject has the state |ψ j with probability p j , then we suppose that the joint probability of the sequence of responses X 1 = x 1 , . . . , X n = x n , where X k = A or X k = B and x k = 0 or x k = 1 for all k = 1, . . . , n, is given by where ρ = j p j |ψ j ψ j |. Then, we have Thus, Postulate 5 is satisfied by our model. The structure of this postulate is discussed in more detail in appendix E.

MIND STATES
In modeling successive question-response experiments, such as the Clinton-Gore experiment, we assume that the subject's mind state is represented by the space Ω = {0, 1} 2 × {0, 1, 2}. Here we suppose that any individual subject has one of the mind states, ω ∈ Ω, and any statistical ensemble of the subjects is characterized by a probability distribution on Ω. Thus, the statistical data under consideration should be explained solely by one of the probability distributions on Ω. The mind state α represents the state in which the subject will answer "yes" for the question A if α = 1 and answer "no" otherwise. The mind state β represents the analogous state for the question B. Thus, the mind state (α, β) represents the subject's prior belief as long as the questions A and B are concerned. For (α, β, γ) ∈ Ω we write δ (α,β,γ) = |α, β, γ α, β, γ| ∈ H. In this way, we identify the mind state (α, β, γ) ∈ Ω with the quantum state |α, β, γ α, β, γ| ∈ H, and any probability distribution µ of the mind states ω ∈ Ω with the quantum statê µ = ω µ(ω)δ ω . The dynamics of producing the answer to the question and preparing the mind state for the next question is supposed to be described as a process of quantum measurement, or equivalently a mathematical object called a quantum instrument, which we believe is the most flexible framework consistent with the psycho-physical parallelism.
Our previously defined two quantum instruments I A and I B describe the measurement on the mind state as follows.
The mind state γ represents the personality of the subject in such a way that if γ = 1, the subject changes his/her mind to prepare the answer for the other question to be the opposite to the previous answer, and if γ = 2, the subject changes his/her mind to prepare the answer for the other question to be the same as the previous answer; on the other hand, if γ = 0, the subject's mind is so robust that the question will not affect his/her mind.
Thus, the state change is the Bayesian update if γ has only the value γ = 0, and yet if the value γ = 1 or γ = 2 is allowed, the statistics of the answers does not follows the Bayesian update rule.
The following considerations considerably simplify our manipulations of instruments I A and I B for arbitrary density operators ρ.

RESPONSE REPLICABILITY EFFECT (RRE)
Here, we consider two properties of successive measurements of observables A and B.
In A-A and A-B-A paradigms, the response to A is repeated (with probability 1). We call this response replicability effect (RRE).
In A-B vs B-A paradigm, the joint probabilities of the two responses are different on a set of states with a positive Lebesgue measure. This is question order effect (QOE).
The psychological problem raised by the Clinton-Gore experiment is as follows. We are given two observables A and B whose joint probability distributions (JPDs) of successive measurements shows QOE and is naturally considered to satisfy RRE. We cannot represent the JPDs as the JPDs of classical random variables A and B defined by Kolmogorov, which does not show QOE, nor the JPDs of the outcomes of successive measurements of non-commuting quantum observables A and B defined by von Neumann and Lüders, which does not satisfy RRE.
We shall show that instruments I A and I B have both Response Replicability Effect (RRE) in this section, and Question Order Effect (QOE) in the next section.  Thus, relation (iii) follows. Relation (iv) follows in a similar way.

QQ-EQUALITY
If two questions, adjacent to each other, are asked in different orders, then the quantum model of measurement order makes an a priori and parameter-free prediction, named the QQ equality [2,3,9]: Note that the formula in [9, P. 9435, left column] is not correct in that the first and second lines are not equal but have the opposite signs; the correct definition here is due to [3]. We have Thus, it is easy to check that the QQ equality holds and we have Theorem 9.1. The instruments I A and I B satisfy the QQ equality.

QQE-RENORMALIZATIONS
We have shown that joint probabilities of responses to questions obtained from our model satisfies QQE, so that if the original data does not satisfy QQE we cannot reproduce those data from our model with arbitrary mind state ρ.

INDEPENDENCE OF PERSONALITY STATE
In the previous sections, we have not assumed that in the probability distribution µ of the mind state, the personality state is independent from the belief state. According to section 4.2 the personality state determines the type of how the belief state is changed by U A and U B . It would be desirable to describe it to be independent from the belief state (α, β).
If the personality state depends on belief states, the same subject responds to questions in various patterns, so that it is difficulty to interpret the role of this parameter, and according to the number of free parameters it is likely that we can always find a suitable input data to explain the QQE renormalized output data by our model. On the other hand, if the personality state is independent of belief states, each individual subject is classified by the personality state and each ensemble of subjects for the same experiment can be classified by the probability distribution of the personality state. Thus, the statistical role of the personality state is quite clear.
However, in this case, the number of free parameter becomes not enough to reproduce all the QQE renormalized output data. Thus, the new assumption that the personality state is independent of the belief state bring about the following interesting problems: (i) What QQE renormalized output data can be reproduced by the current model? (ii) In such output data, what are characteristic features of the probability distribution of the personality state in the input data? (iii) How can we extend the role of the personality state to cover more QQE renormalized output data?
For the case where only personality state is |0 , the model reduces to Bayesian update model, which does not have QOE. We studied the model only with personality state |1 and showed QOE and RRE. In this paper, we shall further show that the model with personality state |0 , |1 , |2 explains the Clinton-Gore experiment.
Here, we assume the independence of the personality state. Thus, we suppose Then, Eqs. (88)-(95) are written as follows.
Thus, if p(Ay) = p(By) and p(Ay) = p(Bn), we can determine q(0), q(1), q(2) by the experimental data: Note that since the output data {p(AaBb), p(BbAa) | a, b = y, n} obtained from our model satisfy the QQ equality, the above relations are consistent, so that q(1) and q(2) are uniquely determined from the experimental data and then we determine q(0) by q(0) + q(1) + q(2) = 1.
If we obtain the personality state q(0), q(1), q(2), the belief state is determined by the following relations.

CLINTON-GORE POLL
In this section, we shall show that the well-known data from Clinton-Gore experiment can be reproduced within ±0.75% of errors from our model. Consider the following data from Clinton-Gore experiment [3,4,9].
The QQ equality is approximately satisfied with good accuracy.
Thus, their QQE-renormalizationp(Aa, Bb),p(Bb, Aa) are expected to approximate the original data p(Aa, Bb), p(Bb, Aa) with good accuracy. For the Clinton-Gore poll, we have and we obtain their QQE-renormalizations as follows. Under the assumption of the independence of the personality state, from Eq. (125) and the relation q(0) + q(1) + q(2) = 1, we can determine q(0), q(1), q(2) by the QQEnormalized data as follows.
Note that the unity of the total probability is satisfied.
Thus, the belief state p(α, β) and the personality state q(γ) are determined from the experimental data. Then, our quantum model with the belief state p(α, β) and the personality state q(γ) accurately reconstructs the QQE-renormalized datap(Aa, Bb), p(Bb, Aa) for a, b = y, n as follows. Therefore, all data of the QQR-renormalizations are accurately reproduced, and we conclude that the our quantum model reproduces the statistics of the Clinton-Gore Poll data almost faithfully (within ±0.75% of errors) with a prior belief state {p(0, 0), . . . , p(1, 1)} independent of the question order. Thus, this model successfully removes the order effect from the data to determine the genuine distribution of the opinion to the Poll. We restrict considerations to POVMs with a discrete domain of definition X = {x 1 , ..., x N }. POVM is a map x → Π A (x), where A is a symbol for the outcome variable, but does not denote any self-adjoint operator nor observable to be measured. Here, for each x ∈ X, Π A (x) is a positive contractive self-adjoint operator (i.e., 0 ≤ Π A (x) ≤ I) (called an effect), and the normalization condition holds, where I is the unit operator. This map defines an operator valued measure on the algebra of all subsets of X, for O ⊂ X, Π A (O) = x∈O Π A (x). The condition (155) characterizes "probability operator valued measures". From (2), we see that the map x → E A (x) is a special sort of POVM, the projection valued measure -PVM. POVMs Π A represent statistics of measurements with the outcome variable A of quantum observables with the following generalization of the Born's rule: We remark that equality (155) implies that x Pr{A = x ρ} = 1.
POVM does not represent state update. The latter is typically determined (nonuniquely) via representation of effects in the form: where V (x) is a linear operator in H; a canonical choice is V (x) = Π A (x) 1/2 and every V (x) is of the form V (x) = U (x)Π A (x) 1/2 with unitary operators U (x) for any x by polar decomposition. Hence, similarly to (13) the normalization condition has the form x V (x) * V (x) = I. The Born rule can be written similarly to (14): It is assumed that the post-measurement state transformation is based on the map: so Now, we remark that the map x → I A (x) given by (158) is a (very special) quantum instrument. We would like to elevate the role of the use of quantum instruments, comparing with the use of just POVMs. An instrument provides both statistics of the measurement-outputs and the rule for the state update, but POVM should always be endowed with ad hoc condition (156).
Finally, we remark that any instrument I A , with the outcome variable A, generates POVM Π A by the rule: However, its state update need not have the form (158). The general form will be give as follows. Recall that any instrument I A has a family {M xj } x,j of measurement operators satisfying Then, from Eq. (160) the POVM Π A generated by the instrument I A is given by It can be shown that for commuting observables A and B there exist an observable C and polynomials f and g such that A = f (C) and B = g(C) [14]. Then, Eq.  Suppose that the object state |ψ just before the measurement is arbitrarily given, i.e., |ψ = α,β,γ c α,β,γ |α, β, γ . Then, by the Born formula the probability distribution of the observable A is defined as By linearity of U A , it follows from Eqs. (31)-(34) that U A : |ψ |00 → α,β c α,β,0 |α, β, 0 + c α,β,1 |α, α ⊥ , 1 + c α,β,2 |α, α, 2 |α, β .

APPENDIX E. ON POSTULATE 5
As we know, a density operator ρ can be decomposed into probabilistic mixtures of pure states in various ways. We show that Postulate 5 is invariant with respect to such decompositions.