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| ===3.3. Non-projective state update: atomic instruments===
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| In general, the statistical properties of any measurement are characterized by
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| # the output probability distribution <math display="inline">Pr\{\text{x}=x\parallel\rho\}</math>, the probability distribution of the output <math display="inline">x</math> of the measurement in the input state <math display="inline">\rho
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| </math>;
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| # the quantum state reduction <math display="inline">\rho\rightarrow\rho_{(X=x)}
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| </math>,the state change from the input state <math display="inline">\rho
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| </math> to the output state <math display="inline">\rho\rightarrow\rho_{(X=x)}
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| </math> conditional upon the outcome <math display="inline">\text{X}=x
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| </math> of the measurement.
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| In von Neumann’s formulation, the statistical properties of any measurement of an observable is uniquely determined by Born’s rule (5) and the projection postulate (6), and they are represented by the map (9), an instrument of von Neumann type. However, von Neumann’s formulation does not reflect the fact that the same observable <math>A</math> represented by the Hermitian operator <math>\hat{A}</math> in <math display="inline">\mathcal{H}</math> can be measured in many ways.8 Formally, such measurement-schemes are represented by quantum instruments.
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| Now, we consider the simplest quantum instruments of non von Neumann type, known as ''atomic instruments.'' We start with recollection of the notion of POVM (probability operator valued measure); we restrict considerations to POVMs with a discrete domain of definition <math display="inline">X=\{x_1....,x_N.....\}</math>. POVM is a map <math display="inline">x\rightarrow \hat{D}(x)</math> such that for each <math display="inline">x\in X</math>,<math>\hat{D}(x)</math> is a positive contractive Hermitian operator (called effect) (i.e.,<math display="inline">\hat{D}(x)^*=\hat{D}(x), 0\leq \langle\psi|\hat{D}(x)\psi\rangle\leq1</math> or any <math display="inline">\psi\in\mathcal{H}</math>), and the normalization condition
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| <math display="inline">\sum_x \hat{D}(x)=I</math>
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| holds, where <math display="inline">I</math>
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| is the unit operator. It is assumed that for any measurement, the output probability distribution <math display="inline">Pr\{\text{x}=x||\rho\}</math> is given by
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| {| width="80%" |
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| | width="33%" |<math display="inline">Pr\{\text{x}=x||\rho\}=Tr [\hat{D}(x)\rho]</math>
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| | width="33%" align="right" |<math>(10)</math>
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| |}
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| where <math display="inline"> \hat{D}(x)</math> is a POVM. For atomic instruments, it is assumed that effects are represented concretely in the form
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| {| width="80%" |
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| | width="33%" |<math display="inline"> \hat{D}(x)=\hat{V}(x)^*\hat{V}(x)</math>
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| | width="33%" align="right" |<math>(11)</math>
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| |}
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| where <math display="inline"> {V}(x)</math> is a linear operator in <math display="inline">\mathcal{H}</math>. Hence, the normalization condition has the form <math display="inline">\sum_x V(x)^*V(x)=I</math>.9 The Born rule can be written similarly to (5):
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| | width="33%" |<math display="inline">Pr\{\text{x}=x||\rho\}=Tr [{V}(x)\rho{V}^*(x)]</math>
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| | width="33%" align="right" |<math>(12)</math>
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| |}
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| It is assumed that the post-measurement state transformation is based on the map:
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| | width="33%" |'''<big>*</big>'''
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| | width="33%" |<math display="inline">\rho\rightarrow\mathcal{L_A(x)\rho=V(X)\rho V^*(x)}</math>
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| | width="33%" align="right" |<math>(13)</math>
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| |}
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| so the quantum state reduction is given by
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| | width="33%" | '''<big>*</big>'''
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| | width="33%" |<math display="inline">\rho\rightarrow\rho_{(\text{x}=x)}=\frac{\mathcal{L}_A(x) \rho}{Tr[\mathcal{L}_A(x)\rho]}</math>
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| | width="33%" align="right" |<math>(14)</math>
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| |}
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| The map <math>x\rightarrow\mathcal{L_A(x)}</math> given by (13) is an atomic quantum instrument. We remark that the Born rule (12) can be written in the form
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| | width="33%" | '''<big>*</big>'''
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| | width="33%" |<math display="inline">Pr\{\text{x}=x||\rho\}=Tr [\Im_A(x)\rho]</math>
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| | width="33%" align="right" |<math>(15)</math>f
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| |}
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| Let <math>\hat{A}</math> be a Hermitian operator in <math display="inline">\mathcal{H}</math>. Consider a POVM <math display="inline"> \hat{D}=\biggl(\hat{D}^A(x)\Biggr)</math> with the domain of definition given by the spectrum of <math>\hat{A}</math>. This POVM represents a measurement of observable <math>A</math> if Born’s rule holds:
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| | width="33%" |<math display="inline">Pr\{\text{A}=x||\rho\}=Tr [\widehat{D}^A(x)\rho]=Tr[\widehat{E}^A(x)\rho]</math>
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| | width="33%" align="right" |<math>(16)</math>
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| |}
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| Thus, in principle, probabilities of outcomes are still encoded in the spectral decomposition of operator <math>\hat{A}</math> or in other words operators <math display="inline"> \biggl(\hat{D}^A(x)\Biggr)</math> should be selected in such a way that they generate the probabilities corresponding to the spectral decomposition of the symbolic representation <math>\hat{A}</math> of observables <math>A</math>, i.e.,<math display="inline"> \biggl(\hat{D}^A(x)\Biggr)</math> is uniquely determined by<math>\hat{A}</math> as <math display="inline"> \hat{D}^A(x)=\hat{E}^A(x)</math>. We can say that this operator carries only information about the probabilities of outcomes, in contrast to the von Neumann scheme, operator <math>\hat{A}</math> does not encode the rule of the state update. For an atomic instrument, measurements of the observable <math>A</math> has the unique output probability distribution by the Born’s rule (16), but has many different quantum state reductions depending of the decomposition of the effect <math display="inline"> \hat{D}(x)=\hat{E}^A(x)=V(x)^*V(x)</math> in such a way that
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| | width="33%" |<math display="inline">\rho\rightarrow\rho_{(\text{A}=x)}=\frac{{V}(x) \rho V(x)^*}{Tr[{V}(x)\rho V(x)^*]}</math>
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| | width="33%" align="right" |<math>(17)</math>
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| |}
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| ----
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| ===3.4. General theory (Davies–Lewis–Ozawa)===
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| Finally, we formulate the general notion of quantum instrument. A superoperator acting in <math display="inline">\mathcal{L}(\mathcal{H})</math> is called positive if it maps the set of positive semi-definite operators into itself. We remark that, for each '''<u><math>x,\Im_A(x)</math></u>''' given by (13) can be considered as linear positive map.
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| Generally any map<math>x\rightarrow\Im_A(x)</math> , where for each <math>x</math>, the map <math>\Im_A(x)</math> is a positive superoperator is called ''Davies–Lewis'' (Davies and Lewis, 1970) quantum instrument.
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| Here index <math display="inline">A</math> denotes the observable coupled to this instrument. The probabilities of <math display="inline">A</math>-outcomes are given by Born’s rule in form (15) and the state-update by transformation (14). However, Yuen (1987) pointed out that the class of Davies–Lewis instruments is too general to exclude physically non-realizable instruments. Ozawa (1984) introduced the important additional condition to ensure that every quantum instrument is physically realizable. This is the condition of complete positivity.
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| A superoperator is called ''completely positive'' if its natural extension <math display="inline">\jmath\otimes I</math> to the tensor product <math display="inline">\mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})=\mathcal{L}(\mathcal{H}\otimes\mathcal{H})</math> is again a positive superoperator on <math display="inline">\mathcal{L}(\mathcal{H})\otimes\mathcal{L}(\mathcal{H})</math>. A map <math>x\rightarrow\Im_A(x)</math> , where for each <math display="inline">x</math>, the map <math>\Im_A(x)</math> is a completely positive superoperator is called ''Davies–Lewis–Ozawa'' (Davies and Lewis, 1970, Ozawa, 1984) quantum instrument or simply quantum instrument. As we shall see in Section 4, complete positivity is a sufficient condition for an instrument to be physically realizable. On the other hand, necessity is derived as follows (Ozawa, 2004).
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| Every observable <math display="inline">A</math> of a system <math display="inline">S</math> is identified with the observable <math display="inline">A\otimes I</math> of a system <math display="inline">S+S'</math> with any system <math display="inline">S'</math> external to<math display="inline">S</math> .10
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| Then, every physically realizable instrument <math>\Im_A</math> measuring <math display="inline">A</math> should be identified with the instrument <math display="inline">\Im_A{_\otimes}_I
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| </math> measuring <math display="inline">A{\otimes}I
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| </math> such that <math display="inline">\Im_A{_\otimes}_I(x)=\Im_A(x)\otimes I
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| </math>. This implies that <math display="inline">\Im_A(x)\otimes I
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| </math> is agin a positive superoperator, so that <math>\Im_A(x)</math> is completely positive.
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| Similarly, any physically realizable instrument <math>\Im_A(x)</math> measuring system <math display="inline">S</math> should have its extended instrument <math display="inline">\Im_A(x)\otimes I
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| </math> measuring system <math display="inline">S+S'</math> for any external system<math display="inline">S'</math>. This is fulfilled only if <math>\Im_A(x)</math> is completely positive. Thus, complete positivity is a necessary condition for <math>\Im_A</math> to describe a physically realizable instrument.
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| ==4. Quantum instruments from the scheme of indirect measurements==
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| The basic model for construction of quantum instruments is based on the scheme of indirect measurements. This scheme formalizes the following situation: measurement’s outputs are generated via interaction of a system <math>S</math> with a measurement apparatus <math>M</math> . This apparatus consists of a complex physical device interacting with <math>S</math> 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 <math>A</math> for the system <math>S</math>. Thus, the indirect measurement scheme involves:
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| # the states of the systems <math>S</math> and the apparatus <math>M</math>
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| # the operator <math>U</math> representing the interaction-dynamics for the system <math>S+M</math>
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| # the meter observable <math>M_A</math> giving outputs of the pointer of the apparatus <math>M</math>.
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| An ''indirect measurement model'', introduced in Ozawa (1984) as a “(general) measuring process”, is a quadruple
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| <math>(H,\sigma,U,M_A)</math>
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| consisting of a Hilbert space <math>\mathcal{H}</math> , a density operator <math>\sigma\in S(\mathcal{H})</math>, a unitary operator <math>U</math> on the tensor product of the state spaces of <math>S</math> and<math>M,U:\mathcal{H}\otimes\mathcal{H}\rightarrow \mathcal{H}\otimes\mathcal{H}</math> and a Hermitian operator <math>M_A</math> on <math>\mathcal{H}</math> . By this measurement model, the Hilbert space <math>\mathcal{H}</math> describes the states of the apparatus <math>M</math>, the unitary operator <math>U</math> describes the time-evolution of the composite system <math>S+M</math>, the density operator <math>\sigma</math> describes the initial state of the apparatus <math>M</math> , and the Hermitian operator <math>M_A</math> describes the meter observable of the apparatus <math>M</math>. Then, the output probability distribution <math>Pr\{A=x\|\sigma\}</math> in the system state <math>\sigma\in S(\mathcal{H})</math> is given by
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| | width="33%" |<math>Pr\{A=x\|\rho\}=Tr[\Bigl(I\otimes E^M{^{_A}(x)\Bigr)}U(\rho \otimes\sigma)U^*]
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| </math>
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| | width="33%" align="right" |<math>(18)</math>
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| |}
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| where <math>E^{M_{A}}(x)</math> is the spectral projection of <math>M_A</math> for the eigenvalue <math>x</math>.
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| The change of the state <math>\sigma</math> of the system <math>S</math> caused by the measurement for the outcome <math>A=x</math> is represented with the aid of the map <math>\Im_A(x)</math> in the space of density operators defined as
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| Tr_\mathcal{H}[\Bigl(I\otimes E^M{^{_A}(x)\Bigr)}U(\rho \otimes\sigma)U^*]</math>
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| | width="33%" align="right" |<math>(19)</math>
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| where <math>Tr_\mathcal{H}</math> is the partial trace over <math>\mathcal{H}</math> . Then, the map <math>x\rightarrow\Im_A(x)</math>turn out to be a quantum instrument. Thus, the statistical properties of the measurement realized by any indirect measurement model <math>(H,\sigma,U,M_A)</math> is described by a quantum measurement. We remark that conversely any quantum instrument can be represented via the indirect measurement model (Ozawa, 1984). Thus, quantum instruments mathematically characterize the statistical properties of all the physically realizable quantum measurements.
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| ==5. Modeling of the process of sensation–perception within indirect measurement scheme==
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| Foundations of theory of ''unconscious inference'' for the formation of visual impressions were set in 19th century by H. von Helmholtz. Although von Helmholtz studied mainly visual sensation–perception, he also applied his theory for other senses up to culmination in theory of social unconscious inference. By von Helmholtz here are two stages of the cognitive process, and they discriminate between ''sensation'' and ''perception'' as follows:
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| * Sensation is a signal which the brain interprets as a sound or visual image, etc.
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| * Perception is something to be interpreted as a preference or selective attention, etc.
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| In the scheme of indirect measurement, sensations represent the states of the sensation system of human and the perception system plays the role of the measurement apparatus . The unitary operator describes the process of interaction between the sensation and perception states. This quantum modeling of the process of sensation–perception was presented in paper (Khrennikov, 2015) with application to bistable perception and experimental data from article (Asano et al., 2014).
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| ==6. Modeling of cognitive effects==
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| In cognitive and social science, the following opinion pool is known as the basic example of the order effect. This is the Clinton–Gore opinion pool (Moore, 2002). In this experiment, American citizens were asked one question at a time, e.g.,
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| :<math>A=</math> “Is Bill Clinton honest and trustworthy?”
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| :<math>B=</math> “Is Al Gore honest and trustworthy?”
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| Two sequential probability distributions were calculated on the basis of the experimental statistical data, <math>p_{A,B}</math> and <math>p_{B,A}</math> (first question<math>A</math> and then question <math>B</math> and vice verse).
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| ===6.1. Order effect for sequential questioning===
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| The statistical data from this experiment demonstrated the ''question order effect'' QOE, dependence of sequential joint probability distribution for answers to the questions on their order <math>p_{(A,B)}\neq p_{(B,A)}</math>. We remark that in the CP-model these probability distributions coincide:
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| <math>p_{A,B}(\alpha,\beta)= P(\omega\in\Omega: A(\omega)= \alpha,B(\omega)=\beta)=p_{A,B}(\beta,\alpha)</math>
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| where <math>\Omega</math> is a sample space <math>P</math> and is a probability measure.
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| QOE stimulates application of the QP-calculus to cognition, see paper (Wang and Busemeyer, 2013). The authors of this paper stressed that noncommutative feature of joint probabilities can be modeled by using noncommutativity of incompatible quantum observables <math>A,B</math> represented by Hermitian operators <math>\widehat{A},\widehat{B}</math> . Observable <math>A</math> represents the Clinton-question and observable <math>B</math> represents Gore-question. In this model, QOE is identical incompatibility–noncommutativity of observables:
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| <math>[\widehat{A},\widehat{B}]\neq0</math>
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| ===6.2. Response replicability effect for sequential questioning=== | | ===6.2. Response replicability effect for sequential questioning=== |