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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.
| | {{:S:Khrennikov08}} |
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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).
| | {{:S:Khrennikov09}} |
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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
| | {{:S:Khrennikov10}} |
| </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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| | {{:S:Khrennikov11}} |
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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>
| | {{:S:Khrennikov12}} |
| # 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>
| | {{:S:Khrennikov13}} |
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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="80%" | | | {{:S:Khrennikov14}} |
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| | width="33%" |
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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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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\mathcal{P}_A(x)\rho=
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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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| |}
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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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|
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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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|
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| where <math>\Omega</math> is a sample space <math>P</math> and is a probability measure.
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|
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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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|
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| ===6.2. Response replicability effect for sequential questioning===
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| The approach based on identification of the order effect with noncommutative representation of questions (Wang and Busemeyer, 2013) was criticized in paper (Khrennikov et al., 2014). To discuss this paper, we recall the notion of ''response replicability.'' Suppose that a person, say John, is asked some question <math>A</math> and suppose that he replies, e.g, “yes”. If immediately after this, he is asked the same question again, then he replies “yes” with probability one. We call this property <math>A-A</math> ''response replicability.'' In quantum physics, <math>A-A</math> response replicability is expressed by ''the projection postulate.''The Clinton–Gore opinion poll as well as typical decision making experiments satisfy <math>A-A</math> response replicability. Decision making has also another feature - <math>A-A</math>''response replicability.'' Suppose that after answering the <math>A</math>-question with say the “yes”-answer, John is asked another question <math>B</math>. He replied to it with some answer. And then he is asked <math>A</math> again. In the aforementioned social opinion pool, John repeats her original answer to <math>A</math>, “yes” (with probability one).
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| This behavioral phenomenon we call <math>A-B-A</math> response replicability. Combination of <math>A-A</math> with <math>A-B-A</math> and <math>B-A-B</math> response replicability is called ''the response replicability effect'' RRE.
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|
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| ===6.3. “QOE+RRE”: described by quantum instruments of non-projective type===
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| In paper (Khrennikov et al., 2014), it was shown that by using the von Neumann calculus it is ''impossible to combine RRE with QOE.'' To generate QOE, Hermitian operators <math>\widehat{A},\widehat{B} </math> should be noncommutative, but the latter destroys<math>A-B-A </math> response replicability of <math>A </math>. This was a rather unexpected result. It made even impression that, although the basic cognitive effects can be quantum-likely modeled separately, their combinations cannot be described by the quantum formalism.
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| However, recently it was shown that theory of quantum instruments provides a simple solution of the combination of QOE and RRE effects, see Ozawa and Khrennikov (2020a) for construction of such instruments. These instruments are of non-projective type. Thus, the essence of QOE is not in the structure of observables, but in the structure of the state transformation generated by measurements’ feedback. QOE is not about the joint measurement and incompatibility (noncommutativity) of observables, but about sequential measurement of observables and sequential (mental-)state update. Quantum instruments which are used in Ozawa and Khrennikov (2020a) to combine QOE and RRE correspond to measurement of observables represented by commuting operators <math>\widehat{A},\widehat{B} </math>. Moreover, it is possible to prove that (under natural mathematical restriction) QOE and RRE can be jointly modeled only with the aid of quantum instruments for commuting observables.
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|
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| ===6.4. Mental realism===
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| Since very beginning of quantum mechanics, noncommutativity of operators <math>\widehat{A},\widehat{B} </math> representing observables <math>A,B </math> was considered as the mathematical representation of their incompatibility. In philosophic terms, this situation is treated as impossibility of the realistic description. In cognitive science, this means that there exist mental states such that an individual cannot assign the definite values to both observables (e.g., questions). The mathematical description of QOE with observables represented by noncommutative operators (in the von Neumann’s scheme) in Wang and Busemeyer (2013) and Wang et al. (2014) made impression that this effect implies rejection of mental realism. The result of Ozawa and Khrennikov (2020a) demonstrates that, in spite of experimentally well documented QOE, the mental realism need not be rejected. QOE can be modeled within the realistic picture mathematically given by the joint probability distribution of observables <math>A </math> and <math>B </math>, but with the noncommutative action of quantum instruments updating the mental state:
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>[\mathcal{J_A(x)},\mathcal{J_B(x)}]=\mathcal{J_A(x)}\mathcal{J_B(x)}-\mathcal{J_B(x)},\mathcal{J_A(x)}\neq0</math>
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| | width="33%" align="right" |<math>(20)</math>
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| |}
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| This is the good place to remark that if, for some state <math>\rho,[\Im_A(x),\Im_B(x)]\rho=0 </math>, then QOE disappears, even if <math>[\Im_A(x),\Im_B(x)]\neq0 </math>. This can be considered as the right formulation of Wang–Bussemeyer statement on connection of QOE with noncommutativity. Instead of noncommutativity of operators <math>\widehat{A} </math> and <math>\widehat{B} </math> symbolically representing quantum obseravbles, one has to speak about noncommutativity of corresponding quantum instruments.
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| ----
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|
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| ==7. Genetics: interference in glucose/lactose metabolism==
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| In paper (Asano et al., 2012a), there was developed a quantum-like model describing the gene regulation of glucose/lactose metabolism in Escherichia coli bacterium.11 There are several types of E. coli characterized by the metabolic system. It was demonstrated that the concrete type of E. coli can be described by the well determined linear operators; we find the invariant operator quantities characterizing each type. Such invariant operator quantities can be calculated from the obtained statistical data. So, the quantum-like representation was reconstructed from experimental data.
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| Let us consider an event system <math>\{Q_+,Q_-\}:Q_+</math> means the event that E. coli activates its lactose operon, that is, the event that -galactosidase is produced through the transcription of mRNA from a gene in lactose operon; <math>Q_-</math>means the event that E. coli does not activates its lactose operon.
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| This system of events corresponds to activation observable that is mathematically represented by a quantum instrument <math>\Im_Q</math>. Consider now another system of events <math>\{D_L,D_G\}</math> where <math>D_L</math> means the event that an E. coli bacterium detects a lactose molecular in cell’s surrounding environment, means <math>D_G</math> detection of a glucose molecular. This system of events corresponds to detection observable <math>D</math> that is represented by a quantum instrument <math>\Im_D</math>.
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| In this model, bacterium’s interaction–reaction with glucose/lactose environment is described as sequential action of two quantum instruments, first detection and then activation. As was shown in Asano et al. (2012a), for each concrete type of E. coli bacterium, these quantum instruments can be reconstructed from the experimental data; in Asano et al. (2012a), reconstruction was performed for W3110-type of E. coli bacterium. The classical FTP with observables <math>A=D</math> and <math>B=Q</math> is violated, the interference term, see (2), was calculated (Asano et al., 2012a).
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| ==8. Open quantum systems: interaction of a biosystem with its environment==
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| As was already emphasized, any biosystem <math>S</math> is fundamentally open. Hence, dynamics of its state has to be modeled via an interaction with surrounding environment <math>
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| \varepsilon</math>. The states of <math>S</math> and <math>
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| \varepsilon</math> are represented in the Hilbert spaces <math>\mathcal{H}</math> and <math>\mathcal{H}</math>. The compound system <math>S+\varepsilon</math> is represented in the tensor product Hilbert spaces . This system is treated as an isolated system and in accordance with quantum theory, dynamics of its pure state can be described by the Schrödinger equation:
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| {| width="80%" |
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| | width="33%" |<math>i\tfrac{d}{dt}\Psi(t)=\widehat{H}\Psi(t)(t), \Psi(0)=\Psi_0</math>
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| | width="33%" align="right" |<math>(21)</math>
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| |}
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| where <math>\psi(t)</math> is the pure state of the system <math>S+\varepsilon</math> and <math>\hat{\mathcal{H}}</math> is its Hamiltonian. This equation implies that the pure state <math>\psi(t)</math> evolves unitarily :<math>\psi(t)=\hat{U}(t)\psi_0</math>. Here <math>\hat{U}(t)=e^{-it\hat{\mathcal{H}}}</math>. Hamiltonian (evolution-generator) describing information interactions has the form <math>\hat{\mathcal{H}}=\hat{\mathcal{H}}_s+\hat{\mathcal{H}}_\varepsilon+{\mathcal{\hat H_{S,\varepsilon}}}</math>, where <math>\hat{\mathcal{H}}_s</math>,<math>\hat{\mathcal{H}}_\varepsilon</math>are Hamiltonians of the systems and <math>{\mathcal{\hat H_{S,\varepsilon}}}</math>is the interaction Hamiltonian.12 This equation implies that evolution of the density operator <math>\hat{\mathcal{R}}(t)</math> of the system <math>S+\varepsilon</math> is described by von Neumann equation:
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| {| width="80%" |
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| | width="33%" |<math>\tfrac{d\widehat{R}}{dt}(t)=-i[\widehat{H},\widehat{R},(t)], \widehat{R}(0)=\widehat{R}_0</math>
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| | width="33%" align="right" |<math>(22)</math>
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| |}
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| However, the state <math>\hat{\mathcal{R}}(t)</math> is too complex for any mathematical analysis: the environment includes too many degrees of freedom. Therefore, we are interested only the state of <math>S</math>; its dynamics is obtained via tracing of the state of <math>S+\varepsilon</math> w.r.t. the degrees of freedom of <math>\varepsilon</math> :
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| {| width="80%" |
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| | width="33%" |<math>\widehat{\rho}(t)=Tr_\mathcal{H}\widehat{R}(t)</math>
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| | width="33%" align="right" |<math>(23)</math>
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| |}
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| Generally this equation, ''the quantum master equation'', is mathematically very complicated. A variety of approximations is used in applications.
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| ===8.1. Quantum Markov model: Gorini–Kossakowski–Sudarshan–Lindbladequation===
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| The simplest approximation of quantum master equation (23) is ''the quantum Markov dynamics'' given by the ''Gorini–Kossakowski–Sudarshan–Lindblad'' (GKSL) equation (Ingarden et al., 1997) (in physics, it is commonly called simply the Lindblad equation; this is the simplest quantum master equation):
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\tfrac{d\widehat{\rho}}{dt}(t)=-i[\widehat{H},\widehat{\rho},(t)]+ \widehat{L}[\widehat{\rho}(t),\widehat{\rho}(0)=\widehat{\rho}_0</math>
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| | width="33%" align="right" |<math>(24)</math>
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| |}
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| where Hermitian operator (Hamiltonian) <math>\widehat{\mathcal{H}}</math> describes the internal dynamics of <math>S</math> and the superoperator <math>\widehat{{L}}</math>, acting in the space of density operators, describes an interaction with environment <math>\varepsilon</math>. This superoperator is often called ''Lindbladian.'' The GKSL-equation is a quantum master equation for Markovian dynamics. In this paper, we have no possibility to explain the notion of quantum Markovianity in more detail. Quantum master equation (23) describes generally non-Markovean dynamics.
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| ----
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|
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| ===8.2. Biological functions in the quantum Markov framework===
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| We turn to the open system dynamics with the GKSL-equation. In our modeling, Hamiltonian <math>\widehat{\mathcal{H}}</math> and Lindbladian <math>\widehat{{L}}</math> represent some special ''biological function'' <math>F</math> (see Khrennikov et al., 2018) for details. Its functioning results from interaction of internal and external information flows. In Sections 10, 11.3, <math>F</math> is some ''psychological function''; in the simplest case <math>F</math> represents a question asked to <math>S</math> (say is a human being). In Section 7, <math>F</math> is the ''gene regulation'' of glucose/lactose metabolism in Escherichia coli bacterium. In Sections 9, 11.2, <math>F</math> represents the process of ''epigenetic mutation''. Symbolically biological function <math>F</math> is represented as a quantum observable: Hermitian operator <math>\widehat{F}</math> with the spectral decomposition <math>\widehat{F}=\sum_xx\widehat{E}^F(x)</math>, where <math>x</math> labels outputs of <math>F</math>. Theory of quantum Markov state-dynamics describes the process of generation of these outputs.
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| In the mathematical model (Asano et al., 2015b, Asano et al., 2017b, Asano et al., 2017a, Asano et al., 2015a, Asano et al., 2012b, Asano et al., 2011, Asano et al., 2012a), the outputs of biological function <math>F</math> are generated via approaching a ''steady state'' of the GKSL-dynamics:
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| {| width="80%" |
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| | width="33%" |<math>\lim_{t \to \infty}\widehat{\rho}(t)=\widehat{\rho}_{steady}</math>
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| | width="33%" align="right" |<math>(25)</math>
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| |}
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| such that it matches the spectral decomposition of <math>\widehat{F}</math>, i.e.,
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| {| width="80%" |
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| | width="33%" |<math>\widehat{\rho}_{steady}=\sum_x p_x\widehat{E}^F(x)</math>
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| | width="33%" align="right" |<math>(26)</math>
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| |}
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| where
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| {| width="80%" |
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| | width="33%" |<math>p_x\geq\sum_xp_x=1</math>
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| | width="33%" align="right" |<math>(26)</math>
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| This means that <math>\widehat{\rho}_{steady}</math> is diagonal in an orthonormal basis consisting of eigenvectors of <math>\widehat{F}</math>. This state, or more precisely, this decomposition of density operator <math>\widehat{\rho}_{steady}</math>, is the classical statistical mixture of the basic information states determining this biological function. The probabilities in state’s decomposition (26) are interpreted statistically.
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|
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| Consider a large ensemble of biosystems with the state <math>\widehat{\rho}_0</math> interacting with environment <math>\varepsilon</math>. (We recall that mathematically the interaction is encoded in the Lindbladian <math>\widehat{{L}}</math>) Resulting from this interaction, biological function <math>F</math> produces output <math>x</math> with probability <math>p_x</math>. We remark that in the operator terms the probability is expressed as <math>p_x=Tr\widehat{\rho}_{steady}\widehat{E}^F(x)</math>
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| This interpretation can be applied even to a single biosystem that meets the same environment many times.
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| It should be noted that limiting state <math>\widehat{\rho}_{steady}</math> expresses the stability with respect to the influence of concrete environment <math>\varepsilon</math>. Of course, in the real world the limit-state would be never approached. The mathematical formula (25) describes the process of stabilization, damping of fluctuations. But, they would be never disappear completely with time.
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| We note that a steady state satisfies the stationary GKSL-equation:
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>i[\widehat{H},\widehat{\rho}_{steady}]= \widehat{L}[\widehat{\rho}_{steady}]</math>
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| | width="33%" align="right" |<math>(27)</math>
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| |}
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|
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| It is also important to point that generally a steady state of the quantum master equation is not unique, it depends on the class of initial conditions.
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|
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| ===8.3. Operation of biological functions through decoherence===
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| To make the previous considerations concrete, let us consider a pure quantum state as the initial state. Suppose that a biological function <math>F</math> is dichotomous, <math>F=0,1
| |
| </math>, and it is symbolically represented by the Hermitian operator that is diagonal in orthonormal basis <math>|0\rangle</math>,<math>|1\rangle</math> . (We consider the two dimensional state space — the qubit space.) Let the initial state has the form of superposition
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| {| width="80%" |
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| | width="33%" |
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| | width="33%" |<math>\psi\rangle=c_0|0\rangle+c_1|1\rangle</math>
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| | width="33%" align="right" |<math>(28)</math>
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| where <math>c_j\in C,|c_0|^2+||c_1|^2=1</math>. The quantum master dynamics is not a pure state dynamics: sooner or later (in fact, very soon), this superposition representing a pure state will be transferred into a density matrix representing a mixed state. Therefore, from the very beginning it is useful to represent superposition (28) in terms of a density matrix:
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| {| width="80%" |
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| | width="33%" |<math>\widehat{\rho}_0=\begin{vmatrix} |c_0|^2 & c_0\bar{c}_1 \\ \bar{c}_0c_1 & |c_1|^2 \end{vmatrix}</math>
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| | width="33%" align="right" |<math>(29)</math>
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| State’s purity, superposition, is characterized by the presence ofnonzero off-diagonal terms.
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| Superposition encodes uncertainty with respect to the concrete state basis, in our case <math>|0\rangle</math>,<math>|1\rangle</math>. Initially biological function <math>F</math> was in the state of uncertainty between two choices <math>x=0,1</math>. This is ''genuine quantum(-like) uncertainty.'' Uncertainty, about possible actions in future. For example, for psychological function (Section 10) <math>F</math> representing answering to some question, say “to buy property” ( <math>F=1</math>) and its negation ( <math>F=0</math>) , a person whose state is described by superposition (28) is uncertain to act with ( <math>F=1</math>) or with ( <math>F=0</math>) . Thus, a superposition-type state describes ''individual uncertainty,'' i.e., uncertainty associated with the individual biosystem and not with an ensemble of biosystems; with the single act of functioning of <math>F</math> and not with a large series of such acts.
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| Resolution of uncertainty with respect to <math>\widehat{F}-basis</math> is characterized by washing off the off-diagonal terms in (29) The quantum dynamics (24) suppresses the off-diagonal terms and, finally, a diagonal density matrix representing a steady state of this dynamical systems is generated:
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| | width="33%" |<math>\widehat{\rho}_0=\begin{vmatrix} p_0 & 0\\ 0 & p_1 \end{vmatrix}</math>
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| | width="33%" align="right" |<math>(30)</math>
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| This is a classical statistical mixture. It describes an ensemble of biosystems; statistically they generate outputs <math>F=\alpha</math> with probabilities <math>p_\alpha</math>. In the same way, the statistical interpretation can be used for a single system that performs <math>F</math>-functioning at different instances of time (for a long time series).
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| In quantum physics, the process of washing off the off-diagonal elements in a density matrix is known as the ''process of decoherence.'' Thus, the described model of can be called operation of biological function through decoherence.
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| ===8.4. Linearity of quantum representation: exponential speed up for biological functioning===
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| The quantum-like modeling does not claim that biosystems are fundamentally quantum. A more natural picture is that they are a complex classical biophysical systems and the quantum-like model provides the information representation of classical biophysical processes, in genes, proteins, cells, brains. One of the advantages of this representation is its linearity. The quantum state space is a complex Hilbert space and dynamical equations are linear differential equations. For finite dimensional state spaces, these are just ordinary differential equations with complex coefficients (so, the reader should not be afraid of such pathetic names as Schrödinger, von Neumann, or Gorini–Kossakowski–Sudarshan–Lindblad equations). The classical biophysical dynamics beyond the quantum information representation is typically nonlinear and very complicated. The use of the linear space representation simplifies the processing structure. There are two viewpoints on this simplification, external and internal. The first one is simplification of mathematical modeling, i.e., simplification of study of bioprocesses (by us, external observers). The second one is more delicate and interesting. We have already pointed to one important specialty of applications of the quantum theory to biology. Here, systems can perform ''self-observations.'' So, in the process of evolution say a cell can “learn” via such self-observations that it is computationally profitable to use the linear quantum-like representation. And now, we come to the main advantage of linearity.
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| The linear dynamics exponentially speeds up information processing. Solutions of the GKSL-equation can be represented in the form <math>\widehat{\rho}(t)=e^{t\widehat{\Gamma}}\widehat{\rho}</math>, where <math>{\widehat{\Gamma}}</math> is the superoperator given by the right-hand side of the GKSL-equation. In the finite dimensional case, decoherence dynamics is expressed via factors of the form <math>e^{t{(ia-b)}}</math>, where <math>b>0</math>. Such factors are exponentially decreasing. Quantum-like linear realization of biological functions is exponentially rapid comparing with nonlinear classical dynamics.
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| The use of the quantum information representation means that generally large clusters of classical biophysical states are encoded by a few quantum states. It means huge information compressing. It also implies increasing of stability in state-processing. Noisy nonlinear classical dynamics is mapped to dynamics driven by linear quantum(-like) equation of say GKSL-type.
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| The latter has essentially simpler structure and via selection of the operator coefficients encoding symbolically interaction within the system <math>S</math> and with its surrounding environment <math>\varepsilon</math>, <math>S</math> can establish dynamics with stabilization regimes leading to steady states.
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| ==9. Epigenetic evolution within theory of open quantum systems==
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| In paper (Asano et al., 2012b), a general model of the epigenetic evolution unifying neo-Darwinian with neo-Lamarckian approaches was created in the framework of theory of open quantum systems. The process of evolution is represented in the form of ''adaptive dynamics'' given by the quantum(-like) master equation describing the dynamics of the information state of epigenome in the process of interaction with surrounding environment. This model of the epigenetic evolution expresses the probabilities for observations which can be done on epigenomes of cells; this (quantum-like) model does not give a detailed description of cellular processes. The quantum operational approach provides a possibility to describe by one model all known types of cellular epigenetic inheritance.
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| To give some hint about the model, we consider one gene, say <math>g</math>. This is the system <math>S</math> in Section 8.1. It interacts with the surrounding environment <math>\varepsilon</math> a cell containing this gene and other cells that send signals to this concrete cell and through it to the gene <math>g</math>. As a consequence of this interaction some epigenetic mutation <math>\mu</math> in the gene <math>g</math> can happen. It would change the level of the <math>g</math>-expression.
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| For the moment, we ignore that there are other genes. In this oversimplified model, the mutation can be described within the two dimensional state space, complex Hilbert space <math>{\mathcal{H}}_{epi}</math> (qubit space). States of <math>g</math> without and with mutation are represented by the orthogonal basis <math>|0\rangle</math>,<math>|1\rangle</math>; these vectors express possible epigenetic changes of the fixed type <math>\mu</math>.
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| A pure quantum information state has the form of superposition<math>|\psi\rangle_{epi}=c_0|0\rangle+c_1|1\rangle</math>.
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| Now, we turn to the general scheme of Section 8.2 with the biological function <math>F</math> expressing <math>\mu</math>-epimutation in one fixed gene. The quantum Markov dynamics (24) resolves uncertainty encoded in superposition <math>|\psi\rangle_{epi}</math> (“modeling epimutations as decoherence”). The classical statistical mixture , <math>{\rho}_{steady}</math>see (30), is approached. Its diagonal elements <math>p_0,p_1</math>give the probabilities of the events: “no <math>\mu</math>-epimutation” and “<math>\mu</math>-epimutation”. These probabilities are interpreted statistically: in a large population of cells, <math>M</math> cells,<math>M\gg1</math> , the number of cells with <math>\mu</math>-epimutation is <math>N_m\approx p_1M</math>. This <math>\mu</math>-epimutation in a cell population would stabilize completely to the steady state only in the infinite time. Therefore in reality there are fluctuations (of decreasing amplitude) in any finite interval of time.
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| Finally, we point to the advantage of the quantum-like dynamics of interaction of genes with environment — dynamics’ linearity implying exponential speed up of the process of epigenetic evolution (Section 8.4).
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| ==10. Connecting electrochemical processes in neural networks with quantum informational processing== | | ==10. Connecting electrochemical processes in neural networks with quantum informational processing== |