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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==
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| As was emphasized in introduction, quantum-like models are formal operational models describing information processing in biosystems. (in contrast to studies in quantum biology — the science about the genuine quantum physical processes in biosystems). Nevertheless, it is interesting to connect the structure quantum information processing in a biosystem with physical and chemical processes in it. This is a problem of high complexity. Paper (Khrennikov et al., 2018) presents an attempt to proceed in this direction for the human brain — the most complicated biosystem (and at the same time the most interesting for scientists). In the framework of quantum information theory, there was modeled information processing by brain’s neural networks. The quantum information formalization of the states of neural networks is coupled with the electrochemical processes in the brain. The key-point is representation of uncertainty generated by the action potential of a neuron as quantum(-like) superposition of the basic mental states corresponding to a neural code, see Fig. 1 for illustration.
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| Consider information processing by a single neuron; this is the system <math>S</math> (see Section 8.2). Its quantum information state corresponding to the neural code ''quiescent and firing,'' <math>0/1</math>, can be represented in the two dimensional complex <math>{\mathcal{H}}_{neuron}</math> Hilbert space (qubit space). At a concrete instant of time neuron’s state can be mathematically described by superposition of two states, labeled by <math>|0\rangle</math>,<math>|1\rangle</math>: <math>|\psi_{neuron}\rangle=c_0|0\rangle+c_1|1\rangle</math>. It is assumed that these states are orthogonal and normalized, i.e., <math>\langle0|1\rangle=0</math> and<math>\langle \alpha|\alpha\rangle=1</math>, <math>\alpha=0,1</math>. The coordinates <math>c_0</math> and <math>c_1</math> with respect to the quiescent-firing basis are complex amplitudes representing potentialities for the neuron <math>S</math> to be quiescent or firing. Superposition represents uncertainty in action potential, “to fire” or “not to fire”. This superposition is quantum information representation of physical, electrochemical uncertainty.
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| Let <math>F</math> be some ''psychological (cognitive) function'' realized by this neuron. (Of course, this is oversimplification, considered, e.g., in the paradigm “grandmother neuron”; see Section 11.3 for modeling of <math>F</math> based on a neural network). We assume that <math>F=0,1</math> is dichotomous. Say <math>F</math> represents some instinct, e.g., aggression: “attack” <math>=1</math>, “not attack” <math>=0</math>.
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| A psychological function can represent answering to some question (or class of questions), solving problems, performing tasks. Mathematically <math>F</math> is represented by the Hermitian operator <math>\widehat{F}</math> that is diagonal in the basis <math>|0\rangle</math>,<math>|1\rangle</math>. The neuron <math>S</math> interacts with the surrounding electrochemical environment <math>\varepsilon</math>. This interaction generates the evolution of neuron’s state and realization of the psychological function <math>F</math>. We model dynamics with the quantum master equation (24). Decoherence transforms the pure state <math>|\psi_{neuron}\rangle</math> into the classical statistical mixture (30), a steady state of this dynamics. This is resolution of the original electrochemical uncertainty in neuron’s action potential.
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| The diagonal elements of <math>\widehat{\rho}_{steady}</math> give the probabilities with the statistical interpretation: in a large ensemble of neurons (individually) interacting with the same environment <math>\varepsilon</math> , say <math>M</math> neurons,<math>M\gg1</math> , the number of neurons which take the decision <math>F=1</math> equals to the diagonal element <math>p_1</math>.
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| We also point to the advantage of the quantum-like dynamics of the interaction of a neuron with its environment — dynamics’ linearity implying exponential speed up of the process of neuron’s state evolution towards a “decision-matrix” given by a steady state (Section 8.4).
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| ==11. Compound biosystems==
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| ===11.1. Entanglement of information states of biosystems===
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| The state space <math>{\mathcal{H}}</math> of the biosystem <math>S</math> consisting of the subsystems <math>S_j,j=1,2,....n</math>, is the tensor product of subsystems’ state spaces<math>{\mathcal{H}}_j</math> , so
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| | width="33%" |'''<big>*</big>'''
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| | width="33%" |<math>\Im=\Im_1\otimes....\otimes\Im_n</math>
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| | width="33%" align="right" |<math>(31)</math>
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| The easiest way to imagine this state space is to consider its coordinate representation with respect to some basis constructed with bases in <math>{\mathcal{H}}_j</math>. For simplicity, consider the case of qubit state spaces <math>{\mathcal{H}}_j</math> let <math>|\alpha\rangle</math>, <math>|\alpha\rangle=0,1</math>, be some orthonormal basis in <math>{\mathcal{H}}_j</math>, i.e., elements of this space are linear combinations of the form <math>|\psi_{j}\rangle=c_0|0\rangle+c_1|1\rangle</math>. (To be completely formal, we have to label basis vectors with the index <math>j</math>, i.e.,<math>|\alpha\rangle\equiv |\alpha\rangle_j</math>. But we shall omit this it.) Then vectors <math>|\alpha_1.....\alpha_n\rangle \equiv |\alpha_1\rangle\otimes....\otimes|\alpha_n\rangle</math> form the orthonormal basis in <math>{\mathcal{H}}</math>, i.e., any state <math>|{\mathcal{\Psi}}\in {\mathcal{H}} </math> can be represented in the form
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| | width="33%" |<math>|\psi\rangle=\sum_{\alpha_j=0,1} C_(\alpha_{1)}....\alpha_n|\alpha_1....\alpha
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| _n\rangle</math>
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| | width="33%" align="right" |<math>(32)</math>
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| and the complex coordinates <math> C_{\alpha_1.....\alpha_n}</math> are normalized: <math> \sum |C_{\alpha_1.....\alpha_n}|^2=1</math>. For example, if <math> n=2</math>, we can consider the state
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| | width="33%" |<math>|\Psi\rangle=(|00\rangle+|11\rangle)/\sqrt{2}</math>
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| | width="33%" align="right" |<math>(33)</math>
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| This is an example of an ''entangled state'', i.e., a state that cannot be factorized in the tensor product of the states of the subsystems. An example of a non-entangled state (up to normalization) is given by
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| | width="33%" |<math>|00\rangle+|01\rangle+|10\rangle+|11\rangle=(|0\rangle+|1\rangle)\otimes (|1\rangle+|0\rangle)</math>
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| Entangled states are basic states for quantum computing that explores state’s inseparability. Acting to one concrete qubit modifies the whole state. For a separable state, by transforming say the first qubit, we change only the state of system <math> S_1</math> . This possibility to change the very complex state of a compound system via change of the local state of a subsystem is considered as the root of superiority of quantum computation over classical one. We remark that the dimension of the tensor product state space is very big, it equals <math> 2^n</math> for <math> n</math> qubit subsystems. In quantum physics, this possibility to manipulate with the compound state (that can have the big dimension) is typically associated with “quantum nonlocality” and ''spooky action at a distance.''But, even in quantum physics this nonlocal interpretation is the source for permanent debates []. In particular, in the recent series of papers [] it was shown that it is possible to proceed without referring to quantum nonlocality and that quantum mechanics can be interpreted as the local physical theory. ''The local viewpoint on the quantum theory is more natural for biological application.''13 For biosystems, spooky action at a distance is really mysterious; for humans, it corresponds to acceptance of parapsychological phenomena.
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| How can one explain generation of state-transformation of the compound system <math> S</math> by “local transformation” of say the state of its subsystem <math> S_1</math>? Here the key-role is played by ''correlations'' that are symbolically encoded in entangled states. For example, consider the compound system <math> S=(S_1,S_2)</math> in the state <math> |\Psi\rangle</math> given by (33). Consider the projection-type observables <math> A_j</math> on <math> A_j</math>represented by Hermitian operators <math>\widehat{A}_j</math> with eigen-vectors <math>|0\rangle</math>,<math>|1\rangle</math> (in qubit spaces <math>{\mathcal{H}}_j</math>). Measurement of say <math> A_1</math> with the output <math> A_1=\alpha</math> induces the state projection onto the vector <math>|\alpha\alpha\rangle</math>.
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| Hence, measurement of <math> A_2</math> will automatically produce the output <math> A_2=\alpha</math>. Thus, the state <math> |\Psi\rangle</math> encodes the exact correlations for these two observables. In the same way, the state
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| | width="33%" |<math>|\Psi\rangle=(|10\rangle+|01\rangle)/\sqrt{2}</math>
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| | width="33%" align="right" |<math>(34)</math>
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| encodes correlations <math> A_1=\alpha</math>, <math> A_2=\alpha</math> (mod 2).
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| So, '<nowiki/>''an entangled state provides the symbolic representation of correlations between states of the subsystems of a compound biosystem'''
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| Theory of open quantum systems operates with mixed states described by density operators. And before to turn to modeling of biological functions for compound systems, we define entanglement for mixed states. Consider the case of tensor product of two Hilbert spaces, i.e., the system <math> S</math> is compound of two subsystems <math> S_1</math> and <math> S_2</math>. A mixed state of <math> S</math> given by <math>\widehat{\rho}</math> is called separable if it can be represented as a convex combination of product states <math>\widehat{\rho}=\sum_k p_k\widehat{\rho}_{1k}\otimes \widehat{\rho}_{2k}</math>, where <math>\widehat{\rho}ik</math>, <math>i=1,2</math>, are the density operator of the subsystem <math> S_i</math> of <math> S</math>. Non-separable states are called entangled. They symbolically represent correlations between subsystems.
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| Quantum dynamics describes the evolution of these correlations. In the framework of open system dynamics, a biological function approaches the steady state via the process of decoherence. As was discussed in Section 8.3, this dynamics resolves uncertainty that was initially present in the state of a biosystem; at the same time, it also washes out the correlations: the steady state which is diagonal in the basis <math> \{|\alpha_1....\alpha_2\rangle\}</math> is separable (disentagled). However, in the process of the state-evolution correlations between subsystems (entanglement) play the crucial role. Their presence leads to transformations of the state of the compound system <math> S</math> via “local transformations” of the states of its subsystems. Such correlated dynamics of the global information state reflects ''consistency of the transformations of the states of subsystems.''
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| Since the quantum-like approach is based on the quantum information representation of systems’ states, we can forget about the physical space location of biosystems and work in the information space given by complex Hilbert space <math>{\mathcal{H}}</math>. In this space, we can introduce the notion of locality based on the fixed tensor product decomposition (31). Operations in its components <math>{\mathcal{H}}_j</math> we can call local (in information space). But, they induce “informationally nonlocal” evolution of the state of the compound system.
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| ===11.2. Entanglement of genes’ epimutations===
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| Now, we come back to the model presented in Section 9 and consider the information state of cell’s epigenome expressing potential epimutations of the chromatin-marking type. Let cell’s genome consists of <math>m</math> genes <math>g_1,....,g_m</math>. For each gene <math>g</math>, consider all its possible epimutations and enumerate them: <math>j_g=1,......k_g</math>. The state of all potential epimutations in the gene <math>g</math> is represented as superposition
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| | width="33%" |<math>|\psi_g\rangle=\sum_{j_g} c_{j_g)}|j_g\rangle</math>
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| | width="33%" align="right" |<math>(35)</math>
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| In the ideal situation – epimutations of the genes are independent – the state of cell’s epigenome is mathematically described by the tensor product of the states <math>|\psi_g\rangle</math> :
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| | width="33%" |<math>|\psi_{epi}\rangle=|\psi_{g_1}\rangle\otimes....\otimes|\psi_{g_{m}}\rangle
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| </math>
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| | width="33%" align="right" |<math>(36)</math>
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| However, in a living biosystem, the most of the genes and proteins are correlated forming a big network system. Therefore, one epimutation affects other genes. In the quantum information framework, this situation is described by entangled states:
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| | width="33%" |<math>|\psi_{epi}\rangle=\sum _{j_{g_1....{j_{g_m}}}} C_{j_{g_1....{j_{g_m}}}}|j_{g_1}...j_{g_m} \rangle
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| </math>
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| | width="33%" align="right" |<math>(37)</math>
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| This form of representation of potential epimutations in the genome of a cell implies that epimutation in one gene is consistent with epimutations in other genes. If the state is entangled (not factorized), then by acting, i.e., through change in the environment, to one gene, say <math>g_1</math>, and inducing some epimutation in it, the cell “can induce” consistent epimutations in other genes.
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| Linearity of the quantum information representation of the biophysical processes in a cell induces the linear state dynamics. This makes the epigenetic evolution very rapid; the off-diagonal elements of the density matrix decrease exponentially quickly. Thus, our quantum-like model justifies the high speed of the epigenetic evolution. If it were based solely on the biophysical representation with nonlinear state dynamics, it would be essentially slower.
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| Modeling based on theory of open systems leads to reconsideration of interrelation between the Darwinian with Lamarckian viewpoint on evolution. Here we concentrated on epimutations, but in the same way we can model mutations (Asano et al., 2015b).
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| ===11.3. Psychological functions===
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| Now, we turn to the model presented in Section 10. A neural network is modeled as a compound quantum system; its state is presented in tensor product of single-neuron state spaces. Brain’s functions perform self-measurements modeled within theory of open quantum systems. (There is no need to consider state’s collapse.) State’s dynamics of some brain’s function (psychological function) <math>F</math> is described by the quantum master equation. Its steady states represent classical statistical mixtures of possible outputs of <math>F</math> (decisions). Thus through interaction with electrochemical environment, <math>F</math> (considered as an open system) resolves uncertainty that was originally encoded in entangled state representing uncertainties in action potentials of neurons and correlations between them.
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| Entanglement plays the crucial role in generating consistency in neurons’ dynamics. As in Section 11.1, suppose that the quantum information representation is based on 0–1 <math>0-1</math> code. Consider a network of <math>n</math> neurons interacting with the surrounding electrochemical environment <math>\varepsilon</math>, including signaling from other neural networks. The information state is given by (32). Entanglement encodes correlations between firing of individual neurons. For example, the state (33) is associated with two neurons firing synchronically and the state (34) with two neurons firing asynchronically.
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| Outputs of the psychological function <math>F</math> based biophysically on a neural network are resulted from consistent state dynamics of individual neurons belonging to this network. As was already emphasized, state’s evolution toward a steady state is very rapid, as a consequence of linearity of the open system dynamics; the off-diagonal elements of the density matrix decrease exponentially quickly.
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| ==12. Concluding remarks==
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| Since 1990th (Khrennikov, 1999), quantum-like modeling outside of physics, especially modeling of cognition and decision making, flowered worldwide. ''Quantum information theory'' (coupled to measurement and open quantum systems theories) is fertile ground for quantum-like flowers. The basic hypothesis presented in this paper is that functioning of biosystems is based on the quantum information representation of their states. This representation is the output of the biological evolution. The latter is considered as the evolution in the information space. So, biosystems react not only to material or energy constraints imposed by the environment, but also to the information constraints. In this paper, biological functions are considered as open information systems interacting with information environment.
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| The quantum-like representation of information provides the possibility to process superpositions. This way of information processing is advantageous as saving computational resources: a biological function <math>F</math> need not to resolve uncertainties encoded in superpositions and to calculate JPDs of all compatible variables involved in the performance of <math>F</math>.
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| Another advantageous feature of quantum-like information processing is its linearity. Transition from nonlinear dynamics of electrochemical states to linear quantum-like dynamics tremendously speeds up state-processing (for gene-expression, epimutations, and generally decision making). In this framework, decision makers are genes, proteins, cells, brains, ecological systems.
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| Biological functions developed ''the ability to perform self-measurements'', to generate outputs of their functioning. We model this ability in the framework of open quantum systems, as decision making through decoherence. We emphasize that this model is free from the ambiguous notion of collapse of the wave function.
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| Correlations inside a biological function as well as between different biological functions and environment are represented linearly by entangled quantum states.
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| We hope that this paper would be useful for biologists (especially working on mathematical modeling) as an introduction to the quantum-like approach to model functioning of biosystems. We also hope that it can attract attention of experts in quantum information theory to the possibility to use its formalism and methodology in biological studies.
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| ==Declaration of Competing Interest==
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| The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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| ==Acknowledgments==
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| This work was partially supported by JSPS, Japan KAKENHI, Nos. 26247016and 17K19970. M.O. acknowledges the support of the IRI-NU collaboration, Japan .
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