# Quantum Information in Neural Systems

## Abstract

**:**

## 1. Introduction

## 2. Basic Postulates of Quantum Mechanics

**Axiom**

**1.**

**Axiom**

**2.**

**Axiom**

**3.**

**Axiom**

**4.**

**Axiom**

**5.**

## 3. Minimal Quantum Toy Model

#### 3.1. Hamiltonian of the Toy Model

#### 3.2. Energy Eigenstates and Eigenvalues of the Toy Model

#### 3.3. Quantum Dynamics of the State Vector

## 4. Quantum Dynamic Timescale

## 5. Quantum Entanglement

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Definition**

**3.**

**Theorem**

**3.**

## 6. Quantum Coherence

**Definition**

**4.**

**Example**

**1.**

**Definition**

**5.**

**Example**

**2.**

**Definition**

**6.**

**Proof.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 7. Measurement of Quantum Observables

#### 7.1. Quantum Observables in Spin $zz$ Basis

#### 7.2. Complementary Observables in Spin $xx$ Basis

## 8. Quantum Dynamics of Initial Quantum Entangled States

**Theorem**

**4.**

**Proof.**

## 9. Quantum Coherence Cannot Bind Conscious Experiences

**Definition**

**7.**

**Definition**

**8.**

**Example**

**6.**

**Theorem**

**5.**

**Proof.**

**Definition**

**9.**

**Example**

**7.**

## 10. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Different levels of organization of physical processes within the central nervous system. At the microscopic scale, the brain cortex is composed of neurons, which form neural networks. The morphology of the rendered pyramidal neuron (NMO_09565) from layer 5 of rat motor cortex (http://NeuroMorpho.Org; accessed on 19 April 2021) reflects the functional specialization of cable-like neuronal projections (dendrites and axon). At the nanoscale, the electric activity of neurons is generated by voltage-gated ion channels, which are inserted in the neuronal plasma membrane. As an example of ion channel is shown a single voltage-gated K

^{+}channel composed of four protein $\alpha $-subunits. Each subunit has six $\alpha $-helices traversing the plasma membrane. The 4th $\alpha $-helix is positively charged and acts as voltage sensor. At the picoscale, individual elementary electric charges within the protein voltage sensor could be modeled as qubits represented by Bloch spheres. For the diameter of each qubit is used the Compton wavelength of electron. Consecutive magnifications from micrometer (μm) to picometer (pm) scale are indicated by × symbol.

**Figure 2.**Expectation values of the projectors $\widehat{\mathcal{P}}\left({\uparrow}_{z}{\uparrow}_{z}\right)$, $\widehat{\mathcal{P}}\left({\uparrow}_{z}{\downarrow}_{z}\right)$, $\widehat{\mathcal{P}}\left({\downarrow}_{z}{\uparrow}_{z}\right)$ and $\widehat{\mathcal{P}}\left({\downarrow}_{z}{\downarrow}_{z}\right)$ corresponding to probabilities of obtaining the given measurement outcomes for the z-spin components of the two qubits. The initial state $|\mathsf{\Psi}(0)\rangle $ at $t=0$ is ${|\uparrow}_{z}{\uparrow}_{z}\rangle $ in panel (

**A**), ${|\uparrow}_{z}{\downarrow}_{z}\rangle $ in panel (

**B**), ${|\downarrow}_{z}{\uparrow}_{z}\rangle $ in panel (

**C**) and ${|\downarrow}_{z}{\downarrow}_{z}\rangle $ in panel (

**D**). The internal Hamiltonians were modeled with ${\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{2}=0.3$ rad/ps. The interaction Hamiltonian was non-zero with ${\omega}_{s}=0.3$ rad/ps. The amount of quantum entanglement at each moment of time was measured using the normalized entanglement number $e\left(\mathsf{\Psi}\right)/e{\left(\mathsf{\Psi}\right)}_{max}$.

**Figure 3.**Expectation values of the projectors $\widehat{\mathcal{P}}\left({\uparrow}_{z}{\uparrow}_{z}\right)$, $\widehat{\mathcal{P}}\left({\uparrow}_{z}{\downarrow}_{z}\right)$, $\widehat{\mathcal{P}}\left({\downarrow}_{z}{\uparrow}_{z}\right)$ and $\widehat{\mathcal{P}}\left({\downarrow}_{z}{\downarrow}_{z}\right)$ corresponding to probabilities of obtaining the given measurement outcomes for the z-spin components of the two qubits. The initial state $|\mathsf{\Psi}(0)\rangle $ at $t=0$ is ${|\uparrow}_{z}{\uparrow}_{z}\rangle $ in panel (

**A**), ${|\uparrow}_{z}{\downarrow}_{z}\rangle $ in panel (

**B**), ${|\downarrow}_{z}{\uparrow}_{z}\rangle $ in panel (

**C**) and ${|\downarrow}_{z}{\downarrow}_{z}\rangle $ in panel (

**D**). The internal Hamiltonians were modeled with ${\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{2}=0.3$ rad/ps. The interaction Hamiltonian was zero with ${\omega}_{s}=0$ rad/ps. The amount of quantum entanglement at each moment of time was measured using the normalized entanglement number $e\left(\mathsf{\Psi}\right)/e{\left(\mathsf{\Psi}\right)}_{max}$.

**Figure 4.**Expectation values of the projectors $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\uparrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\downarrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\uparrow}_{x}\right)$ and $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\downarrow}_{x}\right)$ corresponding to probabilities of obtaining the given measurement outcomes for the x-spin components of the two qubits. The initial state $|\mathsf{\Psi}(0)\rangle $ at $t=0$ is ${|\uparrow}_{x}{\uparrow}_{x}\rangle $ in panel (

**A**), ${|\uparrow}_{x}{\downarrow}_{x}\rangle $ in panel (

**B**), ${|\downarrow}_{x}{\uparrow}_{x}\rangle $ in panel (

**C**) and ${|\downarrow}_{x}{\downarrow}_{x}\rangle $ in panel (

**D**). The internal Hamiltonians were modeled with ${\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{2}=0.3$ rad/ps. The interaction Hamiltonian was non-zero with ${\omega}_{s}=0.3$ rad/ps. The amount of quantum entanglement at each moment of time was measured using the normalized entanglement number $e\left(\mathsf{\Psi}\right)/e{\left(\mathsf{\Psi}\right)}_{max}$.

**Figure 5.**Expectation values of the projectors $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\uparrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\downarrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\uparrow}_{x}\right)$ and $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\downarrow}_{x}\right)$ corresponding to probabilities of obtaining the given measurement outcomes for the x-spin components of the two qubits. The initial state $|\mathsf{\Psi}(0)\rangle $ at $t=0$ is ${|\uparrow}_{x}{\uparrow}_{x}\rangle $ in panel (

**A**), ${|\uparrow}_{x}{\downarrow}_{x}\rangle $ in panel (

**B**), ${|\downarrow}_{x}{\uparrow}_{x}\rangle $ in panel (

**C**) and ${|\downarrow}_{x}{\downarrow}_{x}\rangle $ in panel (

**D**). The internal Hamiltonians were modeled with ${\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{2}=0.3$ rad/ps. The interaction Hamiltonian was zero with ${\omega}_{s}=0$ rad/ps. The amount of quantum entanglement at each moment of time was measured using the normalized entanglement number $e\left(\mathsf{\Psi}\right)/e{\left(\mathsf{\Psi}\right)}_{max}$.

**Figure 6.**Expectation values of the projectors $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\uparrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\downarrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\uparrow}_{x}\right)$ and $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\downarrow}_{x}\right)$ corresponding to probabilities of obtaining the given measurement outcomes for the x-spin components of the two qubits. The initial state $|\mathsf{\Psi}(0)\rangle $ at $t=0$ is $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\uparrow}_{x}{\rangle +|\downarrow}_{z}{\downarrow}_{x}\rangle \right)$ in panel (

**A**), $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\downarrow}_{x}{\rangle +|\downarrow}_{z}{\uparrow}_{x}\rangle \right)$ in panel (

**B**), $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\uparrow}_{x}{\rangle -|\downarrow}_{z}{\downarrow}_{x}\rangle \right)$ in panel (

**C**) and $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\downarrow}_{x}{\rangle -|\downarrow}_{z}{\uparrow}_{x}\rangle \right)$ in panel (

**D**). The internal Hamiltonians were modeled with ${\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{2}=0.3$ rad/ps. The interaction Hamiltonian was non-zero with ${\omega}_{s}=0.3$ rad/ps. The amount of quantum entanglement at each moment of time was measured using the normalized entanglement number $e\left(\mathsf{\Psi}\right)/e{\left(\mathsf{\Psi}\right)}_{max}$.

**Figure 7.**Expectation values of the projectors $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\uparrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\uparrow}_{x}{\downarrow}_{x}\right)$, $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\uparrow}_{x}\right)$ and $\widehat{\mathcal{P}}\left({\downarrow}_{x}{\downarrow}_{x}\right)$ corresponding to probabilities of obtaining the given measurement outcomes for the x-spin components of the two qubits. The initial state $|\mathsf{\Psi}(0)\rangle $ at $t=0$ is $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\uparrow}_{x}{\rangle +|\downarrow}_{z}{\downarrow}_{x}\rangle \right)$ in panel (

**A**), $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\downarrow}_{x}{\rangle +|\downarrow}_{z}{\uparrow}_{x}\rangle \right)$ in panel (

**B**), $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\uparrow}_{x}{\rangle -|\downarrow}_{z}{\downarrow}_{x}\rangle \right)$ in panel (

**C**) and $\frac{1}{\sqrt{2}}\left({|\uparrow}_{z}{\downarrow}_{x}{\rangle -|\downarrow}_{z}{\uparrow}_{x}\rangle \right)$ in panel (

**D**). The internal Hamiltonians were modeled with ${\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{2}=0.3$ rad/ps. The interaction Hamiltonian was zero with ${\omega}_{s}=0$ rad/ps. The amount of quantum entanglement at each moment of time was measured using the normalized entanglement number $e\left(\mathsf{\Psi}\right)/e{\left(\mathsf{\Psi}\right)}_{max}$.

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Georgiev, D.D. Quantum Information in Neural Systems. *Symmetry* **2021**, *13*, 773.
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Georgiev DD. Quantum Information in Neural Systems. *Symmetry*. 2021; 13(5):773.
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Georgiev, Danko D. 2021. "Quantum Information in Neural Systems" *Symmetry* 13, no. 5: 773.
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