# Order-Stability in Complex Biological, Social, and AI-Systems from Quantum Information Theory

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## Abstract

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## 1. Introduction

#### 1.1. Order-Stability as a Distinguishing Feature of Biosystems

#### 1.2. Information Biology and Physics

#### 1.3. Quantum-Like Models

#### 1.4. Order-Stability in a Biosystem Compounded of a Few Subsystems from a Quantum Information Approach

#### 1.5. Other Applications: Social Science, Management, Artificial Intelligence, Information Retrieval

#### 1.6. The Problem of Self-Measurement

## 2. Classical Entropy

#### 2.1. Micro and Macrostates

#### 2.2. Compound Systems

#### 2.3. Stability of Global Order Is Possible Only with Stable Local Orders

#### 2.4. Classical Information Processing in Biosystem

- frequency (temporal),
- statistical (ensemble).

## 3. Quantum Entropy

#### 3.1. A Few Words about the Quantum Formalism

#### 3.2. Features of von Neumann Entropy

- $\mathcal{S}\left(\rho \right)=0$ if and only if $\rho $ is a pure quantum state, that is, $\rho =|\psi \rangle \langle \psi |.$
- For a unitary operator $U,\mathcal{S}\left(U\rho {U}^{\u2606}\right)=\mathcal{S}\left(\rho \right).$
- The maximum of entropy is approached on the state ${\rho}_{\mathrm{disorder}}=I/N$ and $\mathcal{S}\left({\rho}_{\mathrm{disorder}}\right)=lnN,$ where N is the dimension of the state space.

#### 3.3. Information Processing in Classical vs. Genuine Quantum and Quantum-Like Bio and AI Systems

#### 3.4. States of a Compound System and Its Subsystems, Entanglement

#### 3.5. Compound Systems; Quantum Channels

## 4. Stability of Global Order, in Spite of the Increase of Local Disorder

## 5. Complex Systems

## 6. Quantum Channel Preserving Compound System’s Entropy, in Spite of Increasing of Its Subsystems’ Entropies

#### 6.1. Two Subsystems with Qubit State Spaces

#### 6.2. Two Subsystems with N-dimensional State Spaces

## 7. Quantum Measurement Theory: Self-Observations in Biosystems

## 8. The Indirect Measurement Scheme

- the states of the systems $\gamma $ and the apparatus $M;$ they are represented in complex Hilbert spaces $\mathcal{H}$ and $\mathcal{K},$ respectively;
- the unitary operator U representing the interaction-dynamics for the compound system $\Gamma =(\gamma ,M);$
- the meter observable ${M}_{A}$ giving outputs of the pointer of the apparatus $M.$

#### 8.1. Biosystems

#### 8.2. Consciousness

## 9. Concluding Discussion

- Systems equipped with genuine quantum information processing devices, say quantum computers or simulators.
- Systems equipped with classical information processing devices, say classical digital or analog computers, realizing quantum(-like) information processing.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Khrennikov, A.; Watanabe, N.
Order-Stability in Complex Biological, Social, and AI-Systems from Quantum Information Theory. *Entropy* **2021**, *23*, 355.
https://doi.org/10.3390/e23030355

**AMA Style**

Khrennikov A, Watanabe N.
Order-Stability in Complex Biological, Social, and AI-Systems from Quantum Information Theory. *Entropy*. 2021; 23(3):355.
https://doi.org/10.3390/e23030355

**Chicago/Turabian Style**

Khrennikov, Andrei, and Noboru Watanabe.
2021. "Order-Stability in Complex Biological, Social, and AI-Systems from Quantum Information Theory" *Entropy* 23, no. 3: 355.
https://doi.org/10.3390/e23030355