# Quantum Mutual Information, Fragile Systems and Emergence

^{1}

^{2}

^{2}mc), Comisión Chilena de Energía Nuclear, Casilla 188-D, Santiago 8320000, Chile

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Emergent Systems

**Non-reducible phenomenon**: the global state of the emergent system cannot be explainable, and neither is it reducible to its sub-system components.**Downward causation**: emergent high-level properties appear from a non-obvious consequence of low-level properties, but, at the same time, all processes at the lower level of hierarchy are constrained by and act in coherence with the laws of the higher level [14].**Wholeness**: a phenomenon wherein a complex, interesting high-level function appears as a result of combining low-level mechanisms in straightforward ways.**Radical novelty emergence**: a phenomenon wherein a system is designed according to certain principles. Interesting unexpected properties arise from the behavior of sub-system elements [15].

## 3. Fragile Systems

## 4. The Density Operator

#### Pure and Mixed States

## 5. Density Operator Formalism and Emergent Systems

## 6. An Example of a Subadditive System

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**The set of joint probabilities ${p}_{ij}=P(\mathit{s}={\mathit{s}}_{i},{\mathit{s}}^{\prime}={\mathit{s}}_{j}|\mathcal{I})$ (third column) for the fragile system arising from the two-integer example described in Section 6. The fourth column shows the values of the elements of the corresponding density matrix.

i | j | ${\mathit{p}}_{\mathit{ij}}$ | ${\mathit{\rho}}_{\mathit{ij}}$ |
---|---|---|---|

1 | 1 | ¼ | ½ |

1 | 2 | 0 | 0 |

1 | 3 | 0 | 0 |

1 | 4 | ¼ | ½ |

2 | 1 | 0 | 0 |

2 | 2 | 0 | 0 |

2 | 3 | 0 | 0 |

2 | 4 | 0 | 0 |

3 | 1 | 0 | 0 |

3 | 2 | 0 | 0 |

3 | 3 | 0 | 0 |

3 | 4 | 0 | 0 |

4 | 1 | ¼ | ½ |

4 | 2 | 0 | 0 |

4 | 3 | 0 | 0 |

4 | 4 | ¼ | ½ |

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**MDPI and ACS Style**

Navarrete, Y.; Davis, S.
Quantum Mutual Information, Fragile Systems and Emergence. *Entropy* **2022**, *24*, 1676.
https://doi.org/10.3390/e24111676

**AMA Style**

Navarrete Y, Davis S.
Quantum Mutual Information, Fragile Systems and Emergence. *Entropy*. 2022; 24(11):1676.
https://doi.org/10.3390/e24111676

**Chicago/Turabian Style**

Navarrete, Yasmín, and Sergio Davis.
2022. "Quantum Mutual Information, Fragile Systems and Emergence" *Entropy* 24, no. 11: 1676.
https://doi.org/10.3390/e24111676