# Pointer States and Quantum Darwinism with Two-Body Interactions

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## Abstract

**:**

## 1. Introduction

## 2. Structure of the Hamiltonian

#### 2.1. Existence of a Pointer Basis

#### 2.2. Further Conditions for Quantum Darwinism

## 3. Coefficients of the Hamiltonian

#### 3.1. Solving the Dynamics

#### 3.2. Rate and Irreversibility of Information Transfer

#### 3.3. Quantum Darwinism—The Classical Plateau

## 4. Representative Examples

#### 4.1. Continuous Parallel Decoherence Interaction

#### 4.2. Discrete Parallel Decoherence Interaction

#### 4.3. Continuous Orthogonal Decoherence Interaction

#### 4.4. Continuous Parallel Decoherence Interaction with Scrambling

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Hamiltonians with a Pointer Basis

## Appendix B. Average Decoherence Factors

## Appendix C. Mutual Information and Asymptotics

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**Figure 1.**Two-body interactions between $\mathfrak{S}$ (red) and $\mathfrak{E}$ (blue). Lines with ⊗ depict interaction terms. (

**a**) Most general scenario in which $\mathfrak{S}$ might separately interact with all qubits in $\mathfrak{E}$. (

**b**) Scenario with separable interaction, i.e., $\mathfrak{S}$ interacts with $\mathfrak{E}$ through a global tensor product structure; this scenario supports a pointer basis for $\mathfrak{S}$.

**Figure 3.**Mutual information $I(\mathfrak{S}:\mathfrak{F})$ as a function of time and fragment size for an arbitrary separable initial state of 9 qubits ($N=8$), divided by the von Neumann entropy of $\mathfrak{S}$ (averaged over ${10}^{2}$ realizations). (

**a**) CPDI (32): irreversible information transfer and classical objectivity. (

**b**) DPDI (33): non-irreversible (periodic) information transfer. (

**c**) CODI (34): no local information transfer or redundancy. (

**d**) CPDI-S (35): competition of emergence of classical objectivity and scrambling in $\mathfrak{E}$.

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**MDPI and ACS Style**

Duruisseau, P.; Touil, A.; Deffner, S.
Pointer States and Quantum Darwinism with Two-Body Interactions. *Entropy* **2023**, *25*, 1573.
https://doi.org/10.3390/e25121573

**AMA Style**

Duruisseau P, Touil A, Deffner S.
Pointer States and Quantum Darwinism with Two-Body Interactions. *Entropy*. 2023; 25(12):1573.
https://doi.org/10.3390/e25121573

**Chicago/Turabian Style**

Duruisseau, Paul, Akram Touil, and Sebastian Deffner.
2023. "Pointer States and Quantum Darwinism with Two-Body Interactions" *Entropy* 25, no. 12: 1573.
https://doi.org/10.3390/e25121573