# Quantum Darwinism in a Composite System: Objectivity versus Classicality

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## Abstract

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## 1. Introduction

## 2. Composite System Model

## 3. Quantum Darwinism and Objectivity

## 4. Strong Quantum Darwinism

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivations of Equations (6) and (7)

#### Appendix A.1. Direct Approach

#### Appendix A.2. Koashi–Winter Relation

## Appendix B. Testing the Strong Quantum Darwinism Criteria

## References

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**Figure 1.**Schematics of the considered model. A composite system that is made up of two interacting qubits, which are also coupled to a fragmented environment. For a excitation preserving interaction between the system qubits, i.e., ${J}_{x}={J}_{y}$ in Equation (2) and pure dephasing interaction between the system qubits and the environment, the interaction Hamiltonians commute, leading to Equation (4).

**Figure 2.**Both system qubits are prepared in a symmetric initial state with ${\theta}_{{S}_{1}}={\theta}_{{S}_{2}}=\pi /6$ with the interaction parameters ${J}_{SE}=1$, $J=10$, and we consider environments of size $N=6$ (solid curves) and $N=250$ (dashed curves). (

**a**) Dynamics of mutual information between the composite system and a single environmental qubit, $\mathcal{I}({\rho}_{{S}_{1}{S}_{2}:{E}_{k}})$, (darker, black) and the entropy of the composite system, $S({\rho}_{{S}_{1}{S}_{2}})$ (lighter, red). (

**b**) Coherence present in the composite system state, $|{\rho}_{{S}_{1}{S}_{2}}^{1,2}|$ (lighter, blue) and coherences in the state of a single environmental qubit $|{\rho}_{{E}_{k}}^{1,2}|$ (darker, black). (

**c**) Quantum discord shared between the two system qubits, ${D}^{\to}({\rho}_{{S}_{1}{S}_{2}})$ (lighter, orange) and quantum discord between one of the system qubits and a single environmental constituent ${D}^{\to}({\rho}_{{S}_{1}{E}_{k}})$ (darker, green). In panels (

**a**–

**c**) the faint vertical line at $t=\pi /4$ denotes the time at which we have $\mathcal{I}({\rho}_{{S}_{1}{S}_{2}:{E}_{k}})=S({\rho}_{{S}_{1}{S}_{2}})$, i.e., the emergence of Darwinism. (

**d**) $\mathcal{I}({\rho}_{{S}_{1}{S}_{2}:{\mathcal{E}}_{f}})/S({\rho}_{{S}_{1}{S}_{2}})$ vs. the size of the environment fraction f (upper, solid) and $\mathcal{I}({\rho}_{{S}_{1}:{\mathcal{E}}_{f}})//S({\rho}_{{S}_{1}})$ (lower, dashed) at two instants of time, $t=\pi /4$ where perfect redundant encoding is observed (lighter, red) and $t=\pi /4-0.1$ (darker, blue).

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**MDPI and ACS Style**

Çakmak, B.; Müstecaplıoğlu, Ö.E.; Paternostro, M.; Vacchini, B.; Campbell, S.
Quantum Darwinism in a Composite System: Objectivity versus Classicality. *Entropy* **2021**, *23*, 995.
https://doi.org/10.3390/e23080995

**AMA Style**

Çakmak B, Müstecaplıoğlu ÖE, Paternostro M, Vacchini B, Campbell S.
Quantum Darwinism in a Composite System: Objectivity versus Classicality. *Entropy*. 2021; 23(8):995.
https://doi.org/10.3390/e23080995

**Chicago/Turabian Style**

Çakmak, Barış, Özgür E. Müstecaplıoğlu, Mauro Paternostro, Bassano Vacchini, and Steve Campbell.
2021. "Quantum Darwinism in a Composite System: Objectivity versus Classicality" *Entropy* 23, no. 8: 995.
https://doi.org/10.3390/e23080995