# Composite Discordant States and Quantum Darwinism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of Relevant Figures of Merit

## 3. Description of the Model

- 1.
- The coupling between qubit ${S}_{j}$ ($j=1,2$) of the bipartite system and the environment. The interaction model that we consider in this case is$${H}_{j{\mathcal{F}}_{k}}=J{\sigma}_{{S}_{j}}^{z}\otimes \sum _{i}{\sigma}_{i}^{z}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(j=1,2).$$
- 2.
- The simultaneous, yet individual coupling of the system qubits with the elements of the environment. The model to consider in this case reads$${H}_{\left({S}_{1}{S}_{2}\right){\mathcal{F}}_{k}}=J({\sigma}_{{S}_{1}}^{z}+{\sigma}_{{S}_{2}}^{z})\otimes \sum _{i}{\sigma}_{i}^{z}$$
- 3.
- The simultaneous (three-body) interaction of both the system’s qubits and the ${i}^{\mathrm{th}}$ element of the environment. This coupling would be described by the Hamiltonian$${H}_{\left({S}_{1}{S}_{2}{\mathcal{F}}_{k}\right)}=J\sum _{i}{\sigma}_{{S}_{1}}^{z}\otimes {\sigma}_{{S}_{2}}^{z}\otimes {\sigma}_{i}^{z}$$

**Figure 2.**Illustration of the physical configurations implied by the Hamiltonian models in Equation (13) [panel (

**a**)], Equation (14) [panel (

**b**)] and Equation (15) [panel (

**c**)]. Solid straight lines represent Hamiltonian coupling. The subsystems ${S}_{1}$ and ${S}_{2}$ are prepared in the discordant state ${\rho}_{S}$ while each of the environmental elements in $|e\rangle $.

## 4. Results

#### 4.1. ${S}_{1}$ Coupled to the Environment

#### 4.2. ${S}_{2}$ Coupled to the Environment

#### 4.3. Simultaneous Individual Couplings

#### 4.4. Simultaneous Three-Body Interaction

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof.**

**Proof.**

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**Figure 1.**The Mutual Information plateau that emerges when full redundancy of the systems information is achieved. Further fragments reveal no information, with the sharp rise at the end attributed to the quantum correlations only available with the whole environment. The curved rise, as opposed to the sharp edges observed in all the other plots in this work, occurs when the smallest constituents of the environment are incapable of encoding all information regarding the system.

**Figure 3.**Panel (

**a**,

**b**): Mutual information (Holevo information) between subsystem ${S}_{1}$ and and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13). Panel (

**c**): Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and ${S}_{1}$ for the same Hamiltonian model as in panels (

**a**,

**b**). In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time. The horizontal axis on all graphs is in terms of the Mutual Information normalized using the systems of interests entropy.

**Figure 4.**Mutual information, Holevo information and Quantum Discord between subsystem ${S}_{2}$ and and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13). The horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time. The horizontal axis is in terms of the Mutual Information normalized using the systems of interests entropy.

**Figure 5.**Panel (

**a**,

**b**): Mutual information (Holevo information) between the whole system and and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13). Panel (

**c**): Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and the whole system for the same Hamiltonian model as in panels (

**a**,

**b**). In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time. The horizontal axis on all graphs is in terms of the Mutual Information normalized using the systems of interests entropy.

**Figure 6.**Panel (

**a**,

**b**): Mutual information (Holevo information) between the whole system and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13), this time given measurements on the environment. Panel (

**c**): Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and the whole system for the same Hamiltonian model as in panels (

**a**,

**b**), this time given measurements on the environment. In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time. The horizontal axis on all graphs is in terms of the Mutual Information normalized using the systems of interests entropy.

**Figure 7.**Panel (

**a**,

**b**): Mutual information (Holevo information) between ${S}_{1}$ and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13). Panel (

**c**): Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and ${\rho}_{{S}_{1}}$ for the same Hamiltonian model as in panels (

**a**,

**b**). In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time. The horizontal axis on all graphs is in terms of the Mutual Information normalized using the systems of interests entropy.

**Figure 8.**Mutual information, Holevo information and Quantum Discord between ${S}_{2}$ and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13). In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time.

**Figure 9.**Mutual information, Holevo information and Quantum Discord between the whole system and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13). In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time.

**Figure 10.**Panel (

**a**,

**b**): Mutual information (Holevo information) between the whole system and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (14). Panel (

**c**): Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and the whole system for the same Hamiltonian model as in panels (

**a**,

**b**). In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time. The quantities on the vertical axis on all graphs reported int he manuscript are in units of the entropy of the systems of interests.

**Figure 11.**Panel (

**a**): Mutual information, Holevo information and Quantum Discord between ${S}_{2}$ and and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13). Panel (

**b**): Mutual information, Holevo information and Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and the whole system for the same Hamiltonian model as in panels (

**a**). In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time.

**Figure 12.**Panel (

**a**,

**b**): Mutual information (Holevo information) between ${S}_{1}$ and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (14) given measurements on the fragments. Panel (

**c**): Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and ${S}_{1}$ for the same Hamiltonian model as in panels (

**a**,

**b**) given measurements on the fragments. In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time. The quantities on the vertical axis on all graphs are in units of the entropy of the systems of interests.

**Figure 13.**Panel (

**a**): Mutual information, Holevo information and Quantum Discord between ${S}_{2}$ and and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (13) given measurements on the fragments. Panel (

**b**): Mutual information, Holevo information and Quantum Discord between the environmental fragments ${\mathcal{F}}_{k}$ and the whole system for the same Hamiltonian model as in panel (

**a**), given measurements on the fragments. In all panels, the horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time.

**Figure 14.**Mutual information, Holevo information and Quantum Discord between ${S}_{2}$ and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (15). The horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time.

**Figure 15.**Mutual information, Holevo information and Quantum Discord between the whole system and environmental fraction ${\mathcal{F}}_{k}$ for the Hamiltonian model in Equation (15). The horizontal axis shows the size of the environment ${\mathcal{E}}^{\#}$, while for all the simulations considered we have taken $Jt=10$ with t the evolution time.

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Ryan, E.; Paternostro, M.
Composite Discordant States and Quantum Darwinism. *Entropy* **2022**, *24*, 1509.
https://doi.org/10.3390/e24111509

**AMA Style**

Ryan E, Paternostro M.
Composite Discordant States and Quantum Darwinism. *Entropy*. 2022; 24(11):1509.
https://doi.org/10.3390/e24111509

**Chicago/Turabian Style**

Ryan, Eoghan, and Mauro Paternostro.
2022. "Composite Discordant States and Quantum Darwinism" *Entropy* 24, no. 11: 1509.
https://doi.org/10.3390/e24111509