# Non-Perfect Propagation of Information to a Noisy Environment with Self-Evolution

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Aspects of Emergence of Objective Information on Quantum Ground

## 3. Analytical Model for Three Interacting Qubits

#### 3.1. Derivation of Objectivity Parameters

#### 3.2. Generalised Pointer Basis Optimal for SBS

#### 3.3. Marginal Cases

## 4. Central Interaction: Optimization of Spectrum Broadcast Structure for 2 Environmental Qubits

#### Importance of the Basis Choice

## 5. Central Interaction: Optimization of Spectrum Broadcast Structure for 8 Environmental Qubits

## 6. Non-Central Interaction for Eight Qubits

## 7. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SBS | spectrum broadcast structure |

QD | quantum Darwinism |

SQD | strong quantum Darwinism |

C-NOT | controlled-NOT gate |

C-INOT | controlled imperfect-NOT gate |

$\mathcal{F}$ | fidelity |

$I(\mathcal{S}:\mathcal{E})$ | mutual information between the system and part of the environment |

$H(\xb7)$ | von Neumann entropy |

$\chi (\mathcal{S}:\mathcal{E})$ | Holevo information between $\mathcal{S}$ and $\mathcal{E}$ |

## Appendix A

## Appendix B

## Appendix C

## References

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**Figure 1.**Sample results of SBS basis optimization (15) using the Bloch parametrization (19). We consider the state state after time $t=1$, with Hamiltonian (6) with parameters ${\alpha}_{1}={\alpha}_{3}=0$, for different values of ${\alpha}_{2}$ and environmental mixedness p, cf. (9), and perfect CNOT interaction. Figure 1a contains the minimized distance (17) obtained for $\tilde{p}$, ${x}_{\psi}$ and ${y}_{\psi}$ parameters shown in Figure 1b–d, respectively. Note thin Figure 1a is the same as in Figure 2a (seen from a different angle). For ${x}_{\psi}$ in Figure 1c, we used trigonometric transformation, and thus that the value 1 refers to the computational basis. Note that the phases factor ${y}_{\psi}$ of the Bloch qubit strongly fluctuates in the region where the computational basis is optimal, as in that case ${y}_{\psi}$ has no impact on the state. In Figure 1c, the yellow part corresponds to the standard basis and the light purple represents bases complementary to the standard basis. The latter bases are in general different from Hadamard basis, which can be seen by examination of the phases in Figure 1d. Each of the basis in the light purple region represents some generalised pointer basis (see the discussion at the beginning of Section 3.2).

**Figure 2.**SBS distance for C-INOT central interaction with various values of the gate imperfection parameter $\theta $ with 2 environmental qubits. Each value of $\theta =0,\pi /8,\pi /4,0.9\pi /2$ refers to different interactions between the central system and each of the environmental qubits, as given in (4). The axis ${\alpha}_{2}$ describes the strength of the self-evolution of the environmental qubits, see (7b), and p refers to the initial mixedness of the environmental qubits, see (9). The figure illustrates non-monotonic dependence of the distance of the evolved state from the closes SBS state of the form (15) from the parameters ${\alpha}_{2}$ and p. In particular, it can be seen that, in many cases, it is not the smallest value of mixedness that leads to states close to the SBS form but the “optimal” environment mixedness p depends on the value of the self-evolution strength ${\alpha}_{2}$.

**Figure 3.**Illustration of non-monotonicity of SBS distance from the self-evolution of the environment parameter ${\alpha}_{2}$ and environmental mixedness (noise) p. (

**a**) Dependence of the SBS distance as a function of ${\alpha}_{2}$ for ${\alpha}_{1}={\alpha}_{3}=p=0$ for various values of $\theta $. (

**b**) Dependence of the SBS distance as a function of p for ${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}=0$ for various values of $\theta $.

**Figure 4.**SBS distance for interactions with C-INOT for various gate imperfection parameter $\theta $ with 2 environmental qubits. Visible is the dependence of the optimal environment mixedness p on the value of the inter-environmental-evolution strength ${\alpha}_{3}$ for the Hamiltonian ${H}_{3}=2{1\phantom{\rule{-3.30002pt}{0ex}}1}_{2}\otimes {\sigma}_{Z}\otimes {\sigma}_{Z}$ instead of (7c).

**Figure 5.**The difference $\Delta $, see (22), of SBS distance for interaction with C-INOT various gate imperfection parameter $\theta $ with 2 environmental qubits if the SBS is restricted to be in the Hadamard basis subtracted with the SBS distance if the SBS is restricted to be in the computational basis. The warmer color indicates that the evolved state ${\rho}_{S{\mathcal{E}}_{1}\mathrm{comp}}$ is closer to SBS in the computational basis, and the cooler colour is in those regions, where the evolved state is closer to SBS in the Hadamard basis.

**Figure 6.**SBS distance for C-INOT with various gate imperfection parameters $\theta $ with 8 environmental qubits. Each value of $\theta $ refers to a different interaction between the central system and each of the environmental qubits, as given in (4). The axis ${\alpha}_{2}$ describes the strength of the self-evolution of the environmental qubits, see (24), and p refers to the initial mixedness of the environmental qubits, see (9). The figure illustrate non-monotonic dependence of the upper bound (14) on the distance of the actually evolved state from the closes SBS state of the form (15) on the parameters ${\alpha}_{2}$ and p. In particular, it can be seen that, in many cases, it is not the smallest value of mixedness, which leads to states closing (in an upper bound sense) to the SBS form, but the “optimal” environment mixedness p depends on the value of the self-evolution strength ${\alpha}_{2}$.

**Figure 7.**Upper bound on the distance to an SBS state for 8-qubit environment and ${\alpha}_{1}={\alpha}_{2}=0$ as a function of neighbour–neighbour interaction (25) strength ${\alpha}_{3}$ and mixedness p of the environment.

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**MDPI and ACS Style**

Mironowicz, P.; Horodecki, P.; Horodecki, R.
Non-Perfect Propagation of Information to a Noisy Environment with Self-Evolution. *Entropy* **2022**, *24*, 467.
https://doi.org/10.3390/e24040467

**AMA Style**

Mironowicz P, Horodecki P, Horodecki R.
Non-Perfect Propagation of Information to a Noisy Environment with Self-Evolution. *Entropy*. 2022; 24(4):467.
https://doi.org/10.3390/e24040467

**Chicago/Turabian Style**

Mironowicz, Piotr, Paweł Horodecki, and Ryszard Horodecki.
2022. "Non-Perfect Propagation of Information to a Noisy Environment with Self-Evolution" *Entropy* 24, no. 4: 467.
https://doi.org/10.3390/e24040467