# Quantum Non-Markovian Environment-to-System Backflows of Information: Nonoperational vs. Operational Approaches

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## Abstract

**:**

## 1. Introduction

## 2. Quantum Non-Markovianity

#### 2.1. Nonoperational Approach

#### 2.2. Operational Approach

#### 2.3. Bipartite Propagator vs. Single Propagator

## 3. Comparing Both Approaches

#### 3.1. Born–Markov Approximation

#### 3.2. Casual Bystander Environments

#### 3.2.1. Classical Mixture of Quantum Markovian Dynamics

#### 3.2.2. Interaction with Stochastic Classical Degrees of Freedom

#### 3.2.3. Environmental Quantum Degrees of Freedom

#### 3.3. Unitary System–Environment Interactions

## 4. Example

#### 4.1. Depolarizing Dynamics

#### 4.2. Operational vs. Nonoperational Quantum Non-Markovianity

#### 4.3. Environment-to-System Backflow of Information

#### 4.3.1. Slow Modulation of the Stationary Environment State

#### 4.3.2. Quantum Coherent Contributions in the Environment Dynamics

## 5. Summary and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 1.**(

**a**) Decay of the trace distance $d\left(t\right)$ (Equation (38)) corresponding to the models (30a) and (30b). (

**b**) Time dependence of the CPF correlation ${C}_{pf}{(t,t)|}_{\stackrel{\u02d8}{y}}$ in the deterministic scheme [56] corresponding to the same model. The value of the quotient $\varphi /\gamma $ is indicated in each plot.

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Budini, A.A.
Quantum Non-Markovian Environment-to-System Backflows of Information: Nonoperational vs. Operational Approaches. *Entropy* **2022**, *24*, 649.
https://doi.org/10.3390/e24050649

**AMA Style**

Budini AA.
Quantum Non-Markovian Environment-to-System Backflows of Information: Nonoperational vs. Operational Approaches. *Entropy*. 2022; 24(5):649.
https://doi.org/10.3390/e24050649

**Chicago/Turabian Style**

Budini, Adrián A.
2022. "Quantum Non-Markovian Environment-to-System Backflows of Information: Nonoperational vs. Operational Approaches" *Entropy* 24, no. 5: 649.
https://doi.org/10.3390/e24050649