# Correlations, Information Backflow, and Objectivity in a Class of Pure Dephasing Models

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## Abstract

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## 1. Introduction

## 2. Dephasing Models

## 3. Spreading of Correlations

#### 3.1. Quantifiers of Correlations

#### 3.2. Model Dependence of Correlation Formation

## 4. Information Backflow

#### 4.1. Model Dependence of Bounds on Distinguishability Revivals

#### 4.2. Fraction Dependence of Bounds on Distinguishability Revivals

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Amount of correlations shared between the system initially in the plus state $|+\rangle =(1/\sqrt{2})\left(\right|0\rangle +|1\rangle )$ and one of the environmental qubits, evaluated by considering the QJSD${}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, comparing this bipartite state with the product of its marginal as a function of time (in inverse units of the coupling parameter) and of the value c of initial coherence in the environmental states. The quantity is renormalised to the value corresponding to a maximally entangled state. Here and in the following figures, the environment is composed of $N=8$ units. The black and red lines correspond to $c=0$ and $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. (

**b**) Distance between total state and product of its marginals at the reference time $gs=\pi /4$ as quantified by the QJSD${}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, expressed as a function of the fraction of considered environmental qubits and of the value, c, of coherences in the environmental states. The total state includes the system and a fraction, $\mathsf{f}$, of the environmental qubits. (

**c**) The same quantity obtained considering as the quantifier the relative entropy, thus recovering the mutual information, still keeping the normalisation to the value corresponding to the maximally entangled state. In both figures, we see the emergence of a plateau for $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, which is gradually washed out for smaller values of c, namely when moving from a model in which the environmental units have coherences to a fully diagonal state.

**Figure 2.**Bloch sphere representation of the considered pair of initial system states. One state is fixed to be the equatorial plus state $|+\rangle =(1/\sqrt{2})\left(\right|0\rangle +|1\rangle )$ (black dot), while the other element of the pair belongs to the maximum circle and is characterised by the angle $\theta $ (red dot). For $\theta =\pi /2$ it becomes the minus state $|-\rangle =(1/\sqrt{2})\left(\right|0\rangle -|1\rangle )$ and one recovers an orthogonal pair of initial states. For $\theta =0$, it corresponds to the up state $|1\rangle $.

**Figure 3.**Plot of the l.h.s. of Equation (20) showing the revivals of the QJSD${}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ as a function of time and choice of initial system states. The reference time $gt$ is fixed to be $\pi /2$, i.e., after one full period of the evolution, while $gs$ sweeps from 0 to $\pi /2$. The initial pair of system states is given by ${\rho}_{S}^{1}\left(0\right)=|+\rangle \langle +|$ and ${\rho}_{S}^{2}\left(0\right)=|\theta \rangle \langle \theta |$, as shown in Figure 2, with $\theta $ ranging from 0 to $\pi /2$, corresponding to the case of an orthogonal pair and maximizing the revivals.

**Figure 4.**Plot of the different contributions at the r.h.s. of Equation (20), together with their sum, all quantified via the QJSD${}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. They are considered as a function of running time $gs$ and initial pair of states fixed by the angle $\theta $. The first row corresponds, as indicated, to the model determined by $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, the second to $c=0$. For $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, the environmental units have the maximum amount of coherence, while for $c=0$, they start in a maximally mixed state and the reduced environmental state remains unchanged, so that one of the contributions is equal to zero.

**Figure 5.**Plot of the different contributions at the r.h.s. of Equation (20), together with their sum, evaluated for the case in which the total state is replaced by a marginal obtained by tracing out some environmental qubits, so that only a fraction $\mathsf{f}$ is considered. Additionally, in this case, all quantities are expressed via the QJSD${}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. They are considered a function of fraction $\mathsf{f}$ and initial pair of states determined by the angle $\theta $ for a fixed time $gs=\pi /4$. The first row corresponds, as indicated, to the model determined by $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, the second to $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ and the third to $c=0$. For $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, plateaus as a function of $\mathsf{f}$ are clearly observed, replaced for $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ by a weak dependence. For $c=0$, a non-zero value is only obtained when including the whole environment since tracing over any environmental units leads to a factorised state.

**Figure 6.**Behaviour of environmental changes and correlations for the model with $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ plotted as a function of both fraction $\mathsf{f}$ and time $gs$. It immediately appears that a plateau as a function of $\mathsf{f}$ only takes place for the time $gs=\pi /4$, corresponding to full decoherence of the system.

**Figure 7.**(

**Left**) Plot of the l.h.s. of Equation (20) showing the revivals of the QJSD${}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ as a function of choice of initial system states for the fixed times $gt=\pi /2$ and $gs=\pi /4$, the latter corresponding to maximal decoherence of the system. The quantity inherently does not depend on $\mathsf{f}$. (

**Right**) The difference of the l.h.s. and sum of quantities on the r.h.s. of the inequality given by Equation (20) when taking into account only a fraction $\mathsf{f}$ of the environment, for $c=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ (see Figure 5, last figure of the second row). For the values of environmental fraction $\mathsf{f}$ and angle $\theta $ (which determines the pair of initial system states) for which the difference is negative (red), the corresponding sum of environmental changes and correlations is no longer an upper bound for the revivals in the reduced dynamics.

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**MDPI and ACS Style**

Megier, N.; Smirne, A.; Campbell, S.; Vacchini, B.
Correlations, Information Backflow, and Objectivity in a Class of Pure Dephasing Models. *Entropy* **2022**, *24*, 304.
https://doi.org/10.3390/e24020304

**AMA Style**

Megier N, Smirne A, Campbell S, Vacchini B.
Correlations, Information Backflow, and Objectivity in a Class of Pure Dephasing Models. *Entropy*. 2022; 24(2):304.
https://doi.org/10.3390/e24020304

**Chicago/Turabian Style**

Megier, Nina, Andrea Smirne, Steve Campbell, and Bassano Vacchini.
2022. "Correlations, Information Backflow, and Objectivity in a Class of Pure Dephasing Models" *Entropy* 24, no. 2: 304.
https://doi.org/10.3390/e24020304