# Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Non-Markovinity Definition

#### 2.2. HSS-Based Witness of Non-Markovianity

#### 2.3. Quantum Entanglement Measure

#### 2.4. Quantum Correlation Quantifier: Measurement-Induced Disturbance

## 3. Analyzing the Efficiency of the HSS Witness in High-Dimensional Systems with Finite Hilbert Spaces

#### 3.1. Single-Qudit Interacting with a Quantum Environment

#### 3.1.1. Coupling to a Thermal Reservoir

#### 3.1.2. Coupling to a Squeezed Vacuum Reservoir

#### 3.2. Hybrid Qubit–Qutrit System Interacting with Various Quantum and Classical Environments

#### 3.2.1. Coupling to Independent Squeezed Vacuum Reservoirs

**Pure initial state.**We take the hybrid qubit–qutrit system initially in a pure state given by [61]

**Mixed initial state.**The non-Markovianity of the system, as faithfully individuated by quantum correlation measures, may in general depend on the initial state. It is thus important to investigate whether the HSS witness, obtained from the initial pure state of Equation (25) by definition, is capable to identify the non-Markovian character of the system dynamics also when the system starts from a mixed state. We shall study this aspect here and in all the other environmental conditions considered hereafter (see sections below devoted to a mixed initial state).

#### 3.2.2. Coupling to Classical Environments

- (1)
- Local or independent environments $\left(ie\right)$: ${J}_{k}=\nu \ne 0$ and ${J}_{c}=0$;
- (2)
- Non-local or common environments $\left(ce\right)$: ${J}_{k}=0$ and ${J}_{c}=\nu \ne 0$.

**Pure initial state in the presence of independent classical environments.**Here, we assume that each of the qubits and qutrits interact locally with local RTN, while the composite system starts with the pure initial state in Equation (25). For this case, the elements of evolved density matrix are given in Appendix A.2. Then the HSS is obtained as

**Mixed initial state in the presence of independent classical environments.**Now we compare the dynamics of the HSS, obtained from the initial pure state of Equation (25), with the evolution of the negativity and quantum correlation computed for the initial mixed state of Equation (27). The evolved density matrix, the corresponding negativity and quantum correlation are obtained from, respectively, Equations (29)–(31) replacing $\mathcal{F}$ with ${{D}_{2}\left(\tau \right)}^{2}$.

**Mixed initial state in the presence of a common classical environment.**Let us now compare the dynamics of the HSS, obtained as usual from the initial pure state of Equation (25) by definition, with the evolution of the negativity and quantum correlation computed for the initial mixed state of Equation (27), when both the qubit and the qutrit are embedded into a common RTN source in the non-Markovian regime. The elements of the evolved dynamical density matrix are given in Appendix A.3. Then, one can easily determine the HSS as

#### 3.2.3. Composite Classical-Quantum Environments

**Pure initial state.**The elements of the evolved density matrix when starting from the pure state of Equation (25) are given in Appendix A.4, leading to the following expression for the HSS:

**Mixed initial state.**Using Equation (27) as the initial state and computing the evolved state of the system (see Appendix B.4), we find that the the negativity and MID, respectively, are in the form of Equations (30) and (31) with $\mathcal{F}={D}_{2}\left(\tau \right){e}^{-4\gamma \left(t\right)}$. In Figure 8, the dynamics of negativity and MID, obtained for the initial mixed state, has been compared with that of the HSS (computed for the initial pure state) in the non-Markovian regime.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Pure Hybrid Qubit–Qutrit Evolved Density Matrix

#### Appendix A.1. Squeezed Vacuum Reservoirs

#### Appendix A.2. Independent Random Telegraph Noise

#### Appendix A.3. Common Random Telegraph Noise

#### Appendix A.4. Composite Classical-Quantum Environments

## Appendix B. Mixed Hybrid Qubit–Qutrit Evolved Density Matrix

#### Appendix B.1. Squeezed Vacuum Reservoirs

#### Appendix B.2. Independent Random Telegraph Noise

#### Appendix B.3. Common Random Telegraph Noise

#### Appendix B.4. Composite Classical-Quantum Environments

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**Figure 1.**Illustration of the composite qubit(A)-qutrit(B) system; Blue dashed lines represent entanglement between the subsystems. The bipartite system can interact either with independent local environments ${E}_{A}$, ${E}_{B}$ or with a common environment ${E}_{C}$.

**Figure 2.**Evolution of the negativity, MID and HSS as a function of dimensionless time $\tau ={\omega}_{0}t$ when each subsystem of the hybrid qubit–qutrit system, starting from the initial pure state, is independently subject to a squeezed vacuum reservoir. The values of the other parameters are $\alpha =0.1$, ${\omega}_{c}=20{\omega}_{0}$, $r=0.3$, $\varphi =\pi $ and $s=3$.

**Figure 3.**Comparing the evolution of negativity and MID computed for the initial mixed state of the hybrid qubit–qutrit system, when each subsystem is independently coupled to a squeezed vacuum reservoir, with HSS (obtained from the initial pure state) for different values of the entanglement parameter p. In all plots the remaining parameters are $\alpha =0.1$, $s=3$, ${\omega}_{c}=20\phantom{\rule{3.33333pt}{0ex}}{\omega}_{0}$, $r=0.3$.

**Figure 4.**Evolution of negativity, MID and HSS as a function of dimensionless time $\tau =\nu t$ when each subsystem of the hybrid qubit–qutrit system, starting from the initial pure state, is independently subject to a random telegraph noise in non-Markovian regime $q=0.1$.

**Figure 5.**Comparing the evolution of negativity and MID computed for the initial mixed state of the hybrid qubit–qutrit system, when each subsystem is independently coupled to a random telegraph noise, with HSS (obtained from the initial pure state) for different values of the entanglement parameter p in the non-Markovian regime: $q=0.1$.

**Figure 6.**Comparing the evolution of negativity and MID computed for the initial mixed state of the hybrid qubit–qutrit system, when its subsystems are subject to a common RTN source, with HSS (obtained from the initial pure state) for different values of the entanglement parameter p in the non-Markovian regime: $q=0.1$.

**Figure 7.**Evolution of negativity, MID and HSS as a function of dimensionless time $\tau $ when the subsystems of the hybrid qubit–qutrit system, starting from the initial pure state, are independently subject to composite classical-quantum environments. The values of the other parameters are given by $\alpha =0.1$, ${\omega}_{c}=20\phantom{\rule{3.33333pt}{0ex}}{\omega}_{0}$, $r=0.3$, and $\nu =100$.

**Figure 8.**Comparing the evolution of the negativity and MID, computed for the initial mixed state of the hybrid qubit–qutrit system, when the subsystems are independently subject to composite classical-quantum environments, with the HSS obtained from the initial pure state for different values of the entanglement parameter p in the non-Markovian regime: $q=0.1$. The values of the other parameters are given by $\alpha =0.1$, $s=3$, ${\omega}_{c}=20\phantom{\rule{3.33333pt}{0ex}}{\omega}_{0}$, $p=0$ and $v=100$.

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**MDPI and ACS Style**

Mahdavipour, K.; Khazaei Shadfar, M.; Rangani Jahromi, H.; Morandotti, R.; Lo Franco, R.
Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness. *Entropy* **2022**, *24*, 395.
https://doi.org/10.3390/e24030395

**AMA Style**

Mahdavipour K, Khazaei Shadfar M, Rangani Jahromi H, Morandotti R, Lo Franco R.
Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness. *Entropy*. 2022; 24(3):395.
https://doi.org/10.3390/e24030395

**Chicago/Turabian Style**

Mahdavipour, Kobra, Mahshid Khazaei Shadfar, Hossein Rangani Jahromi, Roberto Morandotti, and Rosario Lo Franco.
2022. "Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness" *Entropy* 24, no. 3: 395.
https://doi.org/10.3390/e24030395