# Critical Assessment of Information Back-Flow in Measurement-Free Teleportation

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## Abstract

**:**

## 1. Introduction

## 2. Measurement-Free Teleportation

#### 2.1. Effective Depolarizing-Channel Description

#### 2.2. Distinguishability and Non-Markovianity Resulting from the Dynamics

## 3. Analysis of Non-Markovianity in the Measurement-Free Teleportation Circuit

#### 3.1. Review of Measures of Non-Markovianity

#### 3.1.1. Breuer–Laine–Piilo Measure

#### 3.1.2. Rivas-Huelga-Plenio Measure

#### 3.1.3. Luo-Fu-Song Measure

#### 3.2. Information Back-Flow and Non-Markovianity

## 4. Information Back-Flow and Correlations

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Quantum circuit of the measurement-free teleportation protocol. Each gate into which the circuit is decomposed is labeled as ${G}_{i}$ ($i\in \{1,\dots ,8\}$). They can be grouped into three unitary blocks of operations ${U}_{j}(j=1,2,3)$ as defined in Equations (2)–(5). Here, ${G}_{1,3,6}$ are CNOT gates, ${G}_{2,4,7}$ are Hadamard transforms, while ${G}_{5,8}$ are SWAP gates.

**Figure 2.**Non-Markovianity of the dynamics given by Equation (24). Results are plotted only for $p\ge 0.4$ as all the measures listed in Section 3.1 are zero for $0\le p\le 0.4$. Inset: Non-Markovianity of the effective Hamiltonian in Equation (25) as measured by the ${\mathcal{N}}_{BLP}$ measure.

**Figure 3.**Trace distance between the two states of S as they go through the teleportation circuit. Gate ${G}_{i}$ acts when $i-1<t<i$. We plot only $5\le t\le 8$ as the trace distance remains constant at 1 for $t<5$.

**Figure 4.**Trace distance between the two states of ${E}_{2}$ for the original BBC protocol [3] as time evolves. The dashed lines are boundaries between the gates acting on the system and the environment. Gates ${G}_{1}$–${G}_{4}$ correspond to those in Figure 1, while ${G}_{5}$ is a CNOT operation on $S{E}_{2}$ and ${G}_{6}$ is a Hadamard gate on ${E}_{2}$.

**Figure 5.**Correlations in the splitting S-vs-$\left({E}_{1}{E}_{2}\right)$, as quantified by (

**a**) logarithmic negativity, (

**b**) discord and (

**c**) classical correlations as the system evolves according to Equation (24). Time is denoted t, and p determines the Werner state of the environment at $t=0$. The system is initially in the vacuum state $|0\rangle $.

**Figure 6.**Correlations in the partition $S|{E}_{1}{E}_{2}$ as quantified by (

**a**) logarithmic negativity, (

**b**) discord and (

**c**) classical correlations when the Hamiltonian of the system and environment is given by Equation (25). We take the initial state of the system to be $|0\rangle $ and the environment $W\left(p\right)$. Here t is a dimensionless time. We only show $5\le t\le 8$ as there are no system-environment correlations before $t=5$.

**Figure 7.**Correlations in the partition $S|{E}_{1}{E}_{2}$ as quantified by (

**a**) logarithmic negativity, (

**b**) discord and (

**c**) classical correlations when the Hamiltonian of the system and environment is given by Equation (25). We take the initial state of the system to be $|+\rangle $ and the environment $W\left(p\right)$.

**Table 1.**Summary of the explicit form taken by the operators ${\mathcal{V}}_{jk}^{{\alpha}^{l}}$ acting in the Hilbert space of S [cf. Equation (8)].

${\mathcal{V}}_{\mathit{jk}}^{{\mathit{\alpha}}^{\mathit{l}}}$ | $\mathit{j}=0,\mathit{k}=0$ | $\mathit{j}=0,\mathit{k}=1$ | $\mathit{j}=1,\mathit{k}=0$ | $\mathit{j}=1,\mathit{k}=1$ |
---|---|---|---|---|

${\mathcal{V}}_{jk}^{{\varphi}^{+}}$ | $1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{1}_{S}/2$ | $1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{1}_{S}/2$ | $1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{1}_{S}/2$ | $1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{1}_{S}/2$ |

${\mathcal{V}}_{jk}^{{\varphi}^{-}}$ | ${Z}_{S}/2$ | $-{Z}_{S}/2$ | ${Z}_{S}/2$ | $-{Z}_{S}/2$ |

${\mathcal{V}}_{jk}^{{\psi}^{+}}$ | ${X}_{S}/2$ | ${X}_{S}/2$ | $-{X}_{S}/2$ | $-{X}_{S}/2$ |

${\mathcal{V}}_{jk}^{{\psi}^{-}}$ | $-i{Y}_{S}/2$ | $i{Y}_{S}/2$ | $i{Y}_{S}/2$ | $-i{Y}_{S}/2$ |

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McAleese, H.; Paternostro, M.
Critical Assessment of Information Back-Flow in Measurement-Free Teleportation. *Entropy* **2024**, *26*, 780.
https://doi.org/10.3390/e26090780

**AMA Style**

McAleese H, Paternostro M.
Critical Assessment of Information Back-Flow in Measurement-Free Teleportation. *Entropy*. 2024; 26(9):780.
https://doi.org/10.3390/e26090780

**Chicago/Turabian Style**

McAleese, Hannah, and Mauro Paternostro.
2024. "Critical Assessment of Information Back-Flow in Measurement-Free Teleportation" *Entropy* 26, no. 9: 780.
https://doi.org/10.3390/e26090780