# Symmetries in Teleportation Assisted by N-Channels under Indefinite Causal Order and Post-Measurement

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background of Indefinite Causal Order in Communication

#### 1.2. Approaches to Teleportation under Causal Order Schemes

## 2. Teleportation Algorithm as a Quantum Channel and $N$-Redundant Teleportation Problem

#### 2.1. Quantum Teleportation as a Quantum Channel

#### 2.2. N-Redundant Quantum Teleportation

## 3. Quantum Teleportation Assisted by Indefinite Causal Order with $N$ Channels

## 4. Analysis of Quantum Teleportation Assisted by the First Indefinite Causal Orders

#### 4.1. Teleportation with $N=2$ Teleportation Channels in an Indefinite Causal Order Superposition

#### 4.2. Teleportation with an Increasing Number of Teleportation Channels in an iNdefinite Causal Order Superposition

#### 4.2.1. Case ${p}_{1}={p}_{2}={p}_{3}\equiv p$

#### 4.2.2. Case ${p}_{j}\ll 1$, $j=1,2,3$

#### 4.3. Notable Behavior on the Frontal Face of Parametric Region: Case ${p}_{0}=0$

## 5. An Alternative Procedure Introducing Weak Measurement

#### 5.1. General Case for $N=2$ Assisted by a Weak Measurement

#### 5.2. Cases for $N\ge 2$ Assisted by a Weak Measurement

## 6. Experimental Deployment of Teleportation with Indefinite Causal Order

#### 6.1. Implementation of Weak Measurement to Project $|\chi \rangle $

#### 6.2. An Insight View about Teleportation Implementing Indefinite Causal Orders Experimentally with Light

- If ${p}_{0}=1$ for all $i,j=0,\dots ,3$ cases with a global successful probability of $\frac{1}{16}$.
- If ${p}_{0}=0$ for the cases $i=0,\dots ,3$ and $j=2$ with a global successful probability of $\frac{{({p}_{1}-{p}_{2}+{p}_{3})}^{2}}{64}$.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Formulas for ${\mathcal{P}}_{m}$ and $\mathcal{F}$ for the Case ${p}_{1}={p}_{2}={p}_{3}=p$

## Appendix B. Formulas for ${\mathcal{P}}_{m,N}^{\mathrm{ff},\left\{{p}_{i}\right\}}$ and $\mathcal{F}$ for the Case ${p}_{0}=0$

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**Figure 1.**(

**a**) Traditional teleportation circuit T where $\left|\psi \right.\u232a=\alpha \left|0\right.\u232a+\beta \left|1\right.\u232a$ and ideally $\left|\chi \right.\u232a$ is the Bell state $\left|{\beta}_{00}\right.\u232a=\frac{1}{\sqrt{2}}(\left|00\right.\u232a+\left|11\right.\u232a)$. Measurements refer to one single qubit measurement and the double line to classical communication channels. (

**b**) Modified teleportation circuit considering a Bell states measurement (which are generated by enclosing the gates on (

**a**) within the measurement gadget).

**Figure 2.**Sequential fidelity as function of the number N of channels being applied, and p is the deformation strength in $|\chi \rangle $.

**Figure 3.**$N!$ causal order combinations for N identical teleportation channels ${T}_{i},i=1,2,...,N$ finally conforming a superposition of it. Each one is ruled by the control state above.

**Figure 4.**Fidelity for the case of two channels in indefinite causal order as function of p. The blue dashed upper line corresponds to $|{\psi}_{m}\rangle =|+\rangle $ and ${q}_{0}=\frac{1}{2}$ reaching $\mathcal{F}=1$ in $p=\frac{1}{3}$.

**Figure 5.**Condensed outcomes for the case $N=2$. The respective probability ${\mathcal{P}}_{m}$ of measurements are included as function of ${q}_{0}$ and $\theta $ in $\left|{\psi}_{m}\right.\u232a=cos\frac{\theta}{2}\left|0\right.\u232a+sin\frac{\theta}{2}{e}^{i\varphi}\left|1\right.\u232a$ ($\varphi =0$ in the optimal measurement). Fidelity depends entirely from p, and ${\mathcal{P}}_{m}$ goes down while $p\to \frac{1}{3}$.

**Figure 6.**(

**a**) Best fidelity ${\mathcal{F}}_{2}$ for the two-channels case as function of ${p}_{1},{p}_{2},{p}_{3}$. Each point inside the polyhedron corresponds to their acceptable values and it is coloured in agreement with its fidelity value (see the color-scale besides); the cut of polyhedron region exhibits the inner structure; (

**b**) The corresponding values for measurement probabilities ${\mathcal{P}}_{m}$ denoting disperse values around $0.5$. The upper inset confirms the statistical distribution ${\rho}_{{\mathcal{P}}_{m}}$ exhibiting symmetry around ${\mathcal{P}}_{m}=0.5$.

**Figure 7.**Probability ${\mathcal{P}}_{m}$ to obtain different values of fidelity ${\mathcal{F}}_{N}$ when the measurement states $|{\phi}_{+}\rangle $ or $|{\phi}_{-}\rangle $ are applied for cases (

**a**) $N=2$, (

**b**) $N=3$ and (

**c**) $N=4$. Color-scale bar depicts the respective value for p for $N=2,3,4$.

**Figure 8.**Comparison of fidelity obtained when the channels are applied sequentially (blue) and with indefinite causal order depending on the measurement state $|{\phi}_{m}^{+}\rangle $ (red) and $|{\phi}_{m}^{-}\rangle $ (green), for the cases (

**a**) $N=2$, (

**b**) $N=3$, and (

**c**) $N=4$ (in this last case, the fidelity becomes undefined for $|{\phi}_{m}^{-}\rangle $).

**Figure 9.**Bloch sphere showing under the assumption ${p}_{j}\ll 1$, $j=1,2,3$ for each state: (

**a**) ${\mathsf{\Delta}}_{\theta ,\varphi}^{{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}}$ in color obtained for each syndrome in (27), ${\sigma}_{1}\rho {\sigma}_{1},{\sigma}_{2}\rho {\sigma}_{2},{\sigma}_{3}\rho {\sigma}_{3}$ respectively, and (

**b**) the standard deviation ${\sigma}_{{\mathsf{\Delta}}_{\theta ,\varphi}^{{\alpha}_{1},{\alpha}_{2},{\alpha}_{3}}}$ in (31). Red is the best fidelity in (

**a**) and the lower dispersion in (

**b**).

**Figure 10.**Optimal fidelity using two teleportation channels in indefinite causal order followed by an appropriate measurement $|{\phi}_{m}\rangle $. (

**a**) The best fidelity obtained for certain teleported state if optimal control measurement is obtained, (

**b**) the probability ${\mathcal{P}}_{m}$ of success for the last process, and (

**c**) the statistical distribution for ${\mathcal{F}}_{2}$ and ${\mathcal{P}}_{m}$.

**Figure 11.**(

**a**) Values of ${\mathcal{P}}_{m}$ on ${p}_{0}=0$ face, and (

**b**) its corresponding statistical distribution ${\rho}_{{\mathcal{P}}_{m}}$ for two teleportation channels in indefinite causal order.

**Figure 13.**Distribution of ${\mathcal{P}}_{\mathrm{Tot}}$: (

**a**) as function of $({p}_{1},{p}_{2},{p}_{3})$, and (

**b**) as statistical distribution by itself obtained numerically from the data of (

**a**).

**Figure 14.**Schematic teleportation process assisted by indefinite causal order using N-teleportation channels and weak measurement.

**Figure 15.**(

**a**–

**c**) values of ${\mathcal{P}}_{\mathrm{Tot}}$ as function of $({p}_{1},{p}_{2},{p}_{3})$, for ${N}_{2},{N}_{3}$ and ${N}_{4}$ respectively. (

**d**) Statistical distribution numerically obtained for ${{\mathcal{P}}_{\mathrm{Tot}}}_{2},{{\mathcal{P}}_{\mathrm{Tot}}}_{3}$ and ${{\mathcal{P}}_{\mathrm{Tot}}}_{4}$.

**Figure 16.**(

**a**) Quantum circuit generating the weak measurement on $|\chi \rangle $, and (

**b**) contour plots for the map on the region $({p}_{1},{p}_{2},{p}_{3})$ between those probabilities and $({p}_{1}^{*},{p}_{2}^{*},{p}_{3}^{*})$.

**Figure 17.**Diagram for implementation of teleportation with causal ordering as it is discussed in the text. Photons are split on two different set of paths to superpose the two causal orders of two sequential teleportation process.

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**MDPI and ACS Style**

Cardoso-Isidoro, C.; Delgado, F.
Symmetries in Teleportation Assisted by *N*-Channels under Indefinite Causal Order and Post-Measurement. *Symmetry* **2020**, *12*, 1904.
https://doi.org/10.3390/sym12111904

**AMA Style**

Cardoso-Isidoro C, Delgado F.
Symmetries in Teleportation Assisted by *N*-Channels under Indefinite Causal Order and Post-Measurement. *Symmetry*. 2020; 12(11):1904.
https://doi.org/10.3390/sym12111904

**Chicago/Turabian Style**

Cardoso-Isidoro, Carlos, and Francisco Delgado.
2020. "Symmetries in Teleportation Assisted by *N*-Channels under Indefinite Causal Order and Post-Measurement" *Symmetry* 12, no. 11: 1904.
https://doi.org/10.3390/sym12111904