# Communication Enhancement through Quantum Coherent Control of N Channels in an Indefinite Causal-Order Scenario

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## Abstract

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## 1. Introduction

## 2. Transmission over Multiple Channels in Quantum Superposition of Causal Order

#### The Formalism for a Quantum N-Switch Channel $\phantom{\rule{3.33333pt}{0ex}}S({\mathcal{N}}_{1},{\mathcal{N}}_{2},\dots ,{\mathcal{N}}_{N})$

## 3. The Quantum Switch Matrices for $N=2$ and $N=3$

#### 3.1. Evaluation of $\mathcal{S}$ for $N=2$

#### 3.2. Evaluation of $\mathcal{S}$ for $N=3$

## 4. Holevo Information Limit for Two and Three Channels

- The diagonalization and minimization of ${H}^{\mathrm{min}}\left(\mathcal{S}\right)$ is performed on all possible states given by $\rho $. It is done analytically for $N=2$ channels and arbitrary ${q}_{i}$. For $N=3$ channels, we compute the eigenvalues of the full quantum 3-switch matrix $\mathcal{S}({\mathcal{N}}_{1},{\mathcal{N}}_{2},{\mathcal{N}}_{3})\left(\right)open="("\; close=")">\rho \otimes {\rho}_{c}$ numerically.
- ${\tilde{\rho}}_{c}^{\left(N\right)}$ was analytically calculated following [9].
- We deduce $H\left({\tilde{\rho}}_{c}^{\left(N\right)}\right)$ from the analytical expressions of ${\tilde{\rho}}_{c}^{\left(N\right)}$.

#### 4.1. Holevo Information Limit for $N=2$ Channels

#### 4.1.1. Calculation of ${H}^{\mathrm{min}}$

#### 4.1.2. Derivation of ${\tilde{\rho}}_{c}^{\left(2\right)}$

#### 4.2. Holevo Information for $N=3$ Channels

- For a fixed dimension d, the Holevo information for indefinite causal order is always higher than the one obtained using one of the definite causal order shown in Figure 2. This is especially the case for totally depolarized channels i.e., ${q}_{i}=0,\forall i$. For completely clean channels ($q=1$), the Holevo information for indefinite and definite causal order converges to the same value depending on d (not shown).
- Two regions can be distinguished. In the strongly depolarized region (roughly $q<0.3$ for $N=2$ and $q<0.5$ for $N=3$), the increase of the dimension d of the target system is detrimental to the Holevo information transmitted by the quantum switch. In contrast, in the moderately depolarized region ($q>0.3$ for $N=2$ and $q>0.5$ for $N=3$), the Holevo information increases both with q and d, a maximum (not shown) as expected for completely clean channels.
- In the strongly depolarized region, increasing the number of channels to $N=3$ is definitively advantageous for information extraction. For instance, in the case of totally depolarized channels ($q=0$), the Holevo information is approximately doubled with $N=3$ with respect to $N=2$ for all values of the dimension d calculated up to $d=10$

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Completeness Property for W_{i}

## Appendix B. Relations to Evaluate Coefficients ${\mathit{Q}}_{{\mathit{A}}_{\mathit{z}}}^{\mathit{k},{\mathit{k}}^{\mathbf{\prime}}}$

## Appendix C. Matrices $\mathcal{S}\mathit{z}$ for the Quantum 3-Switch

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**Figure 1.**Concept of the quantum 2-switch. ${\mathcal{N}}_{i}={\mathcal{N}}_{{q}_{i}}^{D}$ is a depolarizing channel applied to the quantum state $\rho $, where $1-{q}_{i}$ is the strength of the depolarization. For two channels, depending on the control system ${\rho}_{c}$, there are 2! possibilities to combine the channels with definite causal order: (

**a**) if ${\rho}_{c}$ is in the state $\left|1\right.\u232a\left.\u23291\right|$, the causal order will be ${\mathcal{N}}_{2}\circ {\mathcal{N}}_{1}$, i.e., ${\mathcal{N}}_{1}$ is before ${\mathcal{N}}_{2}$; (

**b**) on the other hand, if ${\rho}_{c}$ is on the state $\left|2\right.\u232a\left.\u23292\right|$, the causal order will be ${\mathcal{N}}_{1}\circ {\mathcal{N}}_{2}$; (

**c**) however, placing ${\rho}_{c}$ in a superposition of its states (i.e., ${\rho}_{c}=\left|+\right.\u232a\left.\u2329+\right|$, where ${\left|+\right.\u232a}_{c}=\frac{1}{\sqrt{2}}(\left|1\right.\u232a+\left|2\right.\u232a)$) results in the indefinite causal order of ${\mathcal{N}}_{1}$ and ${\mathcal{N}}_{2}$ to become indefinite. In this situation, we said that the quantum channels are in a superposition of causal orders. This device is called a quantum 2-switch [6] whose input and output are $\rho \otimes {\rho}_{c}$ and $\mathcal{S}({\mathcal{N}}_{1},{\mathcal{N}}_{2})(\rho \otimes {\rho}_{c})$, respectively.

**Figure 2.**Concept of the quantum 3-switch. For three channels, depending on ${\rho}_{c}$, we have 3! possibilities to combine the channels in a definite causal order: (

**a**) ${\rho}_{c}=\left|1\right.\u232a\left.\u23291\right|$ encodes a causal order ${\mathcal{N}}_{1}\circ {\mathcal{N}}_{2}\circ {\mathcal{N}}_{3}$, i.e., ${\mathcal{N}}_{3}$ is applied first to $\rho $; (

**b**) ${\rho}_{c}=\left|2\right.\u232a\left.\u23292\right|$ encodes ${\mathcal{N}}_{1}\circ {\mathcal{N}}_{3}\circ {\mathcal{N}}_{2}$; (

**c**) ${\rho}_{c}=\left|3\right.\u232a\left.\u23293\right|$ encodes ${\mathcal{N}}_{2}\circ {\mathcal{N}}_{1}\circ {\mathcal{N}}_{3}$; (

**d**) ${\rho}_{c}=\left|4\right.\u232a\left.\u23294\right|$ encodes ${\mathcal{N}}_{2}\circ {\mathcal{N}}_{3}\circ {\mathcal{N}}_{1}$; (

**e**) ${\rho}_{c}=\left|5\right.\u232a\left.\u23295\right|$ encodes ${\mathcal{N}}_{3}\circ {\mathcal{N}}_{1}\circ {\mathcal{N}}_{2}$; (

**f**) ${\rho}_{c}=\left|6\right.\u232a\left.\u23296\right|$ encodes ${\mathcal{N}}_{3}\circ {\mathcal{N}}_{2}\circ {\mathcal{N}}_{1}$; (

**g**) finally, if ${\rho}_{c}=\left|+\right.\u232a\left.\u2329+\right|$, where $\left|+\right.\u232a=\frac{1}{\sqrt{6}}{\sum}_{k=1}^{6}\left|k\right.\u232a$ we shall have a superposition of six different causal orders. This is an indefinite causal order called quantum 3-switch whose input and output are $\rho \otimes {\rho}_{c}$ and $\mathcal{S}({\mathcal{N}}_{1},{\mathcal{N}}_{2},{\mathcal{N}}_{3})(\rho \otimes {\rho}_{c})$, respectively. Notice that, for each superposition with m different causal orders, there are $\left(\right)$ (with $m=1,2,\dots ,6$) possible combinations of causal orders to build such superposition with $N=3$ channels, where $\left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{n}{r}$ is the binomial coefficient. The input and output of each channel are fixed. The arrows along the wire just indicate that the target system enters in or exits from the channel.

**Figure 3.**Entropy map for two noisy channels. The 3D graphs represent contour surfaces of the Von-Neumann entropy ${H}^{\mathrm{min}}\left({\mathcal{S}}_{2}\right)$ when the depolarizing parameters ${q}_{1},{q}_{2},$ and the probabilities ${P}_{1}={P}_{2}=p$ are varied from 0 to 1. We plot several cases when the dimension d of the target $\rho $ is: (

**a**) $d=2$; (

**b**) $d=3$; (

**c**) $d=10$; and (

**d**) $d=100$. The value of ${H}^{\mathrm{min}}\left({\mathcal{S}}_{2}\right)$ is also depicted by the color in the bar.

**Figure 4.**Transmission map of information for two noisy channels.The 3D graphs represent contour surfaces of the Holevo information ${\chi}_{\mathrm{Q}2\mathrm{S}}$ when the depolarising parameters ${q}_{1},{q}_{2}$ and the probabilities ${P}_{1}={P}_{2}=p$ varied from 0 to 1. We plot several cases for the dimension d of the target system: (

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

**b**) $d=3$, (

**c**) $d=10$ and (

**d**) $d=100$. In all these cases there are thirty contour surfaces of ${\chi}_{\mathrm{Q}2\mathrm{S}}$. The values of ${\chi}_{\mathrm{Q}2\mathrm{S}}$ are shown in the color bars.

**Figure 5.**Transmission of information for $N=2$ and $N=3$ channels. Holevo information as a function of the depolarization strengths ${q}_{i}$ of the channels. We plot the subcases of equal depolarization strengths, i.e., ${q}_{1}={q}_{2}={q}_{3}=q$, with equally weighted probabilities ${P}_{k}$ for indefinite causal orders (solid line) with (

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

**b**) $N=3$ channels. The transmission of information first decreases to a minimal value for Holevo information and then the transmission of information increases with q. For completely depolarizing channels, i.e., $q=0$, the transmission of information is nonzero and decreases as d increases. A comparison is shown between the Holevo information when the channels are in a definite causal order (dashed line). A full superposition of $N!$ causal orders is used.

**Table 1.**Values of the Holevo information ratio ${\chi}_{\mathrm{Q}3S}/{\chi}_{\mathrm{Q}2S}$. The mean value of the ratio is 1.9328 ± 0.0617.

d | ${\mathit{\chi}}_{\mathbf{Q}2\mathit{S}}$ | ${\mathit{\chi}}_{\mathbf{Q}3\mathit{S}}$ | ${\mathit{\chi}}_{\mathbf{Q}2\mathit{S}}/{\mathit{\chi}}_{\mathbf{Q}3\mathit{S}}$ |
---|---|---|---|

2 | 0.0487 | 0.0980 | 2.0123 |

3 | 0.0183 | 0.0339 | 1.8524 |

4 | 0.0085 | 0.0159 | 1.8705 |

5 | 0.0046 | 0.0087 | 1.8913 |

6 | 0.0027 | 0.0053 | 1.9629 |

7 | 0.0018 | 0.0034 | 1.8888 |

8 | 0.0012 | 0.0023 | 1.9166 |

9 | 0.0008 | 0.0016 | 2 |

10 | 0.0006 | 0.0012 | 2 |

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**MDPI and ACS Style**

Procopio, L.M.; Delgado, F.; Enríquez, M.; Belabas, N.; Levenson, J.A.
Communication Enhancement through Quantum Coherent Control of *N* Channels in an Indefinite Causal-Order Scenario. *Entropy* **2019**, *21*, 1012.
https://doi.org/10.3390/e21101012

**AMA Style**

Procopio LM, Delgado F, Enríquez M, Belabas N, Levenson JA.
Communication Enhancement through Quantum Coherent Control of *N* Channels in an Indefinite Causal-Order Scenario. *Entropy*. 2019; 21(10):1012.
https://doi.org/10.3390/e21101012

**Chicago/Turabian Style**

Procopio, Lorenzo M., Francisco Delgado, Marco Enríquez, Nadia Belabas, and Juan Ariel Levenson.
2019. "Communication Enhancement through Quantum Coherent Control of *N* Channels in an Indefinite Causal-Order Scenario" *Entropy* 21, no. 10: 1012.
https://doi.org/10.3390/e21101012