# High-Fidelity Hyperentangled Cluster States of Two-Photon Systems and Their Applications

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## Abstract

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## 1. Introduction

## 2. Model and Hamiltonian

## 3. Generation of Photonic Hyperentangled Cluster States

## 4. Applications

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of a diamond nitrogen-vacancy (NV) center coupling to a MTR. (

**b**) The possible electron energy-level configuration of an NV center inside an MTR. The transition $|-\rangle \to |{A}_{2}\rangle $ is driven by a left circularly polarized photon (denoted by $|L\rangle $), and $|+\rangle \to |{A}_{2}\rangle $ is driven by a right circularly polarized photon (denoted by $|R\rangle $).

**Figure 2.**Schematic diagram showing the two-photon polarization-spatial hyperentangled cluster states. The polarizing beam splitter $PB{S}_{i}(i=1,2)$ is used to transmit the right circularly polarized photon $|R\rangle $ and reflect the left circularly polarized photon $|L\rangle $, respectively. $S{W}_{i}(i=1,2,3,4)$ represents the optical switch and $B{S}_{i}(i=1,2,3)$ represents the $50.50$ beam splitter. The $HW{P}_{i}(i=1,2,3,4)$ represents the half-wave plate which is used to perform a bit-flip operation $X=|R\rangle \langle L|+|L\rangle \langle R|$ on the photon in the polarization degrees of freedom (DOF). $QW{P}_{i}(i=1,2)$ represents the quarter-wave plate which is used to realize a Hadamard operation $|R\rangle \to \frac{1}{\sqrt{2}}(|R\rangle +|L\rangle )$ and $|L\rangle \to \frac{1}{\sqrt{2}}(|R\rangle -|L\rangle )$ on the photon in the polarization DOF. ${k}_{1}$ and ${k}_{2}$$(k=a,b,c,d,e)$ represent different spatial modes.

**Figure 3.**The fidelities of the sixteen hyperentangled cluster states vs the parameter ${g}^{2}/{\kappa}_{c}\gamma $ for the different leakage rates ${\kappa}_{s}/{\kappa}_{c}=0$, ${\kappa}_{s}/{\kappa}_{c}=0.03$, and ${\kappa}_{s}/{\kappa}_{c}=0.06$, respectively.

**Table 1.**The relation between the outcomes of the two NV centers and four symmetry hyperentangled cluster states.

${\mathit{NV}}_{1}$ | ${\mathit{NV}}_{2}$ | Four Symmetry Hyperentangled Cluster States |
---|---|---|

$|{\varphi}^{+}{\rangle}_{1}$ | $|{\varphi}^{-}{\rangle}_{2}$ | $|{\Phi}_{1}{\rangle}_{p}{|{\Phi}_{2}\rangle}_{s}$ |

$|{\varphi}^{+}{\rangle}_{1}$ | $|{\varphi}^{+}{\rangle}_{2}$ | $|{\Phi}_{1}{\rangle}_{p}{|{\Phi}_{1}\rangle}_{s}$ |

$|{\varphi}^{-}{\rangle}_{1}$ | $|{\varphi}^{-}{\rangle}_{2}$ | $|{\Phi}_{2}{\rangle}_{p}{|{\Phi}_{2}\rangle}_{s}$ |

$|{\varphi}^{-}{\rangle}_{1}$ | $|{\varphi}^{+}{\rangle}_{2}$ | $|{\Phi}_{2}{\rangle}_{p}{|{\Phi}_{1}\rangle}_{s}$ |

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**MDPI and ACS Style**

Tan, L.; Zhou, F.; Zhang, L.; Xiang, S.; Song, K.; Zhao, Y.
High-Fidelity Hyperentangled Cluster States of Two-Photon Systems and Their Applications. *Symmetry* **2019**, *11*, 1079.
https://doi.org/10.3390/sym11091079

**AMA Style**

Tan L, Zhou F, Zhang L, Xiang S, Song K, Zhao Y.
High-Fidelity Hyperentangled Cluster States of Two-Photon Systems and Their Applications. *Symmetry*. 2019; 11(9):1079.
https://doi.org/10.3390/sym11091079

**Chicago/Turabian Style**

Tan, Liu, Fang Zhou, Lingxia Zhang, Shaohua Xiang, Kehui Song, and Yujing Zhao.
2019. "High-Fidelity Hyperentangled Cluster States of Two-Photon Systems and Their Applications" *Symmetry* 11, no. 9: 1079.
https://doi.org/10.3390/sym11091079