# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding

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## Abstract

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## 1. Introduction

## 2. Schematic Description

## 3. Experimental Implementation

## 4. Security Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A schematic setup of 3-dimensional quantum key distribution (QKD) with hybrid encoding. Alice uses orbital angular momentum (OAM) modes of a single photon, and Bob controls the phase of each path to encode their information in the single photon. The encoded photon enters into a 3-port interferometer. After single photon interference, a OAM value and existing path of the single photon is measured. SLM: spatial light modulator; BS1: 50:50 beam splitter; BS2: beam splitter of which transmissivity is $1/3$; BS3: beam splitter of which transmissivity is $2/3$; OAM CT: cyclic transformation of OAM modes.

**Figure 2.**Schematic setups of Bob’s two encoding systems. (

**a**) Bob chooses one path to encode his information by using optical switch; (

**b**) Bob encodes his information by control phase shifters, B1 and B2. Details are described in the maintext. BS1: 50:50 beam splitter; BS2: beam splitter of which transmissivity is $1/3$; PS: phase shifter

**Figure 3.**A schematic diagram of experimental setup of three-fold cyclic transformation of OAM modes. There are OAM beam splitters (OAM BSs) which consist of a Mach-Zehnder interferometer with Dove prisms. $\alpha /2$ means relative angle between the two Dove prisms. The first OAM BS ($\alpha =\pi $) and the final OAM BS ($\alpha =-\pi $) change a direction of propagation of a photon whose OAM value is odd and even, respectively. The second OAM BS ($\alpha =\pi /2$) separates a photon whose OAM value is 0 and 2. With OAM holograms, the three-fold cyclic transformation of OAM modes $\{-1,0,1\}$ is accomplished.

**Figure 4.**The secret key rate of the original detector-device-independent QKD (DDI-QKD) (black dotted line), 3d- (red dashed line), 4d- (blue dot-dashed line), and 5d-QKD with hybrid encoding (orange solid line). (

**a**) Plot of the secret key rate r (bits/sifted pulse) vs. state error rate Q; (

**b**) Plot of the secret key rate r (bits/sifted pulse) vs. transmission loss $\eta $ (dB). Dark count rate of single photon detectors is assumed as ${10}^{-5}$ per pulse.

**Figure 5.**The secret key rate of 3d-measurement-device-independent QKD (MDI-QKD) (red dashed line) and 3d-QKD with hybrid encoding (black solid line). Plot of the secret key rate R (bits/total pulse) vs. transmission loss $\eta $ (dB). The secret key rate per total signal is obtained from (the secret key rate per sifted key) × (the signal sifting rate). Details are described in maintext. Dark count rate of single photon detectors is assumed as ${10}^{-5}$ per pulse.

**Figure 6.**The secret key rate of 3d-QKD hybrid encoding (black solid line) and a prepare-and-measure 3d-QKD (red dashed line). Dark count rate of single photon detectors is assumed as ${10}^{-5}$ per pulse.

**Table 1.**An example of Bob’s operation on his encoded information when $d=3$ and the result of the measurement is $|{\Phi}_{3i+j}\rangle $. According to their bases choice and the measurement result, it is necessary to retrieve his information for sharing the same information.

Bases | Bob’s Operation ($|{\mathbf{\Phi}}_{3\mathit{i}+\mathit{j}}\rangle $) |
---|---|

bases 1 (${l}_{x}$, ${p}_{y}$) | $y\to y-i$ (mod 3) |

$1\leftrightarrow 2$ for $j=0$ | |

bases 2 (${\overline{l}}_{x}$, ${\overline{p}}_{y}$) | $0\leftrightarrow 2$ for $j=1$ |

$0\leftrightarrow 1$ for $j=2$ |

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**MDPI and ACS Style**

Jo, Y.; Park, H.S.; Lee, S.-W.; Son, W.
Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding. *Entropy* **2019**, *21*, 80.
https://doi.org/10.3390/e21010080

**AMA Style**

Jo Y, Park HS, Lee S-W, Son W.
Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding. *Entropy*. 2019; 21(1):80.
https://doi.org/10.3390/e21010080

**Chicago/Turabian Style**

Jo, Yonggi, Hee Su Park, Seung-Woo Lee, and Wonmin Son.
2019. "Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding" *Entropy* 21, no. 1: 80.
https://doi.org/10.3390/e21010080