# Distributing Secret Keys with Quantum Continuous Variables: Principle, Security and Implementations

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## Abstract

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## 1. Introduction

## 2. Principle of CVQKD with Coherent States

## 3. Security Analysis

- Composable security against arbitrary attacks, if one can bound the trace distance of Equation (3), without any restriction on the input state ${\rho}_{{A}^{N}{B}^{N}}$ of the protocol.
- Composable security against collective attacks, if one can bound the trace distance of Equation (3) under the restriction that the input state is identically and independently distributed, i.e., ${\rho}_{{A}^{N}{B}^{N}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\rho}_{AB}^{\otimes N}$.
- Security against collective attacks in the asymptotic limit of infinitely many uses of the channel, if one can compute an upper bound on the Holevo information, $\chi (B;E)$ from Equation (1), between the raw key and the adversary, assuming that the quantum state shared by Alice and Bob is known. In the case of CV protocols, one only needs to assume that the covariance matrix of the state is known.

**Table 1.**Current security status of the main one-way continuous-variable quantum key distribution (CVQKD) protocols. PM, prepare and measure.

Protocol | (PM) State Preparation | (PM) Modulation | Bob’s Measurement | Best Currently-Available Security Proofs |
---|---|---|---|---|

[41] | squeezed | Gaussian | homodyne | Finite-size [38,39] |

${K}^{\epsilon}\left(N\right)>0$ for practical N | ||||

${lim}_{N\to \infty}{K}^{\epsilon}\left(N\right)<{K}_{\mathrm{coll}}^{\mathrm{asympt}}$ | ||||

[23] | coherent | Gaussian | heterodyne | Finite-size [24] |

${K}_{\mathrm{coll}}^{\epsilon}\left(N\right)\approx {K}_{\mathrm{coll}}^{\mathrm{asympt}}$ for practical N | ||||

${K}^{\epsilon}\left(N\right)=0$ for practical N [37] | ||||

[22] | coherent | Gaussian | homodyne | asymptotic collective [30,31,42] |

[43] | coherent | Gaussian 1D | homodyne | asymptotic collective [43] |

[44] | squeezed | Gaussian | heterodyne | asymptotic collective [45] |

[46] | thermal | Gaussian | homo/heterodyne | asymptotic collective [47,48,49] |

[50] | squeezed | Gaussian + additional Gaussian | homodyne | asymptotic collective [50] |

[51,52] | coherent | Gaussian | homo/heterodyne + Gaussian post-selection | asymptotic collective [51,52] |

## 4. Experimental Implementations

**Figure 1.**Optical layout of a fiber optic CVQKD system implementing the GG02 [22] protocol with homodyne detection.

**Figure 2.**Experimental results obtained for various CVQKD protocols offering different levels of security. The distance for the results of [62] and [65] has been calculated from data obtained with free-space experiments (expressed in dB) assuming an optical fiber with an attenuation coefficient of 0.2 dB/km, which is standard at telecommunication wavelengths.

## 5. Imperfections and Side Channels in Practical CVQKD

#### 5.1. State Preparation

#### 5.2. Local Oscillator Manipulation

#### 5.3. Detection

## 6. Conclusions and Perspectives

## Acknowledgments

## Conflicts of Interest

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Diamanti, E.; Leverrier, A.
Distributing Secret Keys with Quantum Continuous Variables: Principle, Security and Implementations. *Entropy* **2015**, *17*, 6072-6092.
https://doi.org/10.3390/e17096072

**AMA Style**

Diamanti E, Leverrier A.
Distributing Secret Keys with Quantum Continuous Variables: Principle, Security and Implementations. *Entropy*. 2015; 17(9):6072-6092.
https://doi.org/10.3390/e17096072

**Chicago/Turabian Style**

Diamanti, Eleni, and Anthony Leverrier.
2015. "Distributing Secret Keys with Quantum Continuous Variables: Principle, Security and Implementations" *Entropy* 17, no. 9: 6072-6092.
https://doi.org/10.3390/e17096072