# Continuous Variable Quantum Key Distribution with a Noisy Laser

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

_{X}. For Gaussian states any ρ

_{X}has a corresponding covariance matrix Γ

_{X}, and a mean value $\langle X\rangle $, though mean values can always be displaced out without loss of generality, as they do not contribute to the information content of the state in question. The von Neumann entropy for a Gaussian state ρ

_{X}can be shown to be given by [5]

_{i}is the i’th value in the symplectic spectrum of Γ

_{X}. The symplectic spectrum is calculated by finding the absolute eigenvalues of the matrix iΩΓ

_{X}, where

_{X}. The dimension of Γ

_{X}is 2N × 2N.

_{S}is the variance of the Gaussian distribution used for the signal modulation, V

_{0}is the noise carried by the light and W is the variance of Eve’s EPR mode. The entanglement based protocol is related to the prepare-and-measure scheme such that µ = V

_{S}+ 1. This noise can be separated into technical and intrinsic (quantum) noise, such that V

_{0}= 1 + κ, in units of the shot noise. For a protocol assuming shot-noise limited state generation κ = 0.

_{V}= (1−T )V +TW. Similar expressions are derived in [19] for the symplectic spectra of the conditional states. From these expressions of the eigenvalues one can find the Holevo information for direct and reverse reconciliation

## 3. Experiment

## 4. Conclusions

## Acknowledgments

**PACS classifications:**03.67.Dd; 03.67.Hk; 03.67.-a

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Equivalent entanglement based model used in the security proof. Alice produces a noisy Einstein–Podolsky–Rosen (EPR) state which she sends to Bob. The quantum channel with transmission T is controlled by the eavesdropper who injects an EPR state with variance W.

**Figure 2.**Contour plots of the secure key generation rate for varying preparation noise in shot-noise units (SNUs) and transmission T for (

**a**) reverse reconciliation and (

**b**) direct reconciliation. The error reconciliation efficiency was set to β = 95%, the modulation variance was 32 SNUs and the channel excess noise 0.11. The dashed lines indicate the minimal possible transmission of a channel where a positive secret key rate can still be obtained, in the ideal case for β = 1, no channel excess noise and in the limit of high modulation variance. (a) For no preparation noise (κ = 0), the rate decreases asymptotically to zero as the transmission approaches zero. When the preparation noise increases the security of reverse reconciliation is quickly compromised, to the point where almost unity transmission is required to achieve security. (b) For heterodyne detection and no preparation noise the rate goes to zero at about 79% transmission, due to the extra unit of vacuum introduced by heterodyne detection. The plot shows the robustness of direct reconciliation to preparation noise.

**Figure 3.**Schematic representation of the experiment. A shot-noise limited laser is amplitude- and phase-modulated with two independent white-noise sources to simulate a noisy laser. Subsequently, Alice modulates the noisy laser beam in amplitude and phase using a known modulation and sends it to Bob through the quantum channel who performs heterodyne detection. The quantum channel’s transmission was simulated by an (for coherent states) equivalent reduction of the modulation variances. AM: Amplitude Modulation. PM: Phase Modulation. PD: Photo Detector.

**Figure 4.**Measured data and theory curves for different levels of preparation noise using (

**a**) reverse reconciliation and (

**b**) direct reconciliation in the post-processing. Error reconciliation efficiency β = 95%. Due to our simulation of losses (see main text) the error bars on the channel loss are negligibly small and, thus, not shown in the figure.

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**MDPI and ACS Style**

Jacobsen, C.S.; Gehring, T.; Andersen, U.L. Continuous Variable Quantum Key Distribution with a Noisy Laser. *Entropy* **2015**, *17*, 4654-4663.
https://doi.org/10.3390/e17074654

**AMA Style**

Jacobsen CS, Gehring T, Andersen UL. Continuous Variable Quantum Key Distribution with a Noisy Laser. *Entropy*. 2015; 17(7):4654-4663.
https://doi.org/10.3390/e17074654

**Chicago/Turabian Style**

Jacobsen, Christian S., Tobias Gehring, and Ulrik L. Andersen. 2015. "Continuous Variable Quantum Key Distribution with a Noisy Laser" *Entropy* 17, no. 7: 4654-4663.
https://doi.org/10.3390/e17074654