# 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

## References

- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Braunstein, S.L.; van Loock, P. Quantum information with continuous variables. Rev. Mod. Phys.
**2005**, 77, 513–577. [Google Scholar] - Scarani, V.; Bechmann-Pasquinucci, H.; Cerf, N.J.; Dušek, M.; Lütkenhaus, N.; Peev, M. The security of practical quantum key distribution. Rev. Mod. Phys.
**2009**, 81, 1301–1350. [Google Scholar] - Gisin, N.; Ribordy, G.; Tittel, W.; Zbinden, H. Quantum cryptography. Rev. Mod. Phys.
**2002**, 74, 145–195. [Google Scholar] - Weedbrook, C.; Pirandola, S.; García-Patrón, R.; Cerf, N.J.; Ralph, T.C.; Shapiro, J.H.; Lloyd, S. Gaussian quantum information. Rev. Mod. Phys.
**2012**, 84, 621–669. [Google Scholar] - Renner, R.; Gisin, N.; Kraus, B. Information-theoretic security proof for quantum-key-distribution protocols. Phys. Rev. A
**2005**, 72, 012332. [Google Scholar] - Cerf, N.J.; Levy, M.; van Assche, G. Quantum Distribution of Gaussian Keys with Squeezed States. Phys. Rev. A
**2001**, 63, 052311. [Google Scholar] - Grosshans, F.; Grangier, P. Continuous variable quantum cryptography using coherent states. Phys. Rev. Lett.
**2002**, 88, 057902. [Google Scholar] - Grosshans, F.; van Assche, G.; Wenger, J.; Brouri, R.; Cerf, N.J.; Grangier, P. Quantum key distribution using gaussian-modulated coherent states. Nature
**2003**, 421, 238–241. [Google Scholar] - García-Patrón, R.; Cerf, N.J. Continuous-variable quantum key distribution protocols over noisy channels. Phys. Rev. Lett.
**2009**, 102, 130501. [Google Scholar] - Lance, A.M.; Symul, T.; Bowen, W.P.; Sanders, B.C.; Tyc, T.; Ralph, T.C.; Lam, P.K. Continuous-variable quantum-state sharing via quantum disentanglement. Phys. Rev. A
**2005**, 71, 033814. [Google Scholar] - Madsen, L.S.; Usenko, V.C.; Lassen, M.; Filip, R.; Andersen, U.L. Continuous variable quantum key distribution with modulated entangled states. Nature Commun.
**2012**, 3. [Google Scholar] [CrossRef] - Silberhorn, C.; Ralph, T.C.; Lütkenhaus, N.; Leuchs, G. Continuous Variable Quantum Cryptography: Beating the 3dB Loss Limit. Phys. Rev. Lett.
**2002**, 89, 167901. [Google Scholar] - Grosshans, F.; Grangier, P. Reverse Reconciliation Protocols for Quantum Cryptography with Continuous Variables. Proceedings of the Sixth International Conference on Quantum Communication, Measurement and Computing, Cambridge, MA, USA, 22–26 July 2002; pp. 351–356.
- Jouguet, P.; Kunz-Jacques, S.; Leverrier, A.; Grangier, P.; Diamanti, E. Experimental demonstration of long-distance continuous-variable quantum key distribution. Nature Photon.
**2013**, 7, 378–381. [Google Scholar] - Filip, R. Continuous-variable quantum key distribution with noisy coherent states. Phys. Rev. A
**2008**, 77, 022310. [Google Scholar] - Usenko, V.C.; Filip, R. Feasibility of continuous-variable quantum key distribution with noisy coherent states. Phys. Rev. A
**2010**, 81, 022318. [Google Scholar] - Weedbrook, C.; Pirandola, S.; Lloyd, S.; Ralph, T.C. Quantum cryptography approaching the classical limit. Phys. Rev. Lett.
**2010**, 105, 110501. [Google Scholar] - Weedbrook, C.; Pirandola, S.; Ralph, T. Continuous-variable quantum key distribution using thermal states. Phys. Rev. A
**2012**, 86, 022318. [Google Scholar] - Pirandola, S.; García-Patrón, R.; Braunstein, S.L.; Lloyd, S. Direct and reverse secret-key capacities of a quantum channel. Phys. Rev. Lett.
**2009**, 102, 050503. [Google Scholar] - Pirandola, S. Quantum discord as a resource for quantum cryptography. Sci. Rep.
**2014**, 4. [Google Scholar] [CrossRef] - Weedbrook, C.; Ottaviani, C.; Pirandola, S. Two-way quantum cryptography at different wavelengths. Phys. Rev. A
**2014**, 89, 012309. [Google Scholar] - Ralph, T.C. Security of continuous-variable quantum cryptography. Phys. Rev. A
**2000**, 62, 062306. [Google Scholar] - Reid, M.D. Quantum cryptography with a predetermined key, using continuous-variable Einstein–Podolsky–Rosen correlations. Phys. Rev. A
**2000**, 62, 062308. [Google Scholar] - Hillery, M. Quantum cryptography with squeezed states. Phys. Rev. A
**2000**, 61, 022309. [Google Scholar] - König, R.; Renner, R.; Bariska, A.; Maurer, U. Small accessible quantum information does not imply security. Phys. Rev. Lett.
**2007**, 98, 140502. [Google Scholar] - Leverrier, A.; Grosshans, F.; Grangier, P. Finite-size analysis of a continuous-variable quantum key distribution. Phys. Rev. A
**2010**, 81, 062343. [Google Scholar] - Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed; Wiley: Hoboken, NJ, USA, 2006. [Google Scholar]
- García-Patrón, R.; Cerf, N.J. Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution. Phys. Rev. Lett.
**2006**, 97, 190503. [Google Scholar] - Navascués, M.; Grosshans, F.; Acín, A. Optimality of Gaussian attacks in continuous-variable quantum cryptography. Phys. Rev. Lett.
**2006**, 97, 190502. [Google Scholar] - Pirandola, S.; Braunstein, S.L.; Lloyd, S. Characterization of collective Gaussian attacks and security of coherent-state quantum cryptography. Phys. Rev. Lett.
**2008**, 101, 200504. [Google Scholar]

**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