Abstract

Quantum information processing with continuous variables is a paradigm that has attracted a growing interest over the past years, partly as a consequence of the prospects for high-rate quantum communication systems based on standard optical telecommunication components. In this overview article, we introduce the concept of quantum continuous variables in optics and then turn to the fundamental impossibility of cloning continuous-variable light states, a result that lies at the heart of quantum key distribution. Then we present state-of-the-art quantum key distribution systems relying on continuous variables, focusing mainly on the protocols using Gaussian-modulated coherent light states and emphasizing the current experimental demonstration of these systems. Finally, we briefly review recent security proofs of these cryptographic protocols.

© 2007 Optical Society of America

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2006

M. Heid and N. Lütkenhaus, "Efficiency of coherent-state quantum cryptography in the presence of loss: influence of realistic error correction," Phys. Rev. A 73, 052316 (2006).
[CrossRef]

Ch. Weedbrook, A. M. Lance, W. P. Bowen, Th. Symul, T. C. Ralph, and P. K. Lam, "Coherent-state quantum key distribution without random basis switching," Phys. Rev. A 73, 022316 (2006).
[CrossRef]

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, "Experimental quantum key distribution with decoy states," Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

R. T. Thew, S. Tanzilli, L. Krainer, S. C. Zeller, A. Rochas, I. Rech, S. Cova, H. Zbinden, and N. Gisin, "Low jitter up-conversion detectors for telecom wavelength GHz QKD," New J. Phys. 8, 32 (2006).
[CrossRef]

M. Legré, H. Zbinden, and N. Gisin, "Implementation of continuous variable quantum cryptography in optical fibres using a go-&-return configuration," Quantum Inf. Comput. 6, 326-335 (2006).

R. García-Patrón and N. J. Cerf, "Unconditional optimality of Gaussian attacks against continuous-variable QKD," Phys. Rev. Lett. 97, 190503 (2006).
[CrossRef] [PubMed]

M. Navascués, F. Grosshans, and A. Acín, "Optimality of Gaussian attacks in continuous variable quantum cryptography," Phys. Rev. Lett. 97, 190502 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, "Generating optical Schrödinger kittens for quantum information processing," Science 312, 83-86 (2006).
[CrossRef] [PubMed]

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Moelmer, and E. S. Polzik, "Generation of a superposition of odd photon number states for quantum information networks," Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

2005

K. J. Gordon, V. Fernandez, G. S. Buller, I. Rech, S. D. Cova, and P. D. Townsend, "Quantum key distribution system clocked at 2 GHz," Opt. Express 13, 3015-3020 (2005).
[CrossRef] [PubMed]

R. Namiki and T. Hirano, "Security of continuous-variable quantum cryptography using coherent states: decline of postselection advantage," Phys. Rev. A 72, 024301 (2005).
[CrossRef]

G. Van Assche, S. Iblisdir, and N. J. Cerf, "Secure coherent-state quantum key distribution protocols with efficient reconciliation," Phys. Rev. A 71, 052304 (2005).
[CrossRef]

M. Navascués, J. Bae, J. I. Cirac, M. Levenstein, A. Sapera, and A. Acín, "Quantum key distillation from Gaussian states by Gaussian operations," Phys. Rev. Lett. 94, 010502 (2005).
[CrossRef] [PubMed]

F. Grosshans, "Collective attacks and unconditional security in continuous variable quantum key distribution," Phys. Rev. Lett. 94, 020504 (2005).
[CrossRef] [PubMed]

M. Navascués and A. Acín, "Security bounds for continuous variables quantum key distribution," Phys. Rev. Lett. 94, 020505 (2005).
[CrossRef] [PubMed]

E. Diamanti, H. Takesue, T. Honjo, K. Inoue, and Y. Yamamoto, "Performance of various quantum-key-distribution systems using 1.55-μm up-conversion single-photon detectors," Phys. Rev. A 72, 052311 (2005).
[CrossRef]

M. Curty, O. Gühne, M. Lewenstein, and N. Lütkenhaus, "Detecting two-party quantum correlations in quantum-key-distribution protocols," Phys. Rev. A 71, 022306 (2005).
[CrossRef]

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, "No-switching quantum key distribution using broadband modulated coherent light," Phys. Rev. Lett. 95, 180503 (2005).
[CrossRef] [PubMed]

J. Lodewyck, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, "Controlling excess noise in fiber-optics continuous-variable quantum key distribution," Phys. Rev. A 72, 050303(R) (2005).
[CrossRef]

U. L. Andersen, V. Josse, and G. Leuchs, "Unconditional quantum cloning of coherent states with linear optics," Phys. Rev. Lett. 94, 240503 (2005).
[CrossRef]

2004

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

U. L. Andersen, O. Glöckl, S. Lorenz, G. Leuchs, and R. Filip, "Experimental demonstration of continuous variable quantum erasing," Phys. Rev. Lett. 93, 100403 (2004).
[CrossRef] [PubMed]

F. Grosshans and N. J. Cerf, "Continuous-variable quantum cryptography is secure against non-Gaussian attacks," Phys. Rev. Lett. 92, 047905 (2004).
[CrossRef] [PubMed]

G. Van Assche, J. Cardinal, and N. J. Cerf, "Reconciliation of a quantum-distributed Gaussian key," IEEE Trans. Inf. Theory 50, 394-400 (2004).
[CrossRef]

M. Curty, M. Lewenstein, and N. Lütkenhaus, "Entanglement as a precondition for secure quantum key distribution," Phys. Rev. Lett. 92, 217903 (2004).
[CrossRef] [PubMed]

R. Namiki and T. Hirano, "Practical limitation for continuous-variable quantum cryptography using coherent states," Phys. Rev. Lett. 92, 117901 (2004).
[CrossRef] [PubMed]

S. Lorenz, N. Korolkova, and G. Leuchs, "Continuous-variable quantum key distribution using polarization encoding and post selection," Appl. Phys. B 79, 273-277 (2004).
[CrossRef]

Ch. Weedbrook, A. M. Lance, W. P. Bowen, Th. Symul, T. C. Ralph, and P. K. Lam, "Quantum cryptography without switching," Phys. Rev. Lett. 93, 170504 (2004).
[CrossRef] [PubMed]

I. Devetak and A. Winter, "Relating quantum privacy and quantum coherence: an operational approach," Phys. Rev. Lett. 93, 080501 (2004).
[CrossRef] [PubMed]

S. Iblisdir, G. Van Assche, and N. J. Cerf, "Security of quantum key distribution with coherent states and homodyne detection," Phys. Rev. Lett. 93, 170502 (2004).
[CrossRef] [PubMed]

J. Wenger, R. Brouri, and P. Grangier, "Non-Gaussian statistics from individual pulses of squeezed light," Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef] [PubMed]

J. C. Bienfang, A. J. Gross, A. Mink, B. J. Hershman, A. Nakassis, X. Tang, R. Lu, D. H. Su, C. W. Clark, C. J. Williams, E. W. Hagley, and J. Wen, "Quantum key distribution with 1.25 Gbps clock synchronization," Opt. Express 12, 2011-2016 (2004).
[CrossRef] [PubMed]

2003

F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, "Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables," Quantum Inf. Comput. 3, 535-552 (2003).

R. Namiki and T. Hirano, "Security of quantum cryptography using balanced homodyne detection," Phys. Rev. A 67, 022308 (2003).
[CrossRef]

T. Hirano, H. Yamanaka, M. Ashikaga, I. Konishi, and R. Namiki, "Quantum cryptography using pulsed homodyne detection," Phys. Rev. A 68, 042331 (2003).
[CrossRef]

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, "Quantum key distribution using gaussian-modulated coherent states," Nature 421, 238-241 (2003).
[CrossRef] [PubMed]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, "Quantum computation with optical coherent states," Phys. Rev. A 68, 042319 (2003).
[CrossRef]

2002

F. Grosshans and P. Grangier, "Continuous variable quantum cryptography using coherent states," Phys. Rev. Lett. 88, 057902 (2002).
[CrossRef] [PubMed]

J. Eisert, S. Scheel, and M. B. Plenio, "On the impossibility of distilling Gaussian states with Gaussian operations," Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef] [PubMed]

Ch. Silberhorn, N. Korolkova, and G. Leuchs, "Quantum key distribution with bright entangled beams," Phys. Rev. Lett. 88, 167902 (2002).
[CrossRef] [PubMed]

A. C. Funk and M. G. Raymer, "Quantum key distribution using nonclassical photon-number correlations in macroscopic light pulses," Phys. Rev. A 65, 042307 (2002).
[CrossRef]

C. Silberhorn, T. C. Ralph, N. Lütkenhaus, and G. Leuchs, "Continuous variable quantum cryptography: beating the 3 dB loss limit," Phys. Rev. Lett. 89, 167901 (2002).
[CrossRef] [PubMed]

2001

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. van Loock, and S. Massar, "Optimal cloning of coherent states with a linear amplifier and beam splitters," Phys. Rev. Lett. 86, 4938-4941 (2001).
[CrossRef] [PubMed]

J. Fiurasek, "Optical implementation of continuous-variable quantum cloning machines," Phys. Rev. Lett. 86, 4942-4945 (2001).
[CrossRef] [PubMed]

D. Gottesman and J. Preskill, "Secure quantum key distribution using squeezed states," Phys. Rev. A 63, 022309 (2001).
[CrossRef]

K. Bencheikh, Th. Symul, A. Jankovic, and J. A. Levenson, "Quantum key distribution with continuous variables," J. Mod. Opt. 48, 1903-1920 (2001).
[CrossRef]

R. F. Werner and M. M. Wolf, "Bound entangled Gaussian states," Phys. Rev. Lett. 86, 3658-3661 (2001).
[CrossRef] [PubMed]

N. J. Cerf, M. Lévy, and G. Van Assche, "Quantum distribution of Gaussian keys using squeezed states," Phys. Rev. A 63, 052311 (2001).
[CrossRef]

B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
[CrossRef] [PubMed]

2000

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability criterion for continuous variable systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
[CrossRef] [PubMed]

R. Simon, "Peres-Horodecki separability criterion for continuous variable systems," Phys. Rev. Lett. 84, 2726-2729 (2000).
[CrossRef] [PubMed]

N. J. Cerf and S. Iblisdir, "Optimal N-to-M cloning of conjugate quantum variables," Phys. Rev. A 62, 040301(R) (2000).
[CrossRef]

N. J. Cerf, A. Ipe, and X. Rottenberg, "Cloning of continuous quantum variables," Phys. Rev. Lett. 85, 1754-1757 (2000).
[CrossRef] [PubMed]

P. Horodecki and M. Lewenstein, "Bound entanglement and continuous variables," Phys. Rev. Lett. 85, 2657-2660 (2000).
[CrossRef] [PubMed]

M. Hillery, "Quantum cryptography with squeezed states," Phys. Rev. A 61, 022309 (2000).
[CrossRef]

T. C. Ralph, "Continuous variable quantum cryptography," Phys. Rev. A 61, 010303(R) (2000).
[CrossRef]

M. D. Reid, "Quantum cryptography with a predetermined key, using continuous-variable Einstein-Podolsky-Rosen correlations," Phys. Rev. A 62, 062308 (2000).
[CrossRef]

1998

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef] [PubMed]

P. Grangier, J.-A. Levenson, and J.-Ph. Poizat, "Quantum demolition measurements in optics," Nature 396, 537-542 (1998).
[CrossRef]

Acín, A.

M. Navascués, F. Grosshans, and A. Acín, "Optimality of Gaussian attacks in continuous variable quantum cryptography," Phys. Rev. Lett. 97, 190502 (2006).
[CrossRef] [PubMed]

M. Navascués, J. Bae, J. I. Cirac, M. Levenstein, A. Sapera, and A. Acín, "Quantum key distillation from Gaussian states by Gaussian operations," Phys. Rev. Lett. 94, 010502 (2005).
[CrossRef] [PubMed]

M. Navascués and A. Acín, "Security bounds for continuous variables quantum key distribution," Phys. Rev. Lett. 94, 020505 (2005).
[CrossRef] [PubMed]

Andersen, U. L.

U. L. Andersen, V. Josse, and G. Leuchs, "Unconditional quantum cloning of coherent states with linear optics," Phys. Rev. Lett. 94, 240503 (2005).
[CrossRef]

U. L. Andersen, O. Glöckl, S. Lorenz, G. Leuchs, and R. Filip, "Experimental demonstration of continuous variable quantum erasing," Phys. Rev. Lett. 93, 100403 (2004).
[CrossRef] [PubMed]

Ashikaga, M.

T. Hirano, H. Yamanaka, M. Ashikaga, I. Konishi, and R. Namiki, "Quantum cryptography using pulsed homodyne detection," Phys. Rev. A 68, 042331 (2003).
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A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
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R. García-Patrón and N. J. Cerf, "Unconditional optimality of Gaussian attacks against continuous-variable QKD," Phys. Rev. Lett. 97, 190503 (2006).
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L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability criterion for continuous variable systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
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U. L. Andersen, O. Glöckl, S. Lorenz, G. Leuchs, and R. Filip, "Experimental demonstration of continuous variable quantum erasing," Phys. Rev. Lett. 93, 100403 (2004).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, "Generating optical Schrödinger kittens for quantum information processing," Science 312, 83-86 (2006).
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E. Diamanti, H. Takesue, T. Honjo, K. Inoue, and Y. Yamamoto, "Performance of various quantum-key-distribution systems using 1.55-μm up-conversion single-photon detectors," Phys. Rev. A 72, 052311 (2005).
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N. J. Cerf, A. Ipe, and X. Rottenberg, "Cloning of continuous quantum variables," Phys. Rev. Lett. 85, 1754-1757 (2000).
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B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
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A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
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P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing," arXiv:quant-ph/0512071 (2005).

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S. Lorenz, N. Korolkova, and G. Leuchs, "Continuous-variable quantum key distribution using polarization encoding and post selection," Appl. Phys. B 79, 273-277 (2004).
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G. A. Barbosa, E. Corndorf, and P. Kumar, "Quantum cryptography with coherent-state light: demonstration of a secure data encryption scheme operating at 100 kb/s," in Quantum Electronics and Laser Science (QELS), Vol. 74 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), pp. 189-190.

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Ch. Weedbrook, A. M. Lance, W. P. Bowen, Th. Symul, T. C. Ralph, and P. K. Lam, "Coherent-state quantum key distribution without random basis switching," Phys. Rev. A 73, 022316 (2006).
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Ch. Weedbrook, A. M. Lance, W. P. Bowen, Th. Symul, T. C. Ralph, and P. K. Lam, "Coherent-state quantum key distribution without random basis switching," Phys. Rev. A 73, 022316 (2006).
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A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, "No-switching quantum key distribution using broadband modulated coherent light," Phys. Rev. Lett. 95, 180503 (2005).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, "Generating optical Schrödinger kittens for quantum information processing," Science 312, 83-86 (2006).
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M. Legré, H. Zbinden, and N. Gisin, "Implementation of continuous variable quantum cryptography in optical fibres using a go-&-return configuration," Quantum Inf. Comput. 6, 326-335 (2006).

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U. L. Andersen, V. Josse, and G. Leuchs, "Unconditional quantum cloning of coherent states with linear optics," Phys. Rev. Lett. 94, 240503 (2005).
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P. Grangier, J.-A. Levenson, and J.-Ph. Poizat, "Quantum demolition measurements in optics," Nature 396, 537-542 (1998).
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M. Navascués, J. Bae, J. I. Cirac, M. Levenstein, A. Sapera, and A. Acín, "Quantum key distillation from Gaussian states by Gaussian operations," Phys. Rev. Lett. 94, 010502 (2005).
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N. J. Cerf, M. Lévy, and G. Van Assche, "Quantum distribution of Gaussian keys using squeezed states," Phys. Rev. A 63, 052311 (2001).
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M. Curty, O. Gühne, M. Lewenstein, and N. Lütkenhaus, "Detecting two-party quantum correlations in quantum-key-distribution protocols," Phys. Rev. A 71, 022306 (2005).
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U. L. Andersen, O. Glöckl, S. Lorenz, G. Leuchs, and R. Filip, "Experimental demonstration of continuous variable quantum erasing," Phys. Rev. Lett. 93, 100403 (2004).
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M. Heid and N. Lütkenhaus, "Efficiency of coherent-state quantum cryptography in the presence of loss: influence of realistic error correction," Phys. Rev. A 73, 052316 (2006).
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Figures (5)

Fig. 1
Fig. 1

Quantum cloning for CV: (a) quantum circuit model and (b) a simple implementation using a phase-insensitive optical amplifier followed by a beam splitter.

Fig. 2
Fig. 2

First experimental implementation of CV-QKD using a modulated coherent-state protocol.[8] The operating wavelength is 780 nm . OI, optical isolator; λ 2 , half-wave plate; AOM, acousto-optic modulator; MF, polarization-maintaining single-mode fiber; OD, optical density; EOM, electro-optic amplitude modulator; PBS, polarizer; BS, beam splitter; R and T, reflection transmission coefficients; PZT, piezoelectric transducer. Focal lengths are given in millimeters.

Fig. 3
Fig. 3

CV-QKD platform (Institut d'Optique and Thales Research and Technology). The operating wavelength is 1550 nm , and the setup uses only standard optical telecommunication components (distributed feedback laser, integrated amplitude and phase modulators, p-i-n photodiodes). The transmission channel is currently a coil of 25 km standard optical fiber.

Fig. 4
Fig. 4

Typical secret key rates expected from the fiber CV-QKD setup, assuming a 1 MHz pulse rate. Dashed curves show the rate K net = β I A B I B E (secure against arbitrary individual attacks) for β = 1 (upper), β = 0.93 (middle), β = 0.87 (lower curve). Solid curves show the rate H net = β I A B χ B E (secure against arbitrary collective attacks; see Subsection 6C) for the same values of β.

Fig. 5
Fig. 5

Two different views of the tolerable excess noise as a function of the distance. In the upper panel we plot the maximum value of the excess noise ϵ referred to the input (as used in the present paper and in other publications by our groups), and in the lower panel we plot the maximum value of the product ϵ T effectively measured by Bob (as used, e.g., in Refs. [35, 36, 37, 38, 39, 40]). The various curves refer to (a) entanglement-breaking limit, (b) two-mode squeezed states and reverse reconciliation, (c) coherent states and direct reconciliation, and (d) coherent states and reverse reconciliation.

Equations (10)

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[ x , p ] = 2 i ,
N < 1 .
N = 1 T T + ϵ ,
I A B = 1 2 log 2 ( 1 + V A 1 + χ tot ) ,
I B E = 1 2 log 2 [ T 2 ( 1 + χ tot + V A ) ( χ line + 1 1 + V A ) 1 + T χ hom ( χ line + 1 1 + V A ) ] .
K raw = I A B I B E
K net = β I A B I B E ,
ϵ = 1 .
ϵ < 2 1 T
ϵ < 1 2 1 T + 1 T 2 + 1 4

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