Abstract

We introduce a novel time-frequency quantum key distribution (TFQKD) scheme based on photon pairs entangled in these two conjugate degrees of freedom. The scheme uses spectral detection and phase modulation to enable measurements in the temporal basis by means of time-to-frequency conversion. This allows large-alphabet encoding to be implemented with realistic components. A general security analysis for TFQKD with binned measurements reveals a close connection with finite-dimensional QKD protocols and enables analysis of the effects of dark counts on the secure key size.

© 2013 OSA

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  53. Other phase modulation schemes such as four-wave mixing in waveguides [54] or sum-frequency generation in nonlinear crystals [51] could of course be used. Here we consider electro-optic phase modulation, which can be implemented with off-the-shelf telecoms components efficiently and with low loss.
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2013

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics7, 210–214 (2013).
[CrossRef]

J. Lavoie, J. M. Donohue, L. G. Wright, A. Fedrizzi, and K. J. Resch, “Spectral compression of single photons,” Nat. Photonics7363–366 (2013).
[CrossRef]

J. Schneeloch, C. J. Broadbent, S. P. Walborn, E. G. Cavalcanti, and J. C. Howell, “EPR steering inequalities from entropic uncertainty relations,” arXiv:1303.7432 (2013).

J. Schneeloch, P. B. Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell, “Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements,” Phys. Rev. Lett.110, 130407 (2013).
[CrossRef] [PubMed]

2012

Ł. Rudnicki, S. Walborn, and F. Toscano, “Optimal uncertainty relations for extremely coarse-grained measurements,” Phys. Rev. A85, 042115 (2012).
[CrossRef]

E. Martin-Lopez, A. Laing, T. Lawson, R. Alvarez, X.-Q. Zhou, and J. L. O’Brien, “Experimental realization of Shor’s quantum factoring algorithm using qubit recycling,” Nat. Photonics6, 773–776 (2012).
[CrossRef]

S. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett.108, 130502 (2012).
[CrossRef] [PubMed]

H. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett.108, 130503 (2012).
[CrossRef] [PubMed]

2011

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A83, 063836 (2011).
[CrossRef]

2010

Z. Chang-Hua, P. Chang-Xing, Q. Dong-Xiao, G. Jing-Liang, C. Nan, and Y. Yun-Hui, “A new quantum key distribution scheme based on frequency and time coding,” Chin. Phys. Lett.27, 090301 (2010).
[CrossRef]

H. Takesue, K. Harada, K. Tamaki, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. Itabashi, “Long-distance entanglement-based quantum key distribution experiment using practical detectors,” Opt. Express18, 16777–16787 (2010).
[CrossRef] [PubMed]

L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. P. Huy, “Frequency-bin entangled photons,” Phys. Rev. A82, 013804 (2010).
[CrossRef]

2009

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
[CrossRef] [PubMed]

G. Brida, V. Caricato, M. Fedorov, M. Genovese, M. Gramegna, and S. Kulik, “Characterization of spectral entanglement of spontaneous parametric-down conversion biphotons in femtosecond pulsed regime,” Europhys. Lett.87, 64003 (2009).
[CrossRef]

M. Avenhaus, A. Eckstein, P. J. Mosley, and C. Silberhorn, “Fiber-assisted single-photon spectrograph,” Opt. Lett.34, 2873–2875 (2009).
[CrossRef] [PubMed]

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon.1, 308–437 (2009).
[CrossRef]

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics3, 696–705 (2009).
[CrossRef]

J. H. Shapiro, “Defeating passive eavesdropping with quantum illumination,” Phys. Rev. A80, 022320 (2009).
[CrossRef]

J. M. Renes and J.-C. Boileau, “Conjectured strong complementary information tradeoff,” Phys. Rev. Lett.103, 020402 (2009).
[CrossRef] [PubMed]

L. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B42, 114011 (2009).
[CrossRef]

2008

R. Salem, M. Foster, A. Turner, D. Geraghty, M. Lipson, and A. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33, 1047–1049 (2008).
[CrossRef] [PubMed]

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett.100, 110504 (2008).
[CrossRef] [PubMed]

N. Beaudry, T. Moroder, and N. Lütkenhaus, “Squashing models for optical measurements in quantum communication,” Phys. Rev. Lett.101, 93601 (2008).
[CrossRef]

A. Lita, A. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express16, 3032–3040 (2008).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008).
[CrossRef] [PubMed]

2007

H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-db channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
[CrossRef]

L. Zhang, A. B. U’ren, R. Erdmann, K. O’Donnell, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Generation of highly entangled photon pairs for continuous variable Bell inequality violation,” J. Mod. Opt.54, 707–719 (2007).
[CrossRef]

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett.98, 060503 (2007).
[CrossRef] [PubMed]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
[CrossRef]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Oemer, M. Fuerst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Free-space distribution of entanglement and single photons over 144 km,” Nature Phys.3, 481 (2007).
[CrossRef]

2006

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A73, 063819 (2006).
[CrossRef]

B. Qi, “Single-photon continuous-variable quantum key distribution based on the energy-time uncertainty relation,” Opt. Lett.31, 2795–2797 (2006).
[CrossRef] [PubMed]

I. Ali-Khan and J. C. Howell, “Experimental demonstration of high two-photon time-energy entanglement,” Phys. Rev. A73, 031801 (2006).
[CrossRef]

2004

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

2003

J. Azaña, “Time-to-frequency conversion using a single time lens,” Opt. Commun.217, 205–209 (2003).
[CrossRef]

2002

M. Bourennane, A. Karlsson, G. Björk, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A35, 10065 (2002).
[CrossRef]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett.88, 127902 (2002).
[CrossRef] [PubMed]

M. Krishna and K. Parthasarathy, “An entropic uncertainty principle for quantum measurements,” Sankhyâ A842–851 (2002).

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys.74, 145–195 (2002).
[CrossRef]

2001

L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414, 413–418 (2001).
[CrossRef] [PubMed]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

2000

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

1999

1998

W. P. Grice, R. Erdmann, I. A. Walmsley, and D. Branning, “Spectral distinguishability in ultrafast parametric down-conversion,” Phys. Rev. A57, R2289–R2292 (1998).
[CrossRef]

1997

W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with a broadband pump,” Phys. Rev. A56, 1627–1634 (1997).
[CrossRef]

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, ““Plug and play” systems for quantum cryptography,” Appl. Phys. Lett.70, 793–795 (1997).
[CrossRef]

P. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” Appl. Math. J. Comp26, 1484–1509 (1997).

1994

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett.64, 270–272 (1994).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron.30, 1951–1963 (1994).
[CrossRef]

1991

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67, 661–663 (1991).
[CrossRef] [PubMed]

1990

1989

S. Cova, A. Lacaita, M. Ghioni, G. Ripamonti, and T. Louis, “20-ps Timing resolution with single-photon avalanche diodes,” Rev. Sci. Instrum.60, 1104 (1989).
[CrossRef]

1984

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Figures (3)

Fig. 1
Fig. 1

Time-frequency quantum key distribution with spectrally entangled photon pairs. (a) photon pairs from the source are distributed to Alice and Bob, who have identical measurement setups: spectral correlations are revealed with photon-counting spectrometers (spect); security is certified by measurements in the conjugate arrival-time basis using time-to-frequency conversion (TFC). (b) We assume a source with a Gaussian anti-correlated joint spectral amplitude f (ω,ω′) (top) and a corresponding positively correlated Gaussian arrival time distribution (t,t′) (bottom). Black ruled lines indicate spectral/temporal measurements made by Alice and Bob centred at the values {ωj}/{tj} (see text). (c) Time-to-frequency conversion is implemented with a dispersive element followed by a phase modulator. The phase modulation (black wavy line) is timed so that the pulses from the source (red) arrive during the quadratic portion of the phase modulation.

Fig. 2
Fig. 2

TFQKD in the presence of losses and noise. (a) The source emits mostly the vacuum state |0〉, with ε the small probability to emit a correlated photon pair. For simplicity we consider the source located equidistant from Alice and Bob, connected by lossy channels of length L with attenuation length Latt. The photon detectors comprising the spectrometers have efficiency ηd and suffer dark counts (red stars) with probability d. (b) We compute the secure key size I as a function of the alphabet size IM at a distance L = Latt for a typical protocol with ε = 0.1, ηd = 25% (including coupling losses) and d = 10−6, which are achievable with guided-wave parametric sources [74, 75] and standard APD detectors at near-infrared (NIR) wavelengths [41]. (c) We plot the secure key size as a function of the channel length L, assuming Latt = 2.2 km (corresponding to silica fibre at NIR wavelengths [76]), for a range of alphabet sizes.

Fig. 3
Fig. 3

Schematic of the operator Π j 1 / 2 Π ˜ k 1 / 2, showing the approximate computation of its largest singular value.

Equations (32)

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| ψ = f ( ω , ω ) | ω , ω d ω d ω ,
Π j = F j ( ω ) | ω ω | d ω .
F j ( ω ) = { 1 | ω ω j | δ ω / 2 ; 0 otherwise ,
| ω ( 2 π i ϕ ¨ ) 1 / 2 e i ( ω ω ) 2 / 2 ϕ ¨ | ω d ω .
| ω ( 2 π i ϕ ¨ ) 1 / 2 e i ω 2 / 2 ϕ ¨ e i ( 1 / ϕ ¨ φ ) ω 2 / 2 e i ω ω / ϕ ¨ | ω d ω .
Π ˜ j = ( 2 π ϕ ¨ ) 1 / 2 F ˜ j ( ω ω ) | ω ω | d ω d ω ,
F ˜ j ( x ) = ( 2 π ϕ ¨ ) 1 / 2 F j ( x ) e i x x / ϕ ¨ d x
= δ ω ( 2 π ϕ ¨ ) 1 / 2 e i ω j x / ϕ ¨ sinc ( x δ ω / 2 ϕ ¨ ) .
f ( ω , ω ) = ( π Δ + Δ / 8 ) 1 / 2 exp ( ω 2 / 2 Δ + 2 ω + 2 / 2 Δ 2 ) ,
ϕ ¨ = δ ω δ t = Δ Δ + = β + β M δ ω 2 .
Ω β δ ω .
A = β + β M .
I BA = b a p b a BA log 2 ( p b a BA p b B p a A ) ,
I = I BA I BE ,
H X | Y = y p y Y H X | Y = y , where H X | Y = y = x p x | y XY log 2 p x | y XY
I = H B | E H B | A .
H B ( ρ ) + H ˜ B ( ρ ) B ,
H B | EA + H ˜ B | EA B .
H B | E + H ˜ B | A B .
I B H ˜ B | A H B | A .
B = 2 log 2 C ,
H B I min { H B , log 2 [ 2 π δ ω δ t ] H ˜ B | A H B | A } ,
p b a BA = [ ( 1 p ) δ b a + p / M ] / M ,
p = κ ( M 1 ) κ M + 1 ; where κ = 2 d ( 1 η ) η + M d 2 ( 1 + 1 ε ε η 2 ) .
H ˜ B | A = H B | A = p log 2 ( M 1 ) + h ( p ) ,
I I M 2 p log 2 ( M 1 ) 2 h ( p ) c ,
Π j Π ˜ k = ( 2 π ϕ ¨ ) 1 / 2 F j ( ω ) F ˜ k ( ω ω ) | ω ω | d ω d ω = δ ω 2 π ϕ ¨ F j ( ω ) e i ω k ( ω ω ) / ϕ ¨ sinc [ δ ω 2 ( ω ω ) ϕ ¨ ] | ω ω | d ω d ω .
( ω , ω ) = { δ ω 2 π ϕ ¨ sinc [ δ ω 2 ( ω ω ) ϕ ¨ ] | ω ω j | δ ω / 2 ; 0 otherwise .
λ = δ ω 2 π ϕ ¨ × δ ω 2 π ϕ ¨ δ ω .
p = p incorrect p correct + p incorrect ,
P incorrect = d ¯ 2 ( M 1 ) [ 2 ε η η ¯ ( M 1 ) d + ε η ¯ 2 d 2 M ( M 1 ) + ε ¯ d 2 M ( M 1 ) ] ,
P correct = d ¯ 2 ( M 1 ) [ ε η 2 + 2 ε η η ¯ d + ε η ¯ 2 d 2 M + ε ¯ d 2 M ] ,

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