## Large-alphabet time-frequency entangled quantum key distribution by means of time-to-frequency conversion |

Optics Express, Vol. 21, Issue 13, pp. 15959-15973 (2013)

http://dx.doi.org/10.1364/OE.21.015959

Acrobat PDF (1273 KB)

### 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

## 1. Introduction

1. J. Rarity, P. Tapster, J. Walker, and S. Seward, “Experimental demonstration of single photon rangefinding using parametric downconversion,” Appl. Opt. **29**, 2939–2943 (1990) [CrossRef] [PubMed] .

2. 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. A **83**, 063836 (2011) [CrossRef] .

5. 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. Photonics **6**, 773–776 (2012) [CrossRef] .

6. 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] .

7. J. H. Shapiro, “Defeating passive eavesdropping with quantum illumination,” Phys. Rev. A **80**, 022320 (2009) [CrossRef] .

*quantum key distribution*(QKD) [10–12

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

13. S. Galbraith, *Mathematics of public key cryptography* (Cambridge University, 2012) [CrossRef] .

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

16. “MagiQ (http://magiqtech.com), ID Quantique (www.idquantique.com), QuintessenceLabs (qlabsusa.com), Toshiba (http://www.toshiba-europe.com/research/crl/qig/quantumkeyserver.html),”.

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

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

23. L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature **414**, 413–418 (2001) [CrossRef] [PubMed] .

## 3. Security

59. 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] .

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

*b*

^{th}detector firing along with Alice’s

*a*

^{th}detector, when both measure in the frequency basis. These correlations are quantified by the

*mutual information I*, where

_{BA}*I*is the mutual information between Bob and Eve [66]. We can calculate

_{BE}*I*

_{BA}from the statistics of Alice’s and Bob’s measurements but

*I*

_{BE}is not known and must be bounded by analyzing the correlations in the complementary (temporal) basis. To proceed, we write the mutual information in the form

*I*

_{XY}=

*H*

_{X}−

*H*

_{X|Y}[59

59. 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] .

*x*

^{th}detector of party X fires, given that the

*y*

^{th}detector of party Y fires. We then obtain Next we need to find a lower bound for

*H*

_{B|E}. To do this, we note that the variability of Bob’s measurement results in time and frequency are restricted by Heisenberg’s uncertainty principle. More generally, if Bob receives an arbitrary quantum state

*ρ*, the marginal entropies

*H*

_{B}(

*ρ*) (

*H̃*

_{B}(

*ρ*)) for Bob’s measurements in frequency (time), satisfy the entropic uncertainty relation where

*B*is a number that depends only on Bob’s measurements, not on the state

*ρ*. Alice’s and Eve’s measurements on the state emitted by the source produce a conditional state

*ρ*

_{B|A=}

_{a,}_{E=}

*at Bob’s detectors, so that we have*

_{e}*H*

_{B}(

*ρ*

_{B|A=}

_{a,}_{E=}

*) =*

_{e}*H*

_{B|A=}

_{a,}_{E=}

*, using a natural tripartite extension of the notation in Eq. (14). Substituting this conditional state into Eq. (16) and averaging over the possible outcomes of Alice’s and Eve’s measurements, we obtain Now, since uncertainty (entropy) is only increased by removing a “given” precondition from a conditional distribution, we can remove Alice as a given from the first conditional entropy, and Eve as a given from the second, to get Inserting this into Eq. (15) gives The formula for the entropic bound is where*

_{e}*A*||

**denoting the largest singular value of the operator**

_{∞}*A*[60, 67, 68

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

*C*is well approximated by the quantity

*H*

_{B}of the measurements (since

*H*

_{B|E}≤

*H*

_{B}). That is, regardless of the measurement precision, the secret key cannot exceed the entropy of Bob’s marginal statistics. This is in turn bounded from above by

*H*

_{B}≤

*I*= log

_{M}_{2}

*M*, which is the maximum information content for

*M*-outcome measurements. By design

*M*= (

*β*

_{+}/2

*β*

_{−})

*K*is roughly equal to the Schmidt number of the source, to within a factor on the order of unity. With appropriate choices of

*β*

_{±}and for sufficiently resolving measurements (such that

*δωδ*

*t*∼

*M*

^{−1}), this protocol can therefore be efficient in terms of the information extracted from the quantum states it consumes as a resource. For the modulation scheme described above, we have

*δt*=

*δω/ϕ̈*, which along with Eq. (10) indeed yields

*δωδt*= (

*β*

_{+}

*β*

_{−}

*M*)

^{−1}. However we note that Eq. (21) is a general expression for the size of the secret key for a TFQKD protocol with arbitrary finite resolution in both the time and frequency bases, which is not specific to our proposed modulation scheme and makes no assumptions about the form of the state produced by the source, or the power of Eve’s interventions.

## 4. Practical considerations

*squashing protocols*have been proposed for qubit-based QKD to deal with the effects of multi-pair emission [69

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

*Mutatis mutandis*Eve can make the same kind of intercept-resend attack in the temporal basis. In this way, Eve can gain perfect knowledge of the sifted key, while forcing Alice and Bob to ignore those events where she failed to choose the correct measurement basis. Alice and Bob can prevent this attack by always rejecting photons occupying regions of chronocyclic space outside the sensitivity regions of their detectors, by means of spectral filters and temporal gates [25]. In our proposed protocol this could be accomplished with a combination of a dielectric stack and a Pockels cell preceding each of Alice’s and Bob’s detection systems. See [64] for a more general discussion of the effects of finite detection range on entanglement verification.

*I*on the size

*I*of the alphabet for a typical implementation with APD-type detectors and a channel length with

_{M}*L*=

*L*

_{att}, assuming the values of

*β*

_{±}used previously. The secure key initially grows in proportion to the alphabet, albeit with a magnitude reduced by the binning deficit

*c*≈ 0.086 bits. Here the deficit is small enough that exchanging up to 4 secure key bits per detected photon pair remains feasible with the components described previously. However for

*I*

_{M}> 11 (

*i.e. M*> 2048) the secure key decreases and rapidly falls to zero, since the dark count rate from the growing number of detectors causes significant errors that are attributed to Eve.

## 5. Conclusion

## A. Entropic uncertainty relation for finite-resolution measurements

_{j}}, {Π̃

*} is manifested by the entropic uncertainty relation in Eq. (16), with the bound*

_{k}*F*(

*ω*)

^{1/2}=

*F*(

*ω*). We therefore consider the singular values of the operator

*e*

^{−iωk(ω′−ω″)/ϕ̈}can be dropped since this represents a unitary transformation that does not affect the singular values. Similarly the index

*j*on

*F*is arbitrary since this parameterises a shift in the position of the top-hat function that again does not affect the singular values. The singular values are thus obtained by considering the Schmidt decomposition of the two-dimensional function As shown in Fig. (3), this has the form of a narrow vertical strip cut from a broad diagonal band, and is well approximated by the factorable function ℱ(

_{j}*ω′,ω″*) ≈

*λf*(

*ω′*)

*g*(

*ω″*), where

*f*(

*ω′*) is a mode function with width

*δω*,

*g*(

*ω*

*″*) is a mode function with width 2

*π*

*ϕ̈/*

*δω*, and Numerical calculations confirm that this remains an excellent approximation to the largest singular value of the function in Eq. (27) provided

*δω*/(2

*πϕ̈/δω*) < 0.1, which is guaranteed by Eq. (10) for a large-alphabet protocol with

*M*≫ 1/

*β*

_{+}

*β*

_{−}. Here we note that even though this approximation breaks down for smaller alphabets, we have checked numerically that the formula in Eq. (21) in the main text can still be used, since the condition

*I*≤

*H*

_{B}becomes saturated. Substituting Eq. (28) into Eq. (20) and using

*δt*=

*δω/ϕ̈*, we obtain the bound on the secret key given in Eq. (21). We note that the resulting bound applies quite generally to any TFQKD protocol employing binned frequency and time measurements with resolutions

*δω*,

*δt*.

## B. Dark count error probability

## Acknowledgments

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**OCIS Codes**

(060.4080) Fiber optics and optical communications : Modulation

(060.4230) Fiber optics and optical communications : Multiplexing

(060.4510) Fiber optics and optical communications : Optical communications

(060.5060) Fiber optics and optical communications : Phase modulation

(270.5570) Quantum optics : Quantum detectors

(060.5565) Fiber optics and optical communications : Quantum communications

(270.5568) Quantum optics : Quantum cryptography

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: May 6, 2013

Revised Manuscript: June 19, 2013

Manuscript Accepted: June 20, 2013

Published: June 26, 2013

**Citation**

J. Nunn, L. J. Wright, C. Söller, L. Zhang, I. A. Walmsley, and B. J. Smith, "Large-alphabet time-frequency entangled quantum key distribution by means of time-to-frequency conversion," Opt. Express **21**, 15959-15973 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15959

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### References

- J. Rarity, P. Tapster, J. Walker, and S. Seward, “Experimental demonstration of single photon rangefinding using parametric downconversion,” Appl. Opt.29, 2939–2943 (1990). [CrossRef] [PubMed]
- 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]
- P. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” Appl. Math. J. Comp26, 1484–1509 (1997).
- 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]
- 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]
- 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]
- J. H. Shapiro, “Defeating passive eavesdropping with quantum illumination,” Phys. Rev. A80, 022320 (2009). [CrossRef]
- A. Hayat, X. Xing, A. Feizpour, and A. M. Steinberg, “Multidimensional quantum information based on single-photon temporal wavepackets,” arXiv:1212.1483 (2012).
- P. Rohde, J. Fitzsimons, and A. Gilchrist, “The information capacity of a single photon,” arXiv:1211.1427 (2012).
- J. Mower, Z. Zhang, P. Desjardins, C. Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” arXiv:1210.4501 (2012).
- 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]
- B. Qi, “Single-photon continuous-variable quantum key distribution based on the energy-time uncertainty relation,” Opt. Lett.31, 2795–2797 (2006). [CrossRef] [PubMed]
- S. Galbraith, Mathematics of public key cryptography (Cambridge University, 2012). [CrossRef]
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