We consider two remote parties connected to a relay by two quantum channels.
To generate a secret key, they transmit coherent states to the relay, where the
states are subject to a continuous-variable (CV) Bell detection. We study the
ideal case where Alice's channel is lossless, i.e., the relay is locally
situated in her lab and the Bell detection is performed with unit efficiency.
This configuration allows us to explore the optimal performances achievable by
CV measurement-device-independent (MDI) quantum key distribution (QKD). This
corresponds to the limit of a trusted local relay, where the detection loss can
be re-scaled. Our theoretical analysis is confirmed by an experimental
simulation where 10^-4 secret bits per use can potentially be distributed at
170km assuming ideal reconciliation.Comment: in Proceedings of the SPIE Security + Defence 2015 conference on
  Quantum Information Science and Technology, Toulouse, France (21-24 September
  2015) - Paper 9648-4

Andersen, Ulrik L.

Braunstein, Samuel L.

Gehring, Tobias

Jacobsen, Christian S.

Ottaviani, Carlo

Pirandola, Stefano

Spedalieri, Gaetana

English

arXiv

We consider two remote parties connected to a relay by two quantum channels. To generate a secret key, they transmit coherent states to the relay, where the states are subject to a continuous-variable (CV) Bell detection. We study the ideal case where Alice's channel is lossless, i.e., the relay is locally in her lab and the Bell detection is perfomed with unit efficiency. This configuration allows us to explore the optimal performances achievable by CV measurement-device-independent quantum key distribution. This corresponds to the limit of a trusted local relay, where the detection loss can be re-scaled. Our theoretical analysis is confirmed by an experimental simulation where 10-4 secret bits per use can potentially be distributed at 170km assuming ideal reconciliation

Jacobsen, Christian Scheffmann

Andersen, Ulrik Lund

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 18, 2017Quantum cryptography with an ideal local relaySpedalieri, Gaetana; Ottaviani, Carlo; Braunstein, Samuel L.; Gehring, Tobias; Jacobsen, ChristianScheffmann; Andersen, Ulrik Lund; Pirandola, StefanoPublished in:Electro-Optical and Infrared Systems: Technology and Applications XII; and Quantum Information Science andTechnologyLink to article, DOI:10.1117/12.2202662Publication date:2015Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Spedalieri, G., Ottaviani, C., Braunstein, S. L., Gehring, T., Jacobsen, C. S., Andersen, U. L., & Pirandola, S.(2015). Quantum cryptography with an ideal local relay. In Electro-Optical and Infrared Systems: Technologyand Applications XII; and Quantum Information Science and Technology (Vol. 9648). SPIE - InternationalSociety for Optical Engineering.  (Proceedings of S P I E - International Society for Optical Engineering). DOI:10.1117/12.2202662Quantum cryptography with an ideal local relayGaetana Spedalieria, Carlo Ottaviania, Samuel L. Braunsteina, Tobias Gehringb, Christian S.Jacobsenb, Ulrik L. Andersenb, and Stefano PirandolaaaComputer Science and York Centre for Quantum Technologies,University of York, DeramoreLane, York YO10 5GH, United KingdombDepartment of Physics, Technical University of Denmark, Fysikvej, 2800 Kongens Lyngby,DenmarkABSTRACTWe consider two remote parties connected to a relay by two quantum channels. To generate a secret key, theytransmit coherent states to the relay, where the states are subject to a continuous-variable (CV) Bell detection.We study the ideal case where Alice’s channel is lossless, i.e., the relay is locally in her lab and the Bell detectionis perfomed with unit efficiency. This configuration allows us to explore the optimal performances achievableby CV measurement-device-independent quantum key distribution. This corresponds to the limit of a trustedlocal relay, where the detection loss can be re-scaled. Our theoretical analysis is confirmed by an experimentalsimulation where 10−4 secret bits per use can potentially be distributed at 170km assuming ideal reconciliation.1. INTRODUCTIONQuantum key distribution (QKD)1, 2 is a central area in quantum information science.3, 4 A typical QKD pro-tocol involves two parties, Alice and Bob, who generate secret keys by exchanging quantum systems over aninsecure communication channel. Another scenario involves a swapping-like protocol5 where secret correlationsare established by the measurement of a third untrusted party (relay). This idea of ‘measurement-device inde-pendence’ (MDI)5–15 has been extended to continuous-variable (CV) systems,16, 17 with the possibility of muchhigher key rates.In this paper, we study consider a limit configuration for CV-MDI-QKD, where the relay is in Alice’s laband performs an ideal Bell detection. This is an extrapolation which allows us to investigate the maximalrate/distance performances achievable by CV-MDI-QKD. Experimentally, this is equivalent to consider a localrelay where the loss associated with the various technical imperfections (such as the detector inefficiencies) canbe re-scaled and therefore neglected. As a matter of fact, this limit case corresponds to the case where the lossof the relay is trusted.Our theoretical analysis, confirmed by an experimental simulation, shows that 10−2 secret bits per relay usecan be distributed at 10dB loss in Bob’s channel, equivalent to 50km of standard optical fibre (at the loss rateof 0.2dB/km). Assuming ideal reconciliation, a potential rate of about 10−4 secret bits per relay use can bedistributed over a very lossy link, i.e., 34dB loss corresponding to 170km of fibre.2. PROTOCOLThe scheme is depicted in Fig. 1. At one side, Alice prepares a mode A in a coherent state |α〉 with Gaussian-modulated amplitude α; at the other side, Bob prepares mode B in another coherent state |β〉 with Gaussian-modulated amplitude β (Gaussian distributions have zero mean and large variance). Modes A and B are sentto the relay, which performs a CV Bell detection,18 by mixing the modes in a balanced beam splitter whoseoutput ports are conjugately homodyned with outputs q− and p+. The complex variable γ := (q− + ip+)/√2 isthen communicated to Alice and Bob via a classical public channel. Since γ ' α− β∗, each party may infer thevariable of the other party by postprocessing.Further author information: Send correspondence to G.S. (gae.spedalieri@york.ac.uk)Invited PaperElectro-Optical and Infrared Systems: Technology and Applications XII; and Quantum Information Science and Technology,edited by David A. Huckridge, Reinhard Ebert, Mark T. Gruneisen, Miloslav Dusek, John G. Rarity, Proc. of SPIE Vol. 9648, 96480Z · © 2015 SPIE · CCC code: 0277-786X/15/$18 · doi: 10.1117/12.2202662Proc. of SPIE Vol. 9648  96480Z-1Downloaded From: http://proceedings.spiedigitallibrary.org/ on 11/08/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspxAlice BobA Bq- p+Beam SplitterRelayCoherent StatesCoherent StatesCV Bell DetectionLink LinkFigure 1. Basic protocol. See text for explanations.The most general eavesdropping strategy is a joint attack involving both the relay and the two links.16, 17Here we consider the simple case where Eve attacks Bob’s link B only, by means of a Gaussian attack19 whichintroduces loss and thermal noise. The travelling mode B is mixed with an ancillary mode E by a beam splitterwith transmissivity τ . The ancilla introduces thermal noise with variance ω and belongs to a reservoir of ancillasunder Eve’s control. This kind of entangling-cloner attack4 is repeated for each use of the relay and the outputancillas are finally detected by Eve by means of an optimized collective quantum measurement.3. SECRET-KEY RATEBy specializing the formulas of CV-MDI-QKD,16 we can derive the secret-key rate for the scenario depicted inFig. 2. Assuming ideal reconciliation efficiency and large modulation, the key rate is given byR = h(χ1+τ − 1)− h[τχ−(1+τ)21−τ2]+ log2[2(1+τ)e(1−τ)χ], (1)whereh(x) :=x+ 12log2x+ 12− x− 12log2x− 12, (2)and χ is the equivalent noise, decomposable as χ = χloss + ε, where χloss = 2(1 + τ)/τ is the noise due to loss,while ε is the ‘excess noise’. The maximum theoretical performance of the protocol, with respect to the losspresent in Bob’s link, is reached for ε = 0. In this case, we haveRloss = h[(2− τ)/τ ] + log2[τ/(1− τ)e], (3)which goes to zero only for τ → 0, corresponding to Bob arbitrarily far from the relay. It is easy to converttransmissivity τ to distance d in optical fibre, by considering the standard loss rate of 0.2dB/km.Bobd i s t a n c e dAliceRelayFigure 2. Key-distribution via an ideal local relay. This is locally in Alice’s lab and it is assumed to work perfectly(unit quantum efficiency). By contrast, Bob’s link has transmissivity τ < 1 corresponding to some distance d in standardoptical fibre. Bob’s channel can also be affected by thermal/excess noise.In the scenario of Fig. 2, Bob can be very far from the relay also in the presence of non-zero excess noiseε 6= 0, with potential distances beyond 100 km of simulated fibre. This can be seen from the numerical resultsshown in Fig. 3, where the solid line represents the case of a pure-loss attack (ε = 0), while the dashed curvecorresponds to an attack with non-zero excess noise, in particular ε = 0.02. We can see the robustness of thekey rate with respect to the excess noise.This theoretical analysis is also confirmed by a proof-of-principle experiment where we have realized the localideal relay by suitably re-scaling the loss in Alice’s link in the post-processing of the data. We have reproduced theProc. of SPIE Vol. 9648  96480Z-2Downloaded From: http://proceedings.spiedigitallibrary.org/ on 11/08/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspxextreme asymmetric configuration of Fig. 2, with variable Bob’s transmissivity τ , down to 4×10−4 correspondingto about 170km in standard optical fibre. For every experimental point, we have evaluated the key rate Rassuming ideal reconcilation efficiency ξ = 1. Experimental results are plotted in Fig. 3 and compared with thetheoretical predictions, with excellent agreement. The extrapolated experimental rate approaches the theoreticallimit of the pure-loss attack. Due to imperfections, we have an excess noise ε . 0.02. Note that we can potentiallyreach R ' 10−4 secret bits per relay use over a link with 34dB loss, equivalent to 170km of optical fibre.ææææææææææææææààààààààààààæææææææææææææææææææææææààààààààààæææààà0 50 100 1500.0010.010.11Distance HkmLRateHbitsuseLFigure 3. Secret-key rate R versus Bob’s distance d from the relay. Experimental points refer to ideal reconciliation(ξ = 1, red circles) and realistic reconciliation (ξ ' 0.97, blue squares). For comparison, we also plot the theoretical ratesfor a pure-loss attack (solid line) and a Gaussian attack with excess noise ε = 0.02 (dashed line).Note that the current reconciliation procedures for CV protocols do not have unit efficiency (indeed this isone of the main factors limiting the distance of CV-QKD). By taking this limitation into account20 (ξ ' 0.97),we can still reach remarkably high rates over distances well beyond the typical connection lengths of a network.As we can see from Fig. 3, one can potentially achieve R ' 10−2 secret bits per relay use over a link with 10dBloss, equivalent to 50km of optical fibre.4. CONCLUSIONIn this work, we have explored the maximal performances in terms of rates and distances achievable by CV-MDI-QKD with coherent states. We have considered the extreme configuration where the relay is in Alice’s laband the Bell detection is ideally performed. An important feature of this protocol is the simplicity of the relay,which does not possess any quantum source but just performs a standard optical measurement, with all the heavyprocedures of data post-processing left to the end-users, fulfilling the idea behind the end-to-end principle.21 CVBell detection involves highly efficient photodetectors plus linear optics, whereas the discrete-variable version ofthis measurement needs nonlinear elements to operate deterministically. This feature combined with the use ofcoherent states makes the scheme very attractive, guaranteeing both cheap implementation and extremely highrates.Our study also shows how improvements in the classical reconciliation techniques (from ξ ' 0.97 to ξ =1) have a dramatic impact on the performances of the protocol, which means that the development of moreefficient classical codes for error correction and privacy amplification is a central task in CV-QKD. Finally,future investigations could involve the explicit security analysis of mixed technology environments where someof the connections are established at low frequencies (infrared or microwave) where thermal effects becomeimportant.22–24AcknowledgmentsThis work was supported by EPSRC (Grants EP/J00796X/1 and EP/L011298/1) and the Leverhulme Trust.Proc. of SPIE Vol. 9648  96480Z-3Downloaded From: http://proceedings.spiedigitallibrary.org/ on 11/08/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspxREFERENCES1. Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002).2. Scarani, V., Bechmann-Pasquinucci, H., Cerf, N. J., Dusek, M., Lutkenhaus, N. & Peev, M. The security ofpractical quantum key distribution. Rev. Mod. Phys. 81, 1301 (2009).3. Wilde, M. M. Quantum Information Theory (Cambridge University Press, Cambridge, 2013).4. Weedbrook, C., Pirandola, S., Garcia-Patron, R., Cerf, N. J., Ralph, T. C., Shapiro, J. 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Quantum cryptography with an ideal local relay

Gaetana Spedalieri

Carlo Ottaviani

Samuel L. Braunstein

Tobias Gehring

Christian S. Jacobsen

Ulrik L. Andersen

Stefano Pirandola

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Quantum cryptography with an ideal local relay

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