Most Quantum Key Distribution protocols use a two-dimensional basis such as
HV polarization as first proposed by Bennett and Brassard in 1984. These
protocols are consequently limited to a key generation density of 1 bit per
photon. We increase this key density by encoding information in the transverse
spatial displacement of the used photons. Employing this higher-dimensional
Hilbert space together with modern single-photon-detecting cameras, we
demonstrate a proof-of-principle large-alphabet Quantum Key Distribution
experiment with 1024 symbols and a shared information between sender and
receiver of 7 bit per photon.Comment: 9 pages, 6 figures, Added references, Updated Fig. 1 in the main
  text, Updated Fig.1 in supplementary material, Added section Trojan-horse
  attacks in supplementary material, title changed, Added paragraphs about
  final key rate and overfilling the detector to result sectio

Hooijschuur, P.

Luiten, W. M.

Pinkse, P. W. H.

Tentrup, T. B. H.

van der Meer, R.

English

arXiv

Most quantum key distribution protocols using a two-dimensional basis, such as HV polarization as first proposed by Bennett and Brassard in 1984, are limited to a key generation density of 1 bit per photon. We increase this key density by encoding information in the transverse spatial displacement of the used photons. Employing this higher-dimensional Hilbert space together with modern single- photon-detecting cameras, we demonstrate a proof-of-principle large-alphabet quantum key distribution experiment with 1024 symbols and a shared information between sender and receiver of 7 bit per photon

Tentrup, Tristan Bernhard Horst

Luiten, W.M.

van der Meer, Reinier

Hooijschuur, Peter

Pinkse, Pepijn W.H.

NARCIS 

Large-alphabet quantum key distribution using spatially encoded light

Most quantum key distribution protocols using a two-dimensional basis, such as HV polarization as first proposed by Bennett and Brassard in 1984, are limited to a key generation density of 1 bit per photon. We increase this key density by encoding information in the transverse spatial displacement of the used photons. Employing this higher-dimensional Hilbert space together with modern single- photon-detecting cameras, we demonstrate a proof-of-principle large-alphabet quantum key distribution experiment with 1024 symbols and a shared information between sender and receiver of 7 bit per photon.<br/

Tentrup, T.B.H.

Pinkse, P.W.H.

University of Twente Research Information

Supplemental Material1. DetectionTo detect the photons in a two-dimensional grid, we use an ICCD (Lambert HICAM500S). It consists of an intensifier stage fiber-coupled to a CMOS camera of 1280×1024pixels. The photocathode of the ICCD acts as a gate and is triggered by the heraldphotons at 280 kHz. The delay between the trigger signal and the signal photon wasmeasured to be 91 ns. The gate width of the intensifier is 5 ns. The CMOS camera isread out with 500 frames per second. The variance of the read-out noise of the CMOSis 0.4 counts and a threshold of 5 counts is set to filter the readout noise from the data.Moreover, a threshold on the size and intensity of detection events is set to between 2and 10 pixels and between 1 and 60 counts, respectively, to remove unwanted spuriousion events. This method is similar to [1, 2].2. Trojan-horse attacksThe quantum channel connecting Alice and Bob could be used as a door by aneavesdropper to read out the state of Alice’s and Bob’s devices, making the setupvulnerable against Trojan-horse attacks [3]. To counteract such attacks, the opticaldevices should only be active when the photons are sent. In our setup that could berealized by replacing the mechanical switch of the half-wave plate with a fast electro-optic modulator and the liquid-crystal SLM by a faster digital micromirror device. Theycould be synchronized to the photon arrival. Another countermeasure is to use bandpassfilters and optical isolators at the entrance of Alice’s device. Alice should also useauxiliary detectors to detect any light entering her device to detect attacks. In thefinite-key regime the security of leaky decoy-state BB84 has been investigated [4].3. Intercept-resend attackIn this attack, Eve intercepts a fraction η of the quantum states and performs projectivemeasurements randomly choosing one of the two mutually unbiased bases. Eve resendsher measurement result, introducing an error due to the collapse of the wavefunction.For the security of the protocol, we need to ensure that the information Eve can gain fromintercepting the communication is lower than the mutual information between senderand receiver [5, 6]. Assuming an intercept-resend attack, the information Eve can learnis I(Eve) = η2I(Alice) with η the fraction of intercepted photons and I(Alice) the sentinformation of Alice. Averaged over the compatible bases, we find I(Alice) = 9.4 bit.An eavesdropper can extract a maximum of I(Eve) = η5.72 bit. Therefore, just as in theSupplemental Material 2case of the original BB84, information gain is only possible at the expense of disturbingthe signal [7]. An eavesdropper will be recognized in postprocessing, since she collapsesthe wavefunction and thereby increases the error rate of the key generated by Alice andBob. Alice and Bob will have to compare a random part of their key to decide if theyhave been eavesdropped. Intercepting a fraction of η photons, an attacker introducesan error ofEEve =η2d− 1d, (1)where d is the number of symbols. To calculate the quantum bit error rate of thesifted key, we used the Gray code [8] to encode the x and y position of the symbol ina bit string. In this way, we reduce the bit error rate, since 31.3% of the error is dueto crosstalk to neighboring symbols. In the Gray code, neighboring symbols have aminimum hamming distance of 1. We calculated the averaged quantum bit error rateover all symbols to be QII = 0.078 with a standard deviation of ∆QII = 0.042 for IIconfiguration and QFF = 0.074 and ∆QFF = 0.029 for FF configuration. We assumeAlice and Bob set their threshold to detect eavesdropping to a bit error rate of Q + σ,where Q is the averaged quantum bit error rate. In this case Eve could only intercepta fraction η of the photons.4. Basis guess fidelityIn practical QKD, Alice’s basis choice could leak to an eavesdropper via side channels orimperfect encoding. To include this into the model, we added a guess fidelity of ε. Evecan not guess the basis if ε = 0, while ε = 1 means that Eve knows Alice’s basis choice.In our experiment, we measured ε ∼ 0.15 by a correlation measurement performed withclassical light. Eve can then extract the informationI(Eve) =η2(1 + ε) I(Alice) (2)from what Alice sends. Thereby she adds an additional error ofQEve =12(1 − ε) d− 1d. (3)To detect eavesdropping, Alice and Bob must set an error threshold σ. The error rateintroduced by Eve’s perturbation of the quantum channel has to be lower than thisthreshold to stay unnoticed. The quantum bit error rate including an eavesdropper isQTotal = (1 − η)Q+ η (Q+ (1 −Q)QEve) (4)= Q+ (1 −Q)ηQEve. (5)From the relationQTotal ≤ Q+ σ, (6)the maximum fraction of intercepted photons isηmax =σ(1 −Q)QEve, (7)Supplemental Material 3which depends on the fidelity to guess the correct basis ε and the threshold σ. Theminimum fidelity between Alice and Bob reduces from F toFTotal = F (1 − ηQEve) . (8)Now it is possible to calculate the distance of information between Bob and Eve, whichis a measure for the secure key rate. The information distance is defined asδ = IAB − I(Eve). (9)The amount of information Bob receives depends on the amount of information Alicetransmits and on the channel noise. ThereforeI(Bob) = QTotalI(Alice). (10)Combining equation 2 and 10, the information distance can be written asδ =[(1 −QTotal) −η2(1 + ε)]I(Alice). (11)By substituting ηmax from equation 7 to this expression, the minimum informationdistance can be plotted as a function of ε and σ in figure 1. Compared to equation (5)in the main article the prefactor n/N does not appear, since already a small fractionof the key is enough for parameter estimation. Moreover, the error correction andparameter estimation as well as the uncertainties about Eve’s entropies lower the secretfraction.Figure 1. The minimum secret information distance δ against the basis guess fidelityε for three different thresholds σ.The information distance δ grows monotonically with decreasing threshold σ, butbecomes smaller with increasing ε, as visible in Fig. 1. If Eve knows Alice’s basis choice(ε = 1), her measurements will no longer add noise, which allows here to interceptthe quantum communication without being detected. In comparison to the finite-key-length secret fraction in the case of collective attacks, the values the minimum secretinformation δ is larger.Supplemental Material 4[1] Morris P A, Aspden R S, Bell J E C, Boyd R W and Padgett M J 2015 Nature Communications 61–6 ISSN 2041-1723[2] Aspden, R S and Tasca, D S and Boyd, R W and Padgett, M J 2013 New Journal of Physics 15073032[3] Gisin N, Fasel S, Kraus B, Zbinden H and Ribordy G 2006 Phys. Rev. A 73 022320[4] Wang W, Tamaki K and Curty M 2018 New J. Phys. 20 083027[5] Gisin N, Ribordy G, Tittel W and Zbinden H 2002 Rev. Mod. Phys. 74 145[6] Scarani V, Bechmann-Pasquinucci H, Cerf N J, Dušek M, Lütkenhaus N and Peev M 2009 Rev.Mod. Phys. 81 1301[7] Nielsen M A and Chuang I 2002 Quantum computation and quantum information[8] Gray F 1953 Pulse code communication, us patent 2,632,058

Most quantum key distribution protocols using a two-dimensional basis, such as HV polarization as first proposed by Bennett and Brassard in 1984, are limited to a key generation density of 1 bit per photon. We increase this key density by encoding information in the transverse spatial displacement of the used photons. Employing this higher-dimensional Hilbert space together with modern single-photon-detecting cameras, we demonstrate a proof-of-principle large-alphabet quantum key distribution experiment with 1024 symbols and a shared information between sender and receiver of 7 bit per photon

Spatially encoded light for Large-alphabet Quantum Key Distribution

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University of Twente Research Information