Evaluation of the phase randomness of the light source in quantum key distribution systems with an attenuated laser
aa r X i v : . [ qu a n t - ph ] J u l Evaluation of the phase randomness of the light source inquantum key distribution systems with an attenuated laser
Toshiya Kobayashi, ∗ Akihisa Tomita, and Atsushi Okamoto
Graduate School of Information Science and Technology, Hokkaido UniversityKita 14, Nishi 9, Sapporo 060-0814, Japan
Abstract
The phase randomized light is one of the key assumptions in the security proof of Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol implemented with an attenuatedlaser. Though the assumption has been believed to be satisfied for conventional systems, it shouldbe reexamined for current high speed QKD systems. The phase correlation may be induced bythe overlap of the optical pulses, the interval of which decreases as the clock frequency. Thephase randomness was investigated experimentally by measuring the visibility of interference. Anasymmetric Mach-Zehnder interferometer was used to observe the interference between adjacentpulses from a gain-switched distributed feedback laser diode driven at 10 GHz. Low visibility wasobserved when the minimum drive current was set far below the threshold, while the interferenceemerged when the minimum drive current was close to the threshold. Theoretical evaluation on theimpact of the imperfect phase randomization provides target values for the visibility to guaranteethe phase randomness. The experimental and theoretical results show that secure implementationof decoy BB84 protocol is achievable even for the 10-GHz clock frequency, by using the laser diodeunder proper operating conditions. ∗ Present address: Seiko Epson Corporation . INTRODUCTION Quantum key distribution (QKD) offers an unconditionally secure method to share acryptographic key between remote parties. Bennett-Brassard 1984 (BB84) protocol [1] isone of the most developed QKD protocols, the security proof of which has been well es-tablished [2–5]. Recent researches on the security focus on more practical aspects, such asimperfections in a QKD apparatus. In actual QKD equipment, the device characteristicsdeviate from the ideal ones. Keeping the secure key rate with imperfect devices is an im-portant issue [6, 7]. Since a practical single photon source, assumed in the original BB84protocol, is still unavailable, most QKD experiments have utilized light pulses from a laserdiode (LD) after strong attenuation. The attenuated laser pulses contain two photons ormore with a finite probability. The multiple photon states opened the way to an efficienteavesdropping method called photon number splitting (PNS) attack [8]. Gottesman, Lo,L¨utkenhaus, and Preskill (GLLP [9]) analyzed the security against this imperfection. Animproved protocol called decoy-BB84 [10–12] was proposed to yield better secure key ratethan GLLP. The decoy-BB84 protocol provides not only unconditional security with theattenuated laser light, but also the universal composability [13–16].The strongly attenuated laser light is often called weak coherent light. This term ismisleading, because the security analysis in the GLLP and decoy-BB84 articles assumesthat the light source emits photons in a phase randomized Poissonian state, which is amixture of coherent states with uniformly distributed phases: ρ = 12 π Z π − π dφ | αe iφ ih αe iφ | = e −| α | ∞ X n =0 α n n ! | n ih n | . (1)The state is represented by a diagonal density matrix with respect to photon-number basis.Lo and Preskill [17] showed that, if the photon states were really weak coherent, discrimi-nation of the bases used in the BB84 protocol would be easier. Recently, Tang, et al. [18]showed that the phase information also increases distinguishability between decoy and sig-nal pulses used in the decoy-BB84 protocol. Those reports have issued a warning about thephase correlation among the laser pulses; the phase correlation will increase the informationleakage and thus reduce the secure key rate. Active phase randomization was proposed andimplemented by Zhao, et al. [19]. Effect of partially randomized phase was also examinedfor a plug-and-play system [20]. 2evertheless, most experimentalists have not taken this warning seriously with a fewexceptions [18–20]. Their common belief is that pulses from a gain-switched LD have nophase relationship to other pulses. Therefore, the phase of the light source is automaticallyrandomized, as long as the one-way QKD architecture is employed. The mechanism of thephase randomization is following. In the gain-switched mode, each current pulse excitesthe semiconductor medium from loss to gain. A laser pulse is generated from seed photonsoriginated from spontaneous emission, because the photons from the previous lasing havevanished during the pulse interval. The phase of the spontaneous emission is random, sothat the phase of the laser pulses should vary from one pulse to another. This is truewhen the previously lased photons disappear completely in the interval. However, if thephotons survive until the next excitation, the lasing can be seeded by the remaining photons.Then, the phase of the laser pulse may relate to the previous one, because the stimulatedemission conserves the phase. The effect of the residual photons will emerge significantlyby increasing the pulse repetition rate and narrowing the pulse interval. The state-of-artQKD systems operate at high clock frequencies over 1 GHz, along with the improvement ofthe photon detectors [21–23]. The clock frequency would further increase to meet demandsfor high bit-rate secure communication. The interval time thus decreases down to hundredspicoseconds or even shorter. Furthermore, the drive current may not return to zero inorder to improve modulation response of the laser. It is unclear whether the assumptionof the phase randomized source still holds in QKD systems operated at several GHz-clockfrequencies.In this article, we examine the phase randomness of the light source at 10-GHz clockfrequency. Sec. II A introduces an asymmetric interferometer set-up to measure the phasecorrelation between the adjacent optical pulses. We recall the relation of the phase correla-tion to the visibility of the interference fringe. Sec. II B considers the effects of the partialcoherence in state discrimination, which were analyzed by Lo-Preskill [17] and Tang et al.[18] for perfectly coherent states. We provide target values of the visibility, under which wecan regard the light source as phase randomized. Sec. III shows the measured visibility ofthe interference fringe of the adjacent pulses from a LD operated at 10-GHz clock frequency.We controlled DC bias current to the LD, which determines the effective pulse interval andthe minimum drive current. In sec. IV, we examine the accuracy of the estimated values ofvisibility, and applied corrections to the estimation. We investigate the relation between the3bserved phase correlation and the operating conditions, in terms of the effective photon lifetime of the LDs. II. THEORYA. Relation between visibility and phase correlation
The phase relation of laser light can be characterized with an interferometer. Figure1 illustrates a schematic of an asymmetric interferometer to observe interference betweenthe adjacent optical pulses. We focus on measuring the interference between the adjacentpulses, because the phases between the adjacent pulses are more correlated than those be-tween more temporally-separated pulses. Light pulses generated in the source enter theasymmetric interferometer, where the delay time is adjusted to the pulse period. The ad-jacent optical pulses are combined at the output. A phase modulator is placed in one armof the interferometer to provide a phase difference ϕ between the paths. The signals aredetected by a high-speed photodetector and accumulated by an averager. If a fixed phaserelation between the adjacent pulses exists, the amplitude of the signal takes a definite valueaccording to the phase difference between the optical paths. A clear interference fringe willbe observed as ϕ varies. If the phases between the pulses are random, the interference signaldiffers from pulse to pulse. Then the interference fringe will disappear after accumulation.Visibility of interference Θ, which represents the degree of the phase correlation, is definedby 0 ≤ Θ := I max − I min I max + I min ≤ , (2)where I max and I min stand for the peak (maximum) and the valley (minimum) of the in-terference fringe, respectively. As the phase correlation becomes stronger, the visibility getscloser to one.In the following, we recall a relation between the visibility and the phase correlation [24].The output intensity of the interferometer is given by I ( ϕ ) ∝ E (cid:8) | a A | + | a B | + | a A a B | exp [ i ( θ + ϕ )] + c . c . (cid:9) , (3)where the coefficient E = p ~ ω/ ǫ V carries the dimension of electric field. The complexamplitudes a A and a B represent the fields provided from paths A and B , respectively. In4he asymmetric interferometer, the fields a A and a B correspond to those of the adjacentpulses. The relative phase between the pulses is given by θ . The third and forth terms ofEq. (3) are responsible for the interference. Taking an average over an ensemble, we obtainthe interference terms as h a A a B i (cid:10) e iθ (cid:11) e iϕ + c . c ., (4)where we assume the phase difference ϕ varies slowly, while the relative phase θ is a prob-abilistic variable. Equation (4) shows that the interference visibility is governed by theexpectation value of the relative phase h e iθ i . If the distribution of the phase obeys a Gaus-sian probability density function with the central value θ and the standard deviation σg ( θ, θ ) = 1 √ πσ exp (cid:20) − ( θ − θ ) σ (cid:21) , (5)the expectation value is given by (cid:10) e iθ (cid:11) = Z ∞−∞ e iθ g ( θ, θ ) d ( θ )= exp (cid:20) − σ iθ (cid:21) . (6)The Gaussian probability distribution describes the phase distribution of the LD light well,because the light field vector (phaser) in the phase space is kicked by a number of photonsgenerated by spontaneous emission [25]. The kicks force the field vector to walk randomlyaround the original position. Since the spontaneous emission occurs independently, a numberof kicks results in the Gaussian distribution. Using Eqs. (2)-(6), we relate the visibility tothe standard deviation of the phase distribution asΘ = exp (cid:20) − σ (cid:21) . (7)Here, we assume | a A | = | a B | for simplicity. As expected, the visibility decreases rapidly withincreasing the standard deviation of the phase. The analysis given above uses the classicalcomplex amplitude of the electric field, because the laser field can be well approximated witha classical field, where the effect of the spontaneous emission is introduced by a random kick[24, 25]. The wave properties of the field will not be altered by attenuation. Furthermore,the analysis using quantum operators will provide the similar description.5 IG. 1. A schematic illustration of a measurement apparatus for the phase correlation between theadjacent pulses. The delay time of the asymmetric interferometer is adjusted to the pulse period.The phase difference ϕ between the paths can be modulated. B. Impact of phase correlation
We consider the impact of phase correlation to provide criteria to guarantee the securityof decoy-BB84 with a LD light source. As stated in the introduction, the phase correlationenhances the distinguishability of the states, which are expected to be indistinguishablein the ideal situation [17, 18]. The following calculation will treat two issues on the statediscrimination: one is between the states of different bases, and the other is between thesignal and decoy pulses.Most security proofs of BB84 rely on the assumption that the density matrix of one basisis indistinguishable to another. The distingushability of two density matrices, sometimescalled the imbalance of the quantum coin [9], helps the eavesdropper (Eve) to distinguishthe state encoding. GLLP [9] described the imbalance in terms of the fidelity between thedensity matrices, and analyzed its effects on the security. Though the imbalance of thequantum coin often refers to the state preparation flaws, Lo and Preskill [17] showed thatphase correlation also enhances the distingushability. The imbalance of the quantum coin∆ is given by ∆ = 1 − F ( ρ X , ρ Z )2 , (8)where the fidelity of the density matrices in X -coding and Z -coding is defined by F ( ρ X , ρ Z ) = Tr (cid:16) ρ / Z ρ X ρ / Z (cid:17) / . (9)Further, since Eve may exploit the channel loss, we should recalculate the imbalance to keep6ecurity as ∆ ′ = ∆ ηµ (10)for given transmittance of the channel η and the average photon number µ of the source.Recently, Tamaki, et al. [26] showed that GLLP analysis was too conservative, and proposeda loss-tolerant proof even with state preparation flaws. We here calculate ∆, because it stillprovides a comprehensive measure of the state distinguishability. We can obtain ∆ ′ from ∆by Eq. (10), if we follow the GLLP analysis. The density matrices of the partially phaserandomized coherent states are expressed by ρ Z = 12 Z (cid:0)(cid:12)(cid:12) √ µe iθ (cid:11) F (cid:10) √ µe iθ (cid:12)(cid:12) ⊗ | i S h | + | i F h | ⊗ (cid:12)(cid:12) √ µe iθ (cid:11) S (cid:10) √ µe iθ (cid:12)(cid:12)(cid:1) g ( θ, θ ) dθ = 12 e − µ X M,N µ ( M + N ) / e − ( M − N ) σ / e i ( M − N ) θ √ M ! N ! | M i F h N | ⊗ | i S h | + | i F h | ⊗ X M,N µ ( M + N ) / e − ( M − N ) σ / e i ( M − N ) θ √ M ! N ! | M i S h N | ! (11) ρ X = 12 Z (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)r µ e iθ (cid:29) F (cid:28)r µ e iθ (cid:12)(cid:12)(cid:12)(cid:12) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)r µ e iθ (cid:29) S (cid:28)r µ e iθ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)r µ e iθ (cid:29) F (cid:28)r µ e iθ (cid:12)(cid:12)(cid:12)(cid:12) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) − r µ e iθ (cid:29) S (cid:28) − r µ e iθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) g ( θ, dθ = 12 e − µ X M,N (cid:16) µ (cid:17) ( M + N ) / e − ( M − N ) σ / × X m,n − m − n p ( M − m )! m !( N − n )! n ! | M − m i F h N − n | ⊗ | m i S h n | , (12)where the subscripts F and S denote fast and slow components of the time-bin qubits. Afinite value of θ is assumed for Z coding, while it is set to zero for X coding. Since onlythe relative phase between the two coding affects the distinguishability, this setting will notlose generality. The factor exp[ − ( M − N ) σ /
2] decreases rapidly for large σ , and only the M = N terms survive. In this phase randomized limit, the density matrices (11) and (12)coincide with those of the mixture of M -photon number states after the state preparation.The imbalance of the quantum coin ∆ can be calculated with Eq. (8) and the densitymatrices (11) and (12). The two states ρ Z and ρ X are most distinguishable when the centralvalue of the phase θ = π , and least distinguishable when θ = 0. We calculated thefidelity numerically with the density matrices in the photon-number-state basis truncatedto a finite photon number N max . We changed the number of bases to check the accuracy7f the calculation. Since the average photon number is small, the results converged rapidlyat N max = 8. We thus set N max = 16 in the following calculation. Figure 2 shows thecalculated values of ∆ as a function of the standard deviation of the phase distribution σ for the average photon numbers µ of 0.09 and 0.01. The imbalance of the quantum coindecreases as the standard deviation. For a small standard deviation, that is, less phaserandomized, the imbalance of the quantum coin for θ = π is larger than that for θ = 0.The effect of the central phase difference vanishes for large standard deviation, as the phasesof the states become randomized. As seen in Fig. 2, ∆ converges to finite values for largestandard deviation. The relative errors from the asymptotic values fall below 10 − , whenthe standard deviation exceeds the following values: 2.9 for µ = 0 .
01 and θ = 0, 2.6 for µ = 0 .
01 and θ = π , 3.2 for µ = 0 .
09 and θ = 0, and 2.9 for µ = 0 .
09 and θ = π . Thevalues of standard deviation, 2.6, 2.9, and 3.2, correspond to the visibility of 0.034, 0.015,and 0.006, respectively, which are estimated with Eq. (7). Therefore, target visibility valueswould be 0.015 for µ = 0 .
01 and 0.006 for µ = 0 .
09 in terms of the imbalance of the quantumcoin. The asymptotic value of ∆ for µ = 0 .
09 is larger than that for µ = 0 .
01, due to themultiple photon contribution, which increases the distinguishability between the two states.Decoy method uses the states with different average photon numbers, called signal anddecoy. A key assumption of the decoy method is that Eve cannot distinguish the signalpulses from the decoy. In other words, Eve can measure the photon number contained in apulse, but cannot measure the average photon number of an individual pulse. Then Eve’sstrategy is limited to the one that depends on the photon number of the pulse. The securityproof of the decoy method only needs to consider such limited eavesdropping strategy. Tang,et al. [18] pointed out that phase correlation enables an unambiguous-state-discrimination(USD) measurement to distinguish the signal from the decoy. The final key generated bythe non-phase-randomized system can be compromised by combing the USD measurementand the PNS attack. When the phase is partially randomized, the USD measurement is nolonger possible. However, if Eve allows finite probability to obtain inconclusive results P inc ,she can still increase the probability of the correct decision P C [27]. In the appendix, wederive the optimum positive-operator valued measure (POVM) to discriminate the signalstate from the decoy state for partially coherent states. The results of the POVM enable toextend the analysis presented by Tang, et al. [18].8 -5 -4 -3 -2 =0.01 i m ba l an c e o f quan t u m c o i n Standard deviation of phase =0.09 -1 = FIG. 2. The imbalance of the quantum coin, defined by ∆ = (1 − F ( ρ Z , ρ X )) /
2, as a function ofstandard deviation of phase distribution. Solid lines represent ∆ for the difference of central phasevalue θ = 0, broken lines for θ = π . We investigate the fidelity between the signal and decoy states described by ρ = e − a X m,n a m + n e − ( m − n ) σ / e i ( m − n ) θ √ m ! n ! | m ih n | (13) ρ = e − a X m,n a m + n e − ( m − n ) σ / √ m ! n ! | m ih n | . (14)Following Tang, et al. [18], we consider only the fast component of the time-bin qubits,which carries no information on the key bit value. The density matrices ρ and ρ describepartially phase randomized coherent states with the average photon numbers µ = 2 a and ν = 2 a , respectively. We calculated the fidelity numerically by truncating the number ofbasis to a finite photon number N max = 16, as the imbalance of the quantum coin. Figure3 shows the distinguishability defined by (1 − F ( ρ , ρ )) / ρ and ρ .The distinguishability decreases as the phase randomization, and asymptotically reaches thevalue for the completely phase randomized states. The relative discrepancy between the twobecame less than 10 − for σ > .
5, which corresponds to the visibility of 0.044. This valuewould be a target visibility in terms of the signal-decoy discrimination. The fidelity was9 [ - F ( , ) ]/ Standard deviation of phase
Phase randomized stateCoherent state, = =0.5=0.1 FIG. 3. Distinguishability (1 − F ( ρ , ρ )) / θ = π . Dash-dot line stands for the distinguishability between the completely phase randomizedstates, and dash line between coherent states with θ = π . calculated for θ = π , because the coherent states with the relative phase θ = 0 yield thesame fidelity as the completely phase randomized states. However, the coherent states stillprovide an advantage to the eavesdropper to perform an individual attack as seen in theappendix.In this section, we have derived criteria of the phase randomness and thus the interfer-ence visibility. The target values depend on the average photon numbers. Moreover, theeavesdropping methods are not exhausted with those considered above, so that the targetvalues may be further lowered. Nevertheless, we believe that the present analysis covers awide range of eavesdropping, and the values estimated here should be good indications.10 II. EXPERIMENT
The phase correlation measurement system consists of the interferometer and the pulsedlight source to be tested. We employed the configuration similar to the one depicted in Fig.1. The source was a distributed feedback (DFB) LD (NEL, NLK5C5EBKA,) which wasdesigned for 10-GHz direct modulation to emit optical pulses in a single longitudinal andtransversal mode. It lases around the wavelength of 1560 nm at the threshold current of 9.5mA. The LD was driven by the combination of a 10-GHz sinusoidal current ( I AC ) and a DCbias current ( I DC .) The total current to the LD is expressed by I AC + I DC . The sinusoidalcurrent injected to the laser is expressed by I AC = I pp πf t + φ LD ) , (15)where I pp stands for the peak-to-peak value of the sinusoidal current. The current changesperiodically with the frequency f = 10 GHz and an initial phase φ LD . The sinusoidal signalfrom a pulse-pattern-generator (PPG) was amplified to a fixed amplitude V pp =4.615 V. Weestimated the peak-to-peak AC current to the LD as I pp =92.3 mA, considering the 50-Ωroad resistance. However, in high frequency region such as 10 GHz, the emerging effects ofparasitic impedances of the LD and the circuit may reduce the current injected into the LDactive layer. To correct this effect, we measured the modulation response of the LD witha network analyzer and a 45-GHz band-width photodetector. The resonant-like frequencywas about 12 GHz in this measurement, so that the intrinsic response of the LD affectslittle the modulation response up to 10 GHz. It was found that the optical power responseof the LD decreased by about 1 dB at 10 GHz from that at 100 MHz. Since the opticalpower is proportional to the injected current, we regard the reduction of the response as thedecrease of the current with the same proportion. Then the net current I net is reduced fromthe nominal value I nom by 10 log ( I net /I nom ) = −
1. The net AC current thus swung by I pp = 92 . × .
794 = 73 . − / ≃ . I DC .We define the minimum drive current defined by I min = − I pp / I DC , which refers to thedrive current at the bottom of the AC current. In the following, we use normalized excitationto describe the operating condition. The normalized minimum excitation is defined byΛ = I min − I th I th . (16)11 IG. 4. Waveforms of the LD pulses. The values of the normalized minimum excitation are asfollows: (a) Λ = − .
6, (b) Λ = 0 . .
49, and (d) Λ = 2 . As mentioned, the laser threshold current was I th = 9.5 mA. When Λ >
0, the LD wasalways turned on. When Λ <
0, the LD was turned off during the pulse interval. The turn-off duration increases as the DC bias current decreases, which is obtained as a solution of I AC + I DC = I th with Eq. (15). When Λ < −
1, the LD was reversely biased and no currentwas injected at the minimum. In the present experiment, Λ = 0 and Λ = − I DC =46.15 mA and I DC =36.75 mA, respectively.We employed a commercially available asymmetric Mach-Zehnder interferometer (AMZI)module (Kylia, WT-MINT-M-L) to obtain interference between the adjacent pulses at 10GHz, which was developed as a demodulator for 10-GHz differential phase shift keying(DPSK.) The phase difference between the optical paths was modulated with a phase shifterintegrated in the AMZI module. The signal was accumulated for 256 samples and measuredwith a sampling oscilloscope of 40-GHz optical band-width to observe the interference fringe.The output of the AMZI was attenuated by an optical attenuator to avoid saturation of thephotodetector. The peak and valley intensities of the accumulated interference fringes wererecorded. 12igure 4 shows the observed waveforms of light pulses for (a) Λ = − .
6, (b) Λ = 0 . .
49, and (d) Λ = 2 .
6, where the minimum drive current I min was (a) below thethreshold, (b) near the threshold, (c) above the threshold, and (d) far above the threshold.By setting the I min close to the threshold, sharp and intense pulses were obtained as shown inFig. 4 (b) and (c). When the I min was far above the threshold, the laser output reflected theinput current waveform as in Fig. 4 (d). The LD was no longer operated in the gain-switchedmode in this DC bias region.The observed interference fringes are shown in Fig. 5. Clear interference fringe wasobserved for a large excitation (Λ = 2 .
6) with the visibility close to unity (Θ = 0 . − . / (Averaged Power). Due to the limitation of the device, the range of phasemodulation was only slightly larger than 2 π . The phase difference of the interferometer wasstable enough for the short time to obtain an interference fringe. It was not stable for days,so that the origin of the phase difference varied as shown in Fig. 5(a)-(d.) IV. DISCUSSION
We consider the origins of errors to examine the accuracy of the results obtained in theexperiment. First, the imperfections in the interferometer, such as fluctuation of path length,imbalanced branch ratio of the beam splitters, polarization rotation, and depolarization willreduce the visibility. In fact, we obtained the visibility of only 0.95 with continuous wave(CW) light emitted from the LD excited solely by the DC current of 50 mA, which was farabove the threshold. The LD linewidth implies that the phase of the CW light should be wellconserved in the time scale of 100 ps. Therefore, we should consider the obtained visibilitywas affected mainly by the imperfections in the interferometer. Assuming the imperfectionsare the same throughout the experiment, we should correct the visibility by multiplying1.05.Second, the system noise affects the visibility estimation. In the present experiment,we recorded the observed maximum and minimum values, which included noise. Thus, the13
IG. 5. Interference fringes for several values of the normalized minimum excitation: (a) Λ = − . . .
49, and (d) Λ = 2 . estimated visibility should have been overestimated. This overestimation causes no harm,from the conservative points of view for the security certification. However, it is undesirablefor the practical use, because we may lose some amount of final key by unnecessary privacyamplification. The effect of the noise emerges significantly for small visibilities. To obtainbetter estimation, we examined the results showing low visibilities by magnifying the scale ofintensity, as shown in Fig. 6. The error bars originated mainly from the noise of the samplingoscilloscope. The observed signal-to-noise ratio was about 17 dB, where the average intensitywas normalized to 0.5. The r.m.s. value of the noise suggests that it may be hard to measurethe visibility less than 0.02. Nevertheless, a periodic dependence on the phase difference isseen in Fig. 6 (b). By taking the center values denoted by squares in Fig. 6, we could fitthe interference fringe with I ( ϕ ) = A (1 + Θ cos( ϕ + ϕ )) . (17)The result of the fitting is depicted as a thick solid line in Fig. 6. For excitation of Λ = − . I n t en s i t y [ a r b . un i t] Phase difference [ rad] -1 0 10.480.490.500.510.52 I n t en s i t y [ a r b . un i t] Phase difference [ rad]
FIG. 6. Magnified view of the interference fringe. (a) Λ = − . − .
2. Solid linesdenote the fitted curve. The fitted values of the visibility were 0.004 and 0.02. dependence on the phase difference. The fitted visibility was much less the one estimatedfrom the maximum and minimum intensities, 0.014. For (b), Λ = − .
2, the fitted valueof the visibility was 0.022, while the estimated one was 0.030. The fit was done well, aswe consider the 95 % confidence interval of the fitting value [0.019, 0.025]. Discrepancybetween the fitted and estimated visibility decreases as the visibility increases. On the basisof the above, we conclude that the present experimental set up can detect the visibilitydown to 0.02. The effect of the noise should be reduced by using low noise front-end and byincreasing number of accumulation to obtain lower measurement limit of the visibility.We applied the corrections discussed above. The results are summarized in Fig. 7, wherethe visibility is plotted as a function of the normalized minimum excitation. Figure 7 showsthe visibility increases as the minimum excitation. It raises steeply around Λ = 0, wherethe LD was always turned on. The interference fringe almost disappeared when Λ < − − . I DC =30.95mA,) the visibility was fitted to 0.004, which satisfied the strictest criterion given in sec.II B. Though the fitted value may not be accurate as described above, the visibility satisfiedthe target values 0.015 for the imbalanced coin at µ = 0 .
01, and 0.044 for the decoy statediscrimination. It should be noted that the interference fringe was observed even when theLD was turned off during the pulse interval. When the minimum drive current was set inthe range − < Λ <
0, the light source can be no longer regarded as a phase randomizedin terms of the imbalance of the quantum coin. For example, the visibility reached 0.08 for15 V i s i b ili t y (I min -I th )/I th FIG. 7. Visibility of the interference as a function of the normalized minimum excitation Λ. ForΛ >
0, the LD was always turned on. The LD was reversely biased at the bottom of the pulse,when Λ < − Λ = − .
76. If we care only about the laser waveform (as is common in most applications,) wemay set the bias to the value where the minimum drive current is close to the LD threshold,because it yields the best waveform as seen in Fig. 4 (b) and (c). Unfortunately, the phasesof the pulses are correlated under this operating condition. The observed visibility was0.534 for the case (c), where Λ = 0 . σ = 1 .
12. We need to scarify more bits to guarantee the security offinal key in the privacy amplification under this operating condition.In the following, we consider the dependence of the phase correlation on the minimumexcitation in terms of effective photon life time. As described before, if the photons surviveduring the pulse interval, the phase may correlate with the previous pulses. Typical photonlife time τ ph of a LD cavity is several picoseconds. The effective photon lifetime can beincreased by stimulated emission, even when the excitation is insufficient for lasing. Photondensity S in the cavity will decay approximately as dSdt = (cid:18) Γ g ( n ) − τ ph (cid:19) S + n sp , (18)16here Γ g ( n ) denotes the modal gain for the lasing mode at the carrier density n . The term n sp represents the contribution of the spontaneous emission to the lasing mode. The photonfield is governed by the spontaneous emission, when the photon density decreases to satisfy S ≤ n sp . Then, the phase of the light field become random.The details of the dynamics is described with involved nonlinear coupled equations onphoton density and carrier density. Roughly speaking, though, the photon field loses thephase information after the effective photon life time given by − (Γ g ( n ) − /τ ph ) − , as seen inEq. (18). Since (Γ g ( n ) − /τ ph ) τ ph equals approximately to ( I − I th ) /I th , the effective photonlifetime scales with the inverse of the normalized excitation. When the normalized minimumexcitation Λ exceeds zero, the effective photon life time becomes infinite to negative. Then,the photons of previous pulses remain to contribute the phase correlation. Even when Λis less than zero, the photons may survive during the interval and contribute to the nextlasing. For example, at Λ = − .
33, the effective photon the time life is about three timesas large as the cavity life time at the bottom of the pulse. If we take the cavity life timeas τ ph = 3 ps, the effective photon lifetime increases to 10 ps. This value is comparable tothe turn-off duration of 13 ps calculated from Eq.(15). Therefore, a non-negligible numberof photons are supposed to remain under this condition. In fact, the observed visibility was0.188, indicating some phase correlation. For small excitation satisfying Λ < −
1, the LD isreversely biased, and the effective photon life time should be equal to the cavity life timeat least in the bottom of excitation. The calculated turn-off duration is as long as 30 ps forΛ = − .
6. Under this condition, photons should have disappeared during the pulse interval,and the lasing phase became random, as was observed in the experiment.We see that observed dependence of the visibility on the excitation can be explained withthe relation between the effective photon life time and the turn-off duration. A guide ofthe operating condition can be summarized that the effective photon lifetime should be lessthan the turn-off duration. As described above, this condition is satisfied with Λ < − V. CONCLUSION
In BB84 protocol using an attenuated laser source, the secure key generation rate islowered if the source emits non-phase randomized optical pulses. We evaluated the effect ofthe phase correlation in terms of the imbalance of the quantum coin and the discrimination17f the decoy from the signal pulses, for the partially coherent states. We obtained criteriafor the source to be regarded as phase randomized. The target values for the visibilities were0.006 and 0.015 in terms of the imbalance of the quantum coin at µ = 0 .
09 and at µ = 0 . ν = 0 .
1) from the signal ( µ = 0 . ACKNOWLEDGMENTS
The authors would like to thank Dr. Kiyoshi Tamaki and Dr. Yoshihiro Nambu forhelpful discussions, and Mr. Takahisa Seki for his assistance in the experiment. This workhas been conducted under the commissioned research of National Institute of Informationand Communication Technology (NICT,) ”Secure photonic network technology.”
Appendix A: Optimal discrimination of two density matrices
We here derive the optimum discrimination of the decoy from the signal in partiallycoherent states, which were given in Eq. (14). We optimized the POVM for given valuesof P inc to obtain the highest P C in discriminating the two mixed states. All the densitymatrices here are real and symmetric. The POVMs Π i , i = 0 , , + Π + Π = 1 , (A1)where Π and Π correspond to the conclusive decision that the state is ρ and ρ , respec-tively, while Π represents the inconclusive results. The probability of inconclusive results18s given by P inc = Tr ( ρ Π ) = 1 − Tr ( ρ (Π + Π )) , (A2)and the probability to yield a correct decision is P C = p Tr ( ρ Π ) + (1 − p )Tr ( ρ Π )= 1 − P inc − p Tr ( ρ Π ) − (1 − p )Tr ( ρ Π ) , (A3)where ρ is defined with the probability of ρ ’s occurrence p by ρ = pρ + (1 − p ) ρ . (A4)We applied the iteration method developed by Fiur´aˇsek and Jeˇzek [27] to maximize P C (A3)under the constraints (A1) and (A2) for a given value of P inc using Lagrange multipliers.The iteration was performed with symmetrized equations to keep the POVMs Hermiteand positive semi-definite. We set average photon numbers for the signal and decoy state µ = 0 . ν = 0 .
1, respectively, and assumed the signal and decoy state appear withthe same probability ( p = 1 /
2) for simplicity. The optimization was done for two values ofthe probability of the inconclusive results: P inc = 0 . θ = 0, and P inc = 0 .
712 of θ = π . No USD measurement for θ = 0 exists to satisfy P inc = 0 . P C = 1) is achieved for the coherent states ( σ = 0.)The probabilities of correct decision decrease as the standard deviation of phase increasing.The probabilities converge to the values for completely phase randomized states, when σ exceeds about 2.5, which corresponds to the visibility of 0.044.We observed that the numerical optimization under the condition of P inc = 0 .
712 some-times failed, in particular for large σ . It also returned the probability of correct decisionlower than that for the phase randomized states. Small errors in matrix operations mayresult in such suboptimal results. Nevertheless, we found the optimized POVMs for large σ states were almost the same as those for the phase randomized states. The optimizedPOVM operators were almost diagonal in photon-number state basis. The outcome ρ wasobtained when the observed photon number was larger than a threshold, while the incon-clusive result was obtained when it was smaller than the threshold. The given probability19 P inc =0.983, =0 P inc =0.983, = P inc =0.983, phase random P inc =0.712, = P r obab ili t y o f c o rr e c t de c i s i on Standard deviation of phase P inc =0.712, phase random=0.5=0.1
FIG. 8. Probability of correct decision as a function of the standard deviation of phase. Statediscrimination was done for the partially phase randomized states of average photon number 0.5and 0.1. Squares represent P inc = 0 . , θ = 0, circles P inc = 0 . , θ = π , and triangles P inc = 0 . , θ = π . Dotted line and dashed line refer to the probability of correct decision onthe phase randomized states under the conditions of P inc = 0 .
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