Achieving ultimate noise tolerance in quantum communication
Frédéric Bouchard, Duncan England, Philip J. Bustard, Kate L. Fenwick, Ebrahim Karimi, Khabat Heshami, Benjamin Sussman
AAchieving ultimate noise tolerance in quantum communication
Fr´ed´eric Bouchard, Duncan England, Philip J. Bustard, Kate L. Fenwick,
1, 2 and Benjamin Sussman
1, 2 National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada Department of Physics, University of Ottawa, Advanced Research Complex,25 Templeton Street, Ottawa ON Canada, K1N 6N5
At the fundamental level, quantum communication is ultimately limited by noise. For instance,quantum signals cannot be ampliﬁed without the introduction of noise in the ampliﬁed states. Fur-thermore, photon loss reduces the signal-to-noise ratio, accentuating the eﬀect of noise. Thus, mostof the eﬀorts in quantum communications have been directed towards overcoming noise to achievelonger communication distances, larger secret key rates, or to operate in noisier environmental con-ditions. Here, we propose and experimentally demonstrate a platform for quantum communicationbased on ultrafast optical techniques. In particular, our scheme enables the experimental realizationof high-rates and quantum signal ﬁltering approaching a single spectro-temporal mode, resulting ina dramatic reduction in channel noise. By experimentally realizing a 1-ps optically induced tempo-ral gate, we show that ultrafast time ﬁltering can result in an improvement in noise tolerance by afactor of up to 1200 compared to a 2-ns electronic ﬁlter enabling daytime quantum key distributionor quantum communication in bright ﬁbers.
Since its inception in 1984 , quantum key distri-bution (QKD) has seen a myriad of major conceptualand technological developments. All of these eﬀortswere aimed at achieving quantum communication sys-tems with larger secret key rates , longer achievablecommunication distances [3, 4], or improved security con-siderations [5–7]. At the heart of all these requirements,noise represents one of the main limiting factors. Hence,overcoming noise in quantum communication is one of themost crucial requirements for QKD and, more generally,for entanglement distribution , quantum informationprocessing , and quantum sensing . So far, noisehas been avoided using dedicated ﬁbers (dark ﬁbers) andlimited night-time operation for free-space and satellite-based quantum communication [11, 12]. Provided thatthe quantum states are encoded in the photon’s polariza-tion, all other photonic degrees of freedom, i.e., position,momentum, frequency, and time, can be ﬁltered accord-ingly. In particular, to achieve optimal noise ﬁltering,the sender, typically referred to as
Alice , prepares pho-tons encoded in a single spatial, spectral, and temporal(SST) mode. By doing so, the receiver, typically referredto as
Bob , can avoid detecting noise residing outside ofthe single SST mode by applying appropriate ﬁlters. Intheory, single-mode ﬁltering is the ultimate way to de-crease noise without altering the signal, consequently in-creasing the signal-to-noise ratio directly. However, inpractice, eﬃcient ﬁltering of single photons to a singleSST mode is challenging and has not been demonstratedin the context of quantum communications.In this paper, we experimentally demonstrate ultimatenoise tolerance in quantum communication by realizingan ultrafast temporal ﬁlter that allows the detection ofquantum signals in a nearly single SST mode. The ul-trafast temporal ﬁltering scheme proposed in this workis based on cross-phase modulation via the optical Kerr eﬀect in a single-mode ﬁber (SMF). A bright pump pulseinduces a birefringence in the SMF causing a polarizationrotation of the quantum signal. By placing the SMF be-tween crossed polarizers, the pumped SMF acts as an ul-trafast switch [13, 14] for terahertz-bandwidth photons.As a result, we demonstrate a tremendous increase innoise tolerance without signiﬁcantly altering the quan-tum signal. Our ﬁndings are of particular interest forquantum communication systems operating in extremelynoisy conditions, e.g., the operation of QKD coexistingwith bright classical optical signals , the operationof free-space links in daylight conditions [16–18] or evensatellite-based QKD . For instance, reducing noise by3 orders of magnitude is suﬃcient to allow a QKD systemto transition from full-moon-clear-night to clear-daytimeconditions .Spectral ﬁltering is commonly employed in QKD us-ing interference ﬁlters with bandwidths on the order of afew nanometers. Other techniques may achieve narrowerbandwidths, but typically present certain limitations. Onthe other hand, time ﬁltering can also be jointly employedwith the help of fast detectors and time-tagging devices,where gating times are limited by the timing jitter ofthe single-photon detectors, typically on the order ofnanoseconds, and other electronic devices involved. Withthe recent development of superconducting nanowire sin-gle photon detectors, a signiﬁcant improvement in timingjitters can be achieved, i.e. as low as 50 ps . Unfor-tunately, they require cryogenic cooling (approximately1 K). Finally, despite time gating being an eﬀective wayto limit the amount of noise introduced in the detectionevent, noise counts may have another undesired eﬀect,i.e., to limit the overall signal detection rate due to de-tector dead time and detector saturation. Hence, tempo-ral ﬁltering schemes prior to the single-photon detectorscan be highly desirable to increase overall performanceof QKD in noisy environments.Pulses with minimum time-bandwidth product are a r X i v : . [ qu a n t - ph ] F e b Temporal noiseQuantum signalTemporal profile !!!
Spectral noiseQuantum signalSpectral profile T e m po r a l f ilt e r i ng T e m po r a l a nd s p ec t r a l f ilt e r i ng Quantum signal Temporal noise Quantum signal Spectral noiseQuantum signal Quantum signalTemporal noise Spectral noise N o i s y s i gn a l III ttt
Conceptual scheme of ultrafast temporal ﬁlter-ing.
Quantum signals prepared in a single spectrotemporalmode are found in the presence of spectral and temporal noise.As a ﬁrst step, an ultrafast temporal ﬁlter matching the du-ration of the signal pulse is applied. By doing so, the noise islargely reduced and the quantum signal left untouched. Thebeam is then sent through a spectral ﬁlter that is matched tothe bandwidth of the quantum signal, further increasing thesignal-to-noise ratio. known as Fourier-transform-limited (FTL) and are foundto occupy a single spectrotemporal mode. For instance,a FTL pulse with a spectral bandwidth of 1 nm at acenter wavelength of 800 nm has a pulse duration of ap-proximately 1 ps, which is much lower than the timingjitter of a standard single-photon detector, i.e., approxi-mately 1 ns. Thus, for most QKD experiments reportedto date, an improvement factor of up to 1000 can be ex-pected by using an ultrafast temporal ﬁlter (UTF). Onthe other hand, a 1-ns FTL pulse at a center wavelengthof 800 nm has a spectral bandwidth of approximately1 pm, where highly eﬃcient spectral ﬁltering remainschallenging [21–23]. By employing the optimal combina-tion of modest spectral ﬁltering and ultrafast temporalﬁltering, we achieve ultimate noise tolerance; see Fig. 1.Fibre networks will play a fundamental role in the de-ployment of quantum networks. However, due to theirinevitably lossy nature, their reach has been limited toa few hundred kilometers. Thus, in order to completethe quantum network, satellite-based quantum commu-nications and quantum repeaters will bridge the gap toachieve a quantum network by reaching communicationdistances spanning the entire globe [24–26]. Very re-cently, satellite-based quantum communication enteredthe realm of reality [12, 27–29]. In particular, satellite-based quantum communication represents an instancewhere managing noise is crucial due to the unavoidabledetection of background photons from scattered sunlight.Previous experiments of free-space QKD in daylight con-ditions [16, 17, 30, 31] demonstrated the feasibility of daylight QKD by applying spatial, spectral, and tempo-ral ﬁlters. However, no experiment has so far demon-strated optimal ﬁltering of the spectrotemporal degreeof freedom. By preparing and measuring the exchangedphotons in a single SST mode, we expect a signiﬁcantimprovement in noise tolerance compared to previous re-sults.
To demonstrate the feasibility of our scheme in quan-tum communication, we perform a proof-of-principle ex-periment using our UTF in a polarization-based decoystate BB84 protocol where noise is intentionally intro-duced in the quantum channel to test the noise tol-erance of our technique, see Fig. 2-(a). Weak coher-ent pulses (WCPs) are prepared by attenuating shortpulses at a center wavelength of λ signal = 720 . λ signal = 1 . f rep = 80 MHz. Using a polarizing beam split-ter (PBS) and a half-wave plate (HWP), the polarizationstate of the WCP is prepared at Alice’s stage. Accord-ing to the standard polarization BB84 protocol, Alicerandomly prepares the WCP in one of four polarizationstates, i.e., horizontal, vertical, diagonal, and antidiago-nal. The WCP are then attenuated to the single-photonlevel by using a neutral density ﬁlter placed after the po-larization encoding stage and a combination of a HWPand a PBS prior to the polarization encoding stage. Us-ing the HWP, the mean photon number of Alice’s pre-pared WCP is set to µ , ν , and 0, manually, correspond-ing, respectively, to the signal, decoy, and vacuum pulsesfor the decoy-state protocol . The values of µ and ν are varied for diﬀerent values of channel noise and loss,and subsequently selected oﬄine to optimize the overallsecret key rate for each channel condition.The WCPs are then sent through the quantum chan-nel, where a 10:90 (reﬂection:transmission) beam splitteris used to introduce channel noise. The noise is producedby a second Ti:Sa laser operated in cw mode with a cen-ter wavelength and linewidth of λ CW = 720 . λ CW = 0 .
83 nm. The amount of noise introduced inthe channel is controlled using a combination of HWP,PBS, and neutral density ﬁlter. The polarization of thenoise source is randomly varied using an additional HWP.The incoming photons are then sent through Bob’s de-tection stage, where the polarization is analyzed using aHWP and PBS, and then through the ultrafast tempo-ral ﬁlter. Using a dichroic mirror and a variable delaystage, the signal is made to overlap with a synchronizedpump pulse at a center wavelength of λ pump = 800 nm, Ti:Sa 800 nm OPO 720 nm
PBS λ /2 Isolator SF DM10 cm SMFPDAPDSMF SF GL PBS 10:90 BS λ /2 λ /2 PBS
Bob Alice ND Temporal filter Noise
Coincidence VDDM λ /2 λ /4 (a) (b) (c) Ti:Sa 720 nm ( ps ) S w it c h i ng e ff i c i e n c y
716 718 720 722 724 7260.00.20.40.60.8 Wavelength ( nm ) T r a n s m i ss i on λ /2 λ /2 FIG. 2.
Experimental setup. (a) Experimental setup demonstrating the use of the ultrafast temporal ﬁlter in a quantumcommunication demonstration. Ti:Sa, titanium sapphire laser; λ /2, half-wave plate; λ /4, quarter-wave plate; PBS, polarizingbeam splitter; OPO, optical parametric oscillator; ND, neutral density ﬁlter; BS, beam splitter; DM, dichroic mirror; SF,spectral ﬁlters; VD, variable delay; SMF, single-mode ﬁbre; PD, photodiode; GL, glan-laser polarizer; APD, avalanche photo-diode detectors. (b) Switching eﬃciency as a function of the pump delay demonstrating the temporal proﬁle of the UTF. (c)Spectrum of signal photons after spectral ﬁltering at the receiver’s stage. by coupling both into a 10-cm-long SMF with couplingeﬃciencies of 50 % and 65 %, for the signal and the pump,respectively. In the laboratory, femtosecond-scale timingprecision is guaranteed by the synchronous nature of theWCP and pump pulse. In remote applications, poor syn-chronization will result in temporal jitter. The eﬀect ofthis jitter of the UTF is explored in detail in AppendixF. The switching eﬃciency, η , of the quantum signal isgiven by η = sin (2 θ ) sin (cid:18) ∆ φ (cid:19) , (1)where θ is the angle between the polarization of the sig-nal and the pump, ∆ φ = 8 πn L eﬀ I pump / λ WCP is thenonlinear phase shift induced by the pump in the SMF, n is the nonlinear refractive index of the SMF, L eﬀ is theeﬀective length of the medium, and I pump is the intensityof the pump pulse . Maximal switching eﬃciency isobserved when θ = π/ φ = π . Thus, the polariza-tion of the pump is prepared to be at an angle of 45 ◦ tothe polarization of the quantum signal at the input of theSMF. We note that the UTF is positioned after Bob’s po-larization analysis stage resulting in a ﬁxed signal inputpolarization state. In order to imprint a uniform nonlin-ear phase shift across the WCP, we take advantage of thediﬀerence in group velocity between the quantum signaland the pump pulse inside the SMF. In particular, wecarefully select the pulse duration of the pump and thelength of the ﬁber to allow the pump pulse to fully walkthrough the WCP within the length of the SMF. This isachieved by spectrally ﬁltering the pump with a pair ofangle-tuned bandpass ﬁlters such that ∆ λ pump = 2 . t coinc = 2 . f rep ∆ t coinc = 0 .
16. Note that for more sophis-ticated detectors with improved jitter, the coincidencewindow could be reduced, and the ETF would improveaccordingly. Finally, to assess the temporal response ofour UTF, we scan the variable delay stage varying thetime of arrival of the pump with respect to the signal,see Fig. 2-(b). The switching eﬃciency at optimal de-lay is 99 ± t switch = 0 . ± .
01 ps, so is well matched to the dura-tion of the WCPs. To conﬁrm our experimental results,we simulate the temporal proﬁle of our UTF and obtaina good agreement with the experimental results; see Ap-pendix C.
To assess the feasibility of our temporal ﬁlteringscheme in ultrafast quantum communication, we performa proof-of-principle QKD demonstration, where the ﬁg-ure of merit is given by the secret key rate. In particular,we investigate diﬀerent channel conditions in terms ofnoise and loss to demonstrate diﬀerent regimes where ul-
ETFUTF ( Hz ) L o ss t h r e s ho l d ( d B ) ( Hz ) D i s t a n ce i m p r ov e m e n t f ac t o r N = × Hz N = × Hz N = × Hz N = × Hz Loss ( dB ) S ec r e t k e y r a t e ( bp s ) Loss = = = = Noise counts ( Hz ) S ec r e t k e y r a t e ( bp s ) ( dB ) N o i s e t o l e r a n ce f ac t o r ETFUTF Loss ( dB ) N o i s e t h r e s ho l d ( H z ) (a) (b) ETF UTF (c)
ETF UTF (d) (e) (f) (g) N = × Hz N = × Hz N = × Hz N = × Hz Loss ( dB ) S ec r e t k e y r a t e ( bp s ) FIG. 3.
Secret key rates . (a) Secret key rates as a function of noise counts are reported for diﬀerent channel loss values,5 , , ,
20 dB. (b) The noise threshold, which is deﬁned as the largest amount of noise for which the mean secret key rate islarger than zero by at least one standard deviation, is shown for diﬀerent values of channel loss. (c) The noise improvementfactor corresponding to the ratio of the noise threshold for the case of UTF over ETF is shown for diﬀerent values of channelloss. The secret rates as a function of channel loss are reported for (d) ETF and for (e) UTF for diﬀerent channel noise values, N . (f) The loss threshold, which is deﬁned as the largest amount of loss for which the mean secret key rate is larger thanzero by at least one standard deviation, is shown for diﬀerent values of noise counts. (g) The distance improvement factorcorresponding to the ratio of the loss threshold for the case of UTF over ETF is shown for diﬀerent values of noise counts. Theshaded areas correspond to channel conditions where an improvement of UTF over ETF is observed. trafast quantum communication can oﬀer a considerableadvantage over electronic QKD settings. The maximumeﬀective secret key rate R that could be achieved withthis apparatus is calculated using the standard decoyBB84 postprocessing procedure . We use the follow-ing formula for key generation: R ≥ q ( − Q µ f ( E µ ) H ( E µ ) + Q [1 − H ( e )]) , (2)where q = 1 / Q µ is the gainof signal states, f ( x ) is the error-correction eﬃciency, H ( x ) = − x log ( x ) − (1 − x ) log (1 − x ) is the binaryShannon entropy function, E µ is the quantum bit er-ror rate (QBER), Q is the gain of single-photon states,and e is the error rate of single-photon states. Theexperimentally measured gains and QBER, i.e. Q µ , Q ν , E µ , and E ν , of optimized mean photon numbers µ and ν for the signal and decoy states, respectively, areshown in Appendix B for diﬀerent channel conditions.We use the standard error-correction eﬃciency factor of f ( E µ ) = 1 . R , areshown in Fig. 3-(a) comparing the case of ETF and UTFas a function of channel noise, N , for diﬀerent values ofchannel loss. As can be seen, secret key rates can beachieved in a noise regime that is several orders of mag-nitude larger when operating with UTF (solid curves)compared to ETF (dashed curves). In particular, wecompare the noise threshold, i.e., the largest amount of noise for which the mean secret key rate is larger thanzero by at least one standard deviation, for both tem-poral ﬁltering schemes as a function of channel loss, seeFig. 3-(b). The noise tolerance factor, i.e., ratio of noisethresholds for the UTF to the ETF, is shown in Fig. 3-(c)as a function of channel loss, where a noise improvementfactor in excess of 1200 can be obtained, which agreeswith values obtained from our simulation; see AppendixC. Our results can also be presented in the context of asecond scenario where a ﬁxed amount of noise is presentin the quantum channel, but diﬀerent loss conditions areinvestigated. The secret key rates are shown as a func-tion of channel loss, see Fig. 3-(d) and (e) for variousnoise counts, N . The maximal achievable channel losscan be assessed by considering the loss threshold, seeFig. 3-(f). An improvement in communication distance,i.e., distance improvement factor >
1, occurs for noisecounts starting from 2 . × Hz, see Fig. 3-(g). More-over, a maximal distance improvement factor is achievedat a channel noise of 8 . × Hz with an improvementfactor of 4.2.
IV. DISCUSSION AND OUTLOOK
In our experiment, unit switching eﬃciency is achievedby setting the pump pulse energy to 2 .
47 nJ. At this pulseenergy, 1 . × − noise counts per pulse originating fromthe pump are detected in the single SST mode dedicatedto the quantum signal. In particular, the pump generatesparasitic nonlinearities such as self-phase modulation andtwo-photon absorption. These nonlinear processes maycreate noise photons covering the spectral range of in-terest for the quantum signals. These noise photons areinsigniﬁcant in noisy environments where we expect thisapproach to be most applicable, but become apparentin the low-noise high-loss regime leading to a maximumloss threshold of 21 dB. In Fig. 3, the shaded areas rep-resent conditions where QKD cannot work with ETF,but will with UTF. To extend the advantage region ofUTF, for instance beyond 21 dB of loss, diﬀerent av-enues can be employed to mitigate the eﬀect of pumpnoise, e.g., pump pulse and SMF engineering. Hence,the reported pump noise is not a fundamental limita-tion of our scheme and we expect that lower values ofpump noise can be envisaged with further design eﬀorts.Finally, in our experiment, the signal wavelength is setto λ signal ∼
720 nm. This choice is motivated by theavailable laser source and the desire for low pump noise.Nevertheless, other wavelengths might be of particularinterest, e.g., 1310 and 1550 nm. Our proposed UTFscheme can also be applied to the highly desirable tele-com wavelengths provided a proper design of the ﬁberand pump either using our polarization rotation switchor other switching schemes . Finally, let us compareour scheme involving single SST modes to other noise-tolerant QKD schemes. For the case of high-dimensionalQKD protocols , the advantage in noise tolerance isobserved when a full high-dimensional analysis of mul-timode quantum signals is carried out compared to a“coarse-grained” two-dimensional analysis of the multi-mode signal . Nevertheless, ultimate noise tolerancewill be achieved when the signal is encoded and mea-sured in a single SST mode, preventing noise photonsfrom even entering the measurement apparatus in modesother than that used to communicate quantum signals.In conclusion, we experimentally demonstrate quan-tum communication with quantum signals in a single SSTmode. To show the beneﬁts of our scheme, we have inves-tigate the improvements in terms of noise tolerance anddistance improvement for the case of a standard polariza-tion decoy state BB84 QKD protocol for a wide range ofchannel conditions. We note that our scheme can also begeneralized to other degrees of freedom, see Appendix E.In particular, we observe a noise-tolerance improvementfactor in excess of 1200 and an improvement in distanceby a factor of 4.2 by employing our proposed ultrafast ﬁl-tering scheme compared to electronic ﬁltering. By doingso, we take advantage of ultrashort pulses in the contextof quantum communication. These results can bring day-light free-space quantum communication a step closer toreality. We further expect new features to emanate fromjoining the ﬁelds of ultrafast optics and quantum com-munication.
This work is supported by the National ResearchCouncil’s High Throughput Secure Networks challengeprogram, the Joint Centre for Extreme Photonics,Canada Research Chairs, and Canada First ExcellenceResearch Fund. The authors thank Rune Lausten, De-nis Guay, and Doug Moﬀatt for support and insightfuldiscussions.
APPENDIX A: METHODS
A Ti:sapphire laser at a center wavelength of λ pump =800 nm is used to pump an optical parametric oscil-lator (OPO) at a repetition rate of 80 MHz. A sig-nal beam at a center wavelength of λ signal = 720 . λ OPO = 1441 . λ pump = 2 . λ CW = 720 . λ CW = 0 .
83 nm. Noise and loss are controlled usinghalf-wave plates mounted on motorized rotation stagesfollowed with polarizing beam splitters. The total loss ofthe quantum key distribution channel is varied between12.3 and 40.3 dB, consisting of the channel attenuation, t , varied using a HWP and PBS within a range of 2 to30 dB. Receiving and detecting losses are given by 3.0-dBcoupling loss at the SMF, 0.3-dB loss at the spectral ﬁl-ters, 5.5-dB loss for other optical elements at Bob’s stage,and ﬁnally 1.5-dB loss due to the eﬃciency of the APD.All three beams are coupled to a 10-cm-long single-modeﬁber (SMF) (Thorlabs S630-HP, with FC/PC connec-tors). The pump and the signal beams are coupled intothe ﬁber with a coupling eﬃciency of 65 % and 50 %, re-spectively. An optical isolator is introduced in the pathof the pump pulse prior to the single-mode ﬁber to limitback reﬂection of the bright pump pulse into the oscil-lator. After the SMF, the pump and the signal are sep-arated using a dichroic mirror, where the pump is mea-sured using a fast photodiode to serve as a trigger for theelectronic time gating of the quantum signal measuredusing an avalanche photodiode. A Glan-laser polarizer isthen placed in the path of the quantum signal and setto project onto a polarization state that is orthogonal tothe polarization of the signal at the input of the SMF,where a HWP and a QWP are used to compensate forpolarization rotations induced by the SMF and the DM.A pair of bandpass ﬁlters are then used to ﬁlter out thesignal and noise with a bandwidth of ∆ λ signal = 1 . >
70 dB each) and the shortpass ﬁlter( >
70 dB) provide a total of >
210 dB rejection at thepump wavelength. Finally, the quantum signals are thencoupled to a SMF with a coupling eﬃciency of 70 %, anddetected by an APD. The detected signals from the APDare sent into a time-to-digital converter (TDC) (SwabianInstrument, Time Tagger Ultra) for analysis. We notethat the TDC has a timing jitter of 22-ps FWHM. Sincethe TDC has a maximum data-transfer rate of 65 MHz,which is lower than the repetition rate of the pump, weuse a conditional ﬁlter where all clicks from the fast pho-todiode that are not followed by a detection event fromthe APD are discarded directly at the TDC. The coin-cidence time window is dictated by the timing jitter ofthe APD and is set to ∆ t coinc = 2 ns. The switching ef-ﬁciency is measured as a function of pump pulse energy,see blue curve in Fig. 4, following a quadratic sinusoidalrelation. A switching eﬃciency of 99 ± N o i s e ( - c oun t s / pu l s e ) ( nJ ) S w it c h i ng e ff i c i e n c y FIG. 4.
Eﬃciency and noise characteristics of the ul-trafast temporal ﬁlter . Switching eﬃciency (blue dashedcurve) and noise counts due to the pump pulse (red solidcurve) as a function of the pump pulse energy measured atthe output of the SMF. The switching eﬃciency follows aquadratic sinusoidal relation and the noise curve follows aquadratic relation. The quadratic scaling is typical of third-order nonlinear processes at the origin of the noise countsfrom the pump.
ETFUTFPump noise - - - - - Noise counts ( Hz ) Y FIG. 5.
Background rate . Background rate per pulse, Y , as a function of noise counts that are introduced in thequantum channel by the cw beam. In the case of the UTF,the background rate Y is lower bounded by the pump noisecounts, i.e., 2 . × − , coming from parasitic nonlinear pro-cesses in the SMF. APPENDIX B: EXPERIMENTAL QKDPARAMETERS
The lower bound for the single-photon gain Q andthe upper bound for the single-photon error rate e are,respectively, given by: Q ≥ µ e − µ µν − ν (cid:18) Q ν e ν − Q µ e µ ν µ − µ − ν µ Y (cid:19) , (3) e ≤ E ν Q ν e ν − Y / Q ν/ ( µe − µ ) , (4)where Q is the gain of single-photon states, e is theerror rate of single-photon states, Y is the backgroundrate per pulse, µ and ν are the signal- and decoy-statemean photon numbers, Q µ and Q ν are the signal- anddecoy-state gains, and E µ and E ν are the signal- anddecoy-state quantum bit error rates, see Fig. 6 and Fig. 7.Optimal mean photon numbers µ and ν are set to 0.6 and0.3, respectively.For electronic time ﬁltering, Y is mainly a result ofthe dark counts of the APD, p d = 100 Hz, and the noiseintroduced in the channel. In the case of the ultrafasttime ﬁltering, Y is also a result of noise counts from thepump due to nonlinear processes and the noise photonsintroduced in the channel, see Fig. 5. APPENDIX C: SIMULATION OF THEULTRAFAST SWITCH
We conﬁrm our experimental results by performing asimulation of the ultrafast temporal switch. The intrinsicswitching proﬁle, η , is given by η = sin (2 θ ) sin (cid:18) ∆ φ (cid:19) , (5) (a) (b) Loss = = = = - - - Noise counts ( Hz ) Q (cid:1) Loss = = = = - - - Noise counts ( Hz ) Q (cid:1) FIG. 6.
Gain . Gain for (a) signal pulses µ and (b) decoy pulses ν as a function of noise counts are reported for diﬀerentchannel loss values, t = 5 , , ,
20 dB, respectively blue circles, orange squares, green triangles and red diamond, for the caseof UTF (solid curves and empty markers) and ETF (dashed curves and full markers), respectively on the right and the left.Noise counts correspond to counts detected at the APD within the time window of ∆ t coinc = 1 ns. Loss = = = = ( Hz ) E (cid:1) (a) (b) Loss = = = = ( Hz ) E (cid:1) FIG. 7.
Error rate . Error rate for (a) signal pulses µ and (b) decoy pulses ν as a function of noise counts are reported fordiﬀerent channel loss values, t = 5 , , ,
20 dB, respectively, blue circles, orange squares, green triangles, and red diamond,for the case of UTF (solid curves and empty markers) and ETF (dashed curves and full markers), respectively, on the rightand the left. Noise counts correspond to counts detected at the APD within the time window of ∆ t coinc = 1 ns. where θ deﬁnes the angle between the pump and signalbeam polarizations. Maximal switching occurs when thepump beam polarization is oriented 45 ◦ relative to thesignal beam polarization, so we operate under this condi-tion. Considering the group-velocity mismatch betweenthe pump and the signal, the time-dependent nonlinearphase shift, ∆ φ ( T ), is then given by∆ φ ( T ) = 8 πn λ signal (cid:90) L I pump ( T − d w z ) dz, (6)where n is the nonlinear refractive index of the ﬁber, λ signal is the wavelength of the signal light, and z isthe propagation distance within the ﬁber of length L .The temporal walkoﬀ per unit length experienced by thepump and signal is given by d w = v − gp − v − gs , where v gp and v gs are the pump and signal group velocities, respec-tively. For this reason, we express the intensity proﬁle ofthe pump beam, I pump , in the frame moving with the sig-nal, i.e., T = t − z/v gs , where t is time in the laboratoryframe. Given a set of experimental parameters, i.e. thepump pulse energy (2.47 nJ), center wavelength ( λ pump =800 nm), spectral bandwidth (∆ λ pump = 2 . λ signal = 720 . λ signal = 720 . λ signal = 1 . - - ( ps ) S w it c h i ng t r ace - - ( ps ) I ( t ) / I (a) (b) - Gauss mode order T r a n s m i ss i on - Gauss mode order T r a n s m i ss i on (c) (d) Temporal + Spectral filtering Spectral filtering
Numerical simulation of the ultrafast switch . (a) Simulated switching response for the following input experimen-tal parameters: pump pulse energy (2.47 nJ), center wavelength ( λ pump = 800 nm), spectral bandwidth (∆ λ pump = 2 . λ signal = 720 . λ signal = 1 . explained by the spectral proﬁle of the cw noise photonsand the extra loss removed from the analysis of the ETF.In our experiment, a cw source with ﬁnite linewidth isused as the source of noise. In particular, the measurednoise linewidth is given by ∆ λ noise = 0 .
83 nm, whereasthe spectral ﬁltering is given by ∆ λ ﬁlt = 1 . λ SwitchedCW = 1 .
54 nm. By consideringa cw beam with a narrow linewidth, i.e., << ∆ λ signal , anoise improvement factor of 2360 is expected, matchingour experimental results. APPENDIX D: MEASURING A SINGLESPECTROTEMPORAL MODE
The beneﬁt of our technique lies in the fact that singlespatial and spectrotemporal modes are employed to en-code and decode information. By doing so, noise can onlyenter our measurement apparatus through that single op-tical mode. However, in our experiment we do not ex-actly perform ﬁltering of a single spectrotemporal mode.Indeed, by using a sequence of ﬁltering in the temporalthen spectral domain, we can achieve a ﬁltering down to a very small number of optical modes, see Fig. 8-(c),(d).However, even though a larger signal-to-noise ratio wouldbe achieved by projecting on a single spectrotemporalmode, the currently available techniques to do so stillpossess modest eﬃciencies. Thus by sequentially ﬁlteringboth temporally and spectrally, we can achieve ultimatenoise tolerance by adjusting the level of ﬁltering to op-timize both the signal-to-noise ratio as well as the totaleﬃciency. In order to assess the single-mode characterof our ﬁltering technique, we numerically calculate thetransmission of diﬀerent spectrotemporal modes throughthe sequence of ultrafast temporal and spectral ﬁltering,see Fig. 8-(c), compared to only spectral ﬁltering, seeFig. 8-(d). For Hermite-Gauss modes with mode orderlarger than 4, the transmission for the case of temporaland spectral ﬁlter is already lower than 0.7 %. However,for spectral ﬁltering alone, we see a signiﬁcant contribu-tion to noise due to higher-order modes.
APPENDIX E: IMPLEMENTATION TO OTHERDEGREES OF FREEDOM
We demonstrate the performance of our UTF schemein a polarization-based QKD proof-of-principle experi-ment. However, our technique is not limited to this spe-ciﬁc degree of freedom, i.e. polarization. Other degreesof freedom, such as time bins, phase coding, and spa-tial modes, can be advantageous for various experimen-tal implementations. Hence, achieving ultimate noise re-sistance in quantum communication systems employingthese various degrees of freedom can be critical. Here,we brieﬂy describe how our UTF technique can be gen-eralized to other degrees of freedom.
A common degree of freedom employed in QKD is thetemporal degree of freedom with time-bin encoding. Typ-ically, a signal is prepared as a train of pulses separatedby a time determined by the temporal resolution of thedetectors (timing jitter) or the speed of the source mod-ulators. The information can either be encoded in thetime of arrival (computational basis) or by the relativephase between pulses, which is known as phase codingand will be described in the following section. In thecomputational basis, time-bin encoding is advantageoussince it can propagate through a ﬁber channel with min-imal disturbance and can be straightforwardly detectedwith a single-photon detector. Moreover, the numberof time bins can be extended beyond the standard two-dimensional conﬁguration where larger information ca-pacity could be achieved and the UTF could still be em-ployed. However, for noise considerations, lower dimen-sions result in larger noise tolerance. Hence, the timeseparation between the two time-bins may be selected atwill to match experimental requirements, e.g., nanosec-onds time separation, but the time bin pulses themselvesmust be in near single spectrotemporal modes, as de-scribed in the main text. By doing so, only two opticalmodes are involved in the generation and detection of thesignals dramatically reducing the eﬀect of noise.The ultrafast temporal ﬁlter can then be applied tothe time-bin state by shaping the pump pulse to matchthe pulse shape of the signal, i.e. a train of pump pulsesaligned with the train of signal pulses. The pump willthen switch only the two optical modes occupied by thesignal. After going through the UTF, the signal is de-tected by a single-photon detector where the time of ar-rival of the signal is recorded and used to generate theraw key in the QKD protocol.
A fundamental concept in QKD is the necessity to en-code and decode information in conjugate bases. If time-bin encoding represents the temporal computational ba-sis, the conjugate bases are achieved via phase codingwhere information is encoded in the relative phase of thesignal pulses. There exist various phase-coding proto-cols, e.g., diﬀerential phase shift, coherent one-way, andround-robin diﬀerential phase shift. In all these proto-cols, the phase is ultimately measured using some sortof interferometric apparatus. In the simplest case, the relative phase between two time bins can be measuredusing an unbalanced Mach-Zehnder interferometer andtwo single-photon detectors, i.e. one at each output portof the interferometer. The relative phase between thepulses will dictate whether constructive or destructiveinterference will take place at the output beam splitterresulting in a detection event in the ﬁrst or the secondoutput port, respectively.As in the case of time-bin encoding, the signal must begenerated in near-single spectrotemporal mode pulses atdiﬀerent arrival times. The UTF can then be introducedbefore the interferometer to ﬁlter out noise outside theoptical modes considered. Since the UTF acts only onthe intensity of the signal states and does not alter theirphase, the phase information between the time bins isleft undisturbed by the UTF.
Similar to time bins, information can be encoded anddecoded in frequency bins. Encoding in the frequency de-gree of freedom can still be achieved in the presence of ourUTF with a near-unity eﬃciency. In practice, the inten-sity of the pump is adjusted to achieve a 100 % switchingeﬃciency of the quantum signal at a certain wavelength.For the case of a dual-wavelength implementation a fewnanometers apart, switching eﬃciency in excess of 99 %can still be achieved over the various wavelengths.
Finally, another degree of freedom that has receivedattention for implementation in a QKD system are trans-verse spatial modes. In particular, orbital angular mo-mentum (OAM) states of light have been demonstratedin QKD implementations over free-space channels. Atﬁrst glance, our UTF may not seem compatible withhigher-order spatial modes since it requires coupling thesignal into a single-mode ﬁber. However, the UTF cansimply be introduced after the measurement of the spa-tial modes.Indeed, most spatial mode techniques are based on thefact that a single mode ﬁber can act as a spatial-modeﬁlter for the fundamental Gaussian mode. Thus, if ahigher-order spatial mode can be turned into the funda-mental Gaussian via phase manipulation (using a spatiallight modulator or refractive elements) then coupled toa single-mode ﬁber, that higher-order mode can be mea-sured. Thus, since the last step of a spatial mode mea-surement scheme is coupling to a single mode ﬁber, theUTF can be introduced at that stage with high eﬃciencydue to the spatial mode matching.0
UTF, (cid:1) t = (cid:1) t = (cid:1) t = (cid:1) t = Loss ( dB ) S ec r e t k e y r a t e ( bp s ) UTF, (cid:1) t = (cid:1) t = (cid:1) t = (cid:1) t = Loss ( dB ) S ec r e t k e y r a t e ( bp s ) UTF, (cid:1) t = (cid:1) t = (cid:1) t = (cid:1) t = Loss ( dB ) S ec r e t k e y r a t e ( bp s ) UTF, (cid:1) t = (cid:1) t = (cid:1) t = (cid:1) t = Loss ( dB ) S ec r e t k e y r a t e ( bp s ) (a) (b)(c) (d) Noise counts: Hz Noise counts: Hz Noise counts: Hz Noise counts: Hz FIG. 9.
Numerical simulation of the ultrafast switch in the presence of temporal ﬂuctuations . APPENDIX F: ULTRAFAST TEMPORALFILTERING IN THE PRESENCE OFTEMPORAL FLUCTUATIONS
In our proposed scheme, ﬂuctuations in the time ofarrival and polarization of the pulses from the sender tothe receiver can result in a serious limitation for practicalimplementations. Indeed, temporal ﬂuctuations can arisefrom diﬀerent causes, e.g., atmospheric turbulence, pulsebroadening, or low synchronization precision. Althoughnot a fundamental restriction, temporal ﬂuctuations mayhinder the practicality and eﬃciency of our scheme usingUTF for quantum communication in noisy conditions.We investigate this issue by performing a simulation of anoisy QKD channel in the presence of temporal ﬂuctua-tions.In our simulation, we consider that Alice is preparingpulses that are 1 ps in duration and that Bob appliesa 1-ps temporal ﬁlter. Four diﬀerent scenarios are con-sidered, namely temporal ﬂuctuations from the channelresulting in pulse duration ∆ t of 1, 10, 100, and 500 psat the receiver. As a comparison, we consider the case of electronic time ﬁltering with an electronic gating time of1 ns where temporal ﬂuctuations below 1 ns will have noeﬀect on the secret key rate. Moreover, we consider inour simulation a polarization visibility of 0.99, a detectoreﬃciency of 0.8, detector dark counts of 100 Hz, a repe-tition rate of 80 MHz, and for simplicity we are assumingsingle-photon pulses.In particular, we evaluate four diﬀerent noise scenar-ios, see Fig. 9. In the ﬁrst case, see Fig. 9-(a), a noisecount rate of 9 . × Hz is considered, where the lossthreshold of ETF is similar to the loss threshold of UTFwith temporal ﬂuctuations ∆ t = 10 ps. This means thatfor this particular level of noise, if the temporal ﬂuctua-tions are larger than ∆ t = 10 ps, the ETF outperformsthe use of UTF. However, for larger noise levels, e.g., anoise count rate of 8 × Hz, see Fig. 9-(d), the use ofUTF even in the presence of temporal ﬂuctuations with∆ t = 500 ps, a larger loss threshold is observed. Inter-mediate noise levels are also simulated.In summary, temporal ﬂuctuations can aﬀect the per-formance of QKD in the presence of UTF. Nevertheless,for large enough levels of noise, as is the case in day-light conditions, UTF can still be beneﬁcial even in thepresence of temporal ﬂuctuations.  C. Bennett and G. Brassard, Conf. on Computers, Sys-tems and Signal Processing (Bangalore, India, Dec.1984) , 175 (1984). N. T. Islam, C. C. W. Lim, C. Cahall, J. Kim, and D. J.Gauthier, Provably secure and high-rate quantum key distribution with time-bin qudits, Sci. Adv. , e1701491(2017). H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang,H. Chen, M.-J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. 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