lmpact of Fiber Dispersion on Performance of Entanglement-Based Dispersive Optics Quantum Key Distribution

Jing-Yuan Liu | Xu Liu | Wei Zhang| Yi-Dong Huang

Abstract—Dispersive optics quantum key distribution (DO-QKD) based on energy-time entangled photon pairs is an important QKD scheme.In DO-QKD,the arrival time of photons is used in key generation and security analysis,which would be greatly affected by fiber dispersion.In this work,we establish a theoretical model of the entanglement-based DO-QKD system,considering the protocol,physical processes (such as fiber transmission and single-photon detection),and the analysis of security tests.Based on this theoretical model,we investigate the influence of chromatic dispersion introduced by transmission fibers on the performance of DO-QKD.By analyzing the benefits and costs of dispersion compensation,the system performance under G.652 and G.655 optical fibers are shown,respectively.The results show that dispersion compensation is unnecessary for DO-QKD systems in campus networks and even metro networks.Whereas,it is still required in DO-QKD systems with longer fiber transmission distances.

Index Terms—Dispersion compensation,dispersive optics quantum key distribution (DO-QKD),fiber chromatic dispersion,quantum networks.

1.lntroduction

Quantum key distribution (QKD) enables two remote parties to generate secure and random cryptographic keys,with unconditional security guaranteed by quantum mechanics[1]-[3].Quantum secure communications based on QKD have great potential in modern communications[4].There are mainly two schemes in the experimental implementation of QKD,which are the prepare-and-measurement scheme[5],[6]and the entanglement-based scheme[7]-[9].The entanglement-based QKD scheme naturally establishes network connections between users,showing advantages in the implementation of quantum networks[10]-[12].The entangled photon pairs from the quantum light source are distributed to Alice and Bob,respectively.And the coincidence events between the two users are used to generate cryptographic keys.In addition,the highdimensional encoding process can increase the information carried per coincidence event,which utilizes these quantum resources more effectively[13],[14].

Dispersive optics quantum key distribution (DO-QKD) is a recently proposed QKD protocol[15].In entanglement-based DO-QKD,the energy-time entangled photon pairs are generated from the quantum light source[9],[15],[16].Normal and anomalous dispersion components are introduced at Alice’s and Bob’s sides,constructing time and frequency bases.The arrival time of photons at the time base is further used for key generation.The high-dimensional encoding process is introduced in post-processing,in order to maximize the key generation rate under technical constraints.The security analysis of DO-QKD can be realized by the measurements of unbiased time-frequency bases,which has been proven to be secure against collective attacks[15].However,chromatic dispersion introduced by optical fibers will lead to the broadening of the coincidence peak,consequently affecting the generation of secret keys.In our previous entanglement-based DO-QKD experiment over optical fibers[9],the dispersion compensation module (DCM) is used to compensate for fiber dispersion.Nevertheless,there is still a lack of analysis on the benefits and costs of the dispersion compensation.Recently,entanglement-based DO-QKD was applied to realize QKD networks through entanglement distribution based on passive beam splitting and wavelength division multiplexing[10],[17].Dispersion compensation was not applied since the fiber links were quite short in these experiments.However,it can be expected that fiber dispersion would be a serious problem in a large-scale network and the effects of dispersion compensation should be considered seriously.

In this work,we focus on the impact of fiber dispersion on the performance of the DO-QKD system.Firstly,we realize the theoretical modeling of the DO-QKD system.Based on the established model,the influence of fiber dispersion on the DO-QKD system is analyzed.Furthermore,we investigate the benefits and costs of dispersion compensation,and concluded that dispersion compensation is unnecessary at short distances.Using G.652 fibers,the DO-QKD system without DCM has better performance,when the fiber length is less than 9 km,while the range can be increased to 30 km using G.655 fibers.The results show the potential of realizing a dispersion-compensation-free DO-QKD network on campus,or even metro scale.

2.Theoretical Model of DO-QKD System

The sketch of a DO-QKD system is shown inFig.1,where SMF is the single mode fiber;DCM is the dispersion compensation module;ND is normal dispersion;AD is anomalous dispersion;BS is the 50:50 beam splitter.The telecom-band energy-time entangled photon pairs are generated by a quantum light source.Then,the signal and idler photons of the photon pairs are distributed to Alice and Bob through optical fibers,respectively.At Alice’s side,the photons are separated into two paths by a beam splitter.In one path(A1),photons are detected by a single-photon detector directly and the times of these single-photon events are recorded.In the other path (A2),photons are detected after they pass through a dispersive component with normal dispersion.The similar setup is placed at Bob’s side,with two paths denoted by B1 and B2.The only difference is that the dispersive component of Bob has anomalous dispersion,while,its absolute value is the same as that of the dispersive component of Alice.Path A1 and path B1 form the time base of the measurements for the photon pairs.In this base,the coincidence events show strong temporal correlation,which is indicated by a narrow coincidence peak,and they are used to generate keys.The coincidence events of path A2 and path B2 also show a narrow coincidence peak due to the nonlocal dispersion compensation effect of energy-time entangled photon pairs[16],[18].They form the frequency base,which contributes to security tests.

Fig.1.Sketch of an entanglement-based DO-QKD system.

The raw keys are generated by time encoding of the coincidence events in the time base.The largealphabet processing method[13]is applied,which is shown in the upper part ofFig.2.Photons are recorded at both sides in a continuous stream of time,which is divided into consecutive time frames.A time frame includesM(M=2N) consecutive slots and a slot includesItime bins.In the bin sifting process,Alice and Bob firstly communicate their recorded frame numbers to each other which contain one single-photon event through a public channel,and keep the frames in which both sides have single-photon events.Then,for each retained frame,they check the time bin numbers of the single-photon events,and only keep the frames that the single-photon events at the two sides have the same bin number.Finally,Alice and Bob generate the raw keys by the slot numbers of the single-photon events in these frames.In this process,Alice and Bob avoid communications of the slot numbers,hence,Eve cannot get the information of the keys directly from the communications in the public channel.

Fig.2.Schematic diagram of the raw key generation rate and quantum bit error rate (QBER) in high-dimensional coding format of DO-QKD,when (a) Alice records signal-photon events and (b) Alice records noise events.A frame has 2N slots and a slot has I bins.In the figure,N=2 and I=3.

We establish a theoretical model to investigate the performance of DO-QKD systems,considering the characteristics of the quantum light source,optical fiber transmission,and process of single-photon detection.The modeling method is described below.

2.1.Raw Key Generation Rate and QBER

There are some key parameters to describe the DO-QKD system shown inFig.1.For the quantum light source,there are the generation rate of entangled photon pairsR,and the generation rates of noise photons on signal and idler sides,RsandRi.They can be obtained in the experiment by measuring the single-side photon count rates of signal and idler photons,and the coincidence count rate[19].For the fiber transmission,there are the fiber lengthL,fiber loss coefficientα,and dispersion parameterβ2.The loss of DCM is denoted byland its dispersion value is assumed to compensate that of the transmission fibers exactly.For singlephoton detection,there are the single-photon detection efficiencyηd,dark count rate of the detectordd,and the timing jitter of the detectors (including the electronic circuits for counting)σd.In this model,all the detectors are assumed to have the same performance.

We start with the analysis of the types of single-photon events recorded by Alice.For each event of Alice,she may record a signal photon which is one of an entangled photon pair,or a noise event (including noise photons and events of dark counts).If we set the arrival time of Alice’s events ast0,the probability distribution of the single-photon events recorded by Bob neart0could be derived.We number the slots in each frame bym(m=1,2,…,2N) and the bins in each slot byi(i=1,2,…,I).Let us consider the single-photon events of Alice recorded in a specific bin with a slot number ofm0and a bin number ofi0.Since the single-photon events of Alice randomly fall into the time bin,we firstly derive the contributions of the events at Alice under a specifict0in the bin to the raw key rate and QBER.Then we calculate the average raw key rate and QBER as the final results by considering all the events at Alice in the bin,i.e.,the contributions of events at Alice under differentt0in the bin are calculated and accumulated.

For the case that Alice records signal photon,and ideally,the energy-time entangled photon pairs would ideally reach both sides simultaneously.However,the temporal correlation of the photon pairs decreases due to the timing jitters of the detectors,hence the arrival time of photons on each side will be uncertain.When we set the recorded arrival time of Alice’s photons as a fixed valuet0,the uncertainty of the recorded arrival time of Bob’s photons tot0is determined by the timing jitters of detectors on both sides.Its temporal distribution follows the approximate Gaussian distribution with the mean oft0,accompanied by a floor due to noise photons and dark counts,as shown inFig.2 (a).This temporal distribution at Bob’s side is expressed as

It can be derived from the known parameters of the source,transmission,and detectors.The varianceσinhis mainly caused by the timing jitters of the single-photon detectors on both sides:

whereσ0is the intrinsic correlation time between the two photons in a pair,after they pass through the filters.Usually,σ0is far smaller than the time jitters of single-photon detectors in typical DO-QKD experiments[9].

For the case that Alice records noise events,Bob’s recorded events have no temporal correlation with Alice’s records,forming accidental coincidence count events as shown inFig.2 (b).The temporal distribution of Bob’s events is expressed as

Considering the two cases,we calculate the total number of correct coincidence eventsX1,when Bob’s recorded time is in the same bin witht0.We also calculate the number of error coincidence eventsX2,m(m≠m0),when Bob’s recorded time has the same bin number witht0,but in themth slot (see details in Appendix A).The raw key generation ratebis considered to be the total number of coincidence events multiplied by the encoding dimensionN,which is expressed as

If there is no bin sifting process,it is obvious that the adjacent slots between Alice and Bob are prone to misjudgment,resulting in bit errors,as shown in the white triangles inFig.2.By sifting and keeping the coincidence counts of the same time bin number,the errors caused by this situation can be greatly eliminated,so QBER of the system is effectively reduced.However,there are still bit errors in the case of the same bin number but different slots,as shown in the gray triangles inFig.2.And QBER is expressed as

whereemis the bit error ratio when Bob records an event in themth slot.Since one coincidence event generatesNbits in the high-dimensional time coding format,emrepresents the ratio of the error bit number to the total bit number (N),if the recorded single-photon events of Alice and Bob are in different slots.

Then the average raw key generation rate and QBER are calculated considering allt0in the bin.Although the above analysis only considers the single-photon events of Alice recorded in a specific time bin,the same results could be expected if it is extended to other time bins.Hence,(4) and (5) can be used as final results to calculate the performance of an entanglement-based DO-QKD system.

2.2.Security Analysis and Secret Key Rate

Alice and Bob perform the arrival-time measurements under unbiased time-frequency bases with outcomes described by random variablesTAandTB.The security analysis can be performed by analyzing the numerical characteristics of the random variables (see details in Appendix B) and calculating the timefrequency covariance matrix[15].The secret key rate extracted per coincidence count is expressed by the secret key capacity[20]:

whereβis the reconciliation efficiency,which is set to be 90% in this work.I(A;B) is the Shannon information between Alice and Bob,andχ(A;E) is Eve’s Holevo information.

2.3.lmpact of Fiber Dispersion on DO-QKD

The chromatic dispersion introduced by transmission fibers will broaden the coincidence peak,which will greatly deteriorate the secret key rate.Considering the influence of fiber dispersion,the variance of the temporal distribution of Bob’s photons in the first case is modified,which is expressed as

Nonlocal dispersion cancellation can eliminate most of the broadening caused by dispersion.However,the residual dispersion that has not been well compensated still causes broadening[15],[18],[21],which is characterized by

The relationship between dispersion parametersDandβ2is expressed asD=When there is residual dispersion,the varianceσinhin the above formulas is replaced withσtot.

3.Results and Discussion

3.1.Simulation Results under Experimental Conditions

Based on the above analysis,the performance of the DO-QKD system under encoding parametersN=4 andI=3 is simulated.The main parameters used in the simulation (in accordance with the experiment in [9])are shown inTable 1,whereRandRs,irefer to the corresponding generation rates at the output of the quantum light source.

Table 1:Major parameters used in the simulation of the DO-QKD system

When the bin width is 160 ps that is the same with the experiment in [9],the raw key generation rate is 151 kbps with QBER of 3.69%.Through security analysis,in the case of 20-km transmission with dispersion compensation,the Shannon information between Alice and BobI(A;B) is established as 3.83 bit per coincidence (bpc),and Eve’s Holevo informationχ(A;E) is 0.01 bpc.As a result,the secret key capacity ΔIis 3.44 bpc,leading to the secret key rate of 129.9 kbps.The simulation results are mostly consistent with the experimental results in the previous work[9].It is believed that the model can be used to describe and further analyze the DO-QKD system.

3.2.lmpact of DCM on DO-QKD

In order to investigate the impact of DCM on DO-QKD,the performance of the system under 20-km fiber transmission is shown inFig.3.The solid lines are the results of using DCM (with a loss of 3 dB) in the system,while the dashed lines show those without DCM.Without dispersion compensation,σtotof the temporal distribution at Bob’s side under 20-km fiber transmission is broadened from 63.7 ps to 81.6 ps.The bin sifting process is affected as follows:The error bits caused by coincidence counts in different slots increase,hence QBER of the system increases.The bin width at the minimum QBER becomes wider,because the larger bin width is needed to cover the broadened temporal distribution.By introducing the three-level optimization format[9],we optimize the parameters of the bin sifting process,further reducing QBER and increasing the raw key rate.The bin widthτafter the three-level optimization (under the QBER upper bound of 5%) is 260 ps with DCM,while it reduces to 230 ps without DCM.Since the loss introduced by DCM is avoided,the raw key rate is higher than that with dispersion compensation.However,the raw key rate is not necessarily related to QKD performance.It is necessary to perform security analysis on the system to obtain the secret key rate.

Fig.3.Influence of dispersion compensation on (a) QBER and (b) raw key generation rate b.And high-dimensional encoding parameters:N=4 and I=3.

The security analysis shows that the secret key capacity ΔI<0 in the case of 20-km transmission without dispersion compensation.It deteriorates greatly due to the uncompensated fiber dispersion.In this situation,no secret keys can be extracted from raw ones after error correction and privacy amplification.Therefore,dispersion compensation must be performed in the DO-QKD system under 20-km optical fiber transmission.

3.3.Results under Different Transmission Conditions

Chromatic dispersion leads to the broadening of the temporal distribution of Bob’s photons,which results in the increase of QBER and decrease of the secret key rate.DCM can compensate for the dispersion,whereas it introduces additional losses on the optical links,which would reduce the coincidence counts available for key generation.Hence,the benefits and costs of dispersion compensation should be analyzed to obtain a higher secret key rate.There is a theoretical transmission range in which the performance of the DO-QKD system without dispersion compensation is better than that with dispersion compensation.In this case without dispersion compensation,the increase in the effective coincidence counts makes up for the decrease in the secret key capacity in terms of secret key generation.

The calculation results of a DO-QKD system with parameters set asTable 1(except the variable fiber lengthLand dispersion parameterD) are shown inFig.4.Fig.4 (a)shows the calculated secret key rate under increasing fiber length when the G.652 fiber (D=17 ps/km/nm at the wavelength of 1550 nm) is applied in the system.The black squares are the results that DCM (with a loss of 3 dB) is used in the system.The white circles show those without DCM.It can be seen that when the transmission distance is relatively short,the system without DCM has better performance.However,the performance of the system with DCM is better when the transmission fiber is over 9 km,since the performance deterioration due to the fiber dispersion is larger than the impact of the loss of DCM.It proves that dispersion compensation is unnecessary in DO-QKD for the short-distance transmission.

Fig.4.Secret key rates (optimized with the QBER upper bound of 5%) under increasing fiber length:(a) transmission fiber is G.652 fiber and (b) transmission fiber is G.655 fiber.

Fig.4 (b)shows the calculated secret key rate under the increasing fiber length when the G.655 fiber(D=5 ps/km/nm with the wavelength of 1550 nm) is applied.It can be seen that the results are similar to those inFig.4 (a).Whereas,the system without DCM has better performance when the fiber length is less than 30 km due to the lower dispersion parameter of the G.655 fiber.

4.Conclusions

In this work,we theoretically investigate the influence of chromatic dispersion introduced by the transmission fiber on the performance of the DO-QKD system.We firstly establish a theoretical model of the entanglement-based DO-QKD system,and proved the feasibility of the model.Fiber dispersion will greatly deteriorate the performance of the DO-QKD system.However,it is shown that dispersion compensation is unnecessary when the transmission distance is relatively short.Using the G.652 fiber,the DO-QKD system without DCM has better performance when the fiber length is less than 9 km,while this range increases to 30 km using the G.655 fiber due to its lower dispersion parameter.The above results provide theoretical guidance for practical applications of DO-QKD.It shows that in the application scenarios of campus networks and community access networks,the dispersion compensation in DO-QKD is unnecessary.If transmission fibers with lower dispersion are applied,such as the G.655 fiber,DO-QKD without dispersion compensation even can be used in metro networks.

Acknowledgment

The authors would like to express their appreciation for the support of Tsinghua Initiative Scientific Research Program.

Disclosures

The authors declare no conflicts of interest.

Appendix A:Number of Coincidence Events between Alice and Bob

For the cases that Alice records signal photons and noise events,the temporal distribution at Bob’s side is expressed as the two distribution functionsB1(t) andB2(t).It is regarded as a correct coincidence event when the time recorded by Bob is in the same bin witht0.The number of correct coincidence events is denoted byX1:

wheretm0,iandtm0,erepresent the initial and end time of the bin containingt0,which is in them0th slot.The integrals with respect totare shown in the black areas inFig.2.

Other coincidence events are regarded as error coincidence events.The number of error coincidence events in themth slot is denoted byX2,m(m≠m0):

wheretm,iandtm,erepresent the initial time and end time of the bin with the same bin number oft0in themth slot.The integrals with respect totare shown in the gray areas inFig.2.Hence,the total number of error coincidence events is expressed as

Appendix B:Numerical Characteristics of Random Variables TA and TB

We assume that Alice and Bob both detect a single-photon event in a given time frame.At Alice’s and Bob’s sides,the arrival-time measurement parameters in the time base are described as

When Alice and Bob receive a pair of entangled photons,there is a strong temporal correlation between the recorded events of them.The probability of this case over all the coincidence counts isP1.In other cases,there is no temporal correlation between the recorded events of both sides.We have

whereRcois the coincidence count rate,Racis the accidental coincidence count rate in the time frame with widthT,andT≡6σtot.They are expressed by the known parameters as

At Alice’s side,we assume that the arrival timeTAfollows the uniform distribution in the timeT,with zero mean.The numerical characteristics ofTAare expressed as

whereE[?] means the mean of a random variable and V ar[?] means the variance of a random variable.

At Bob’s side,the time differenceXbetweenTA1andTB1follows the Gaussian distribution with zero mean and varianceσtot.TB2also follows the uniform distribution in the timeT,with zero mean.The random variablesTA1andX,TA2andTB2are independent.Since both the situations at Bob’s side have zero mean,the numerical characteristics ofTBare simplified as:

Based on the above analysis,the statistical characteristics between the random variables of both sides are derived as

where Cov[?,?] means the covariance between two random variables.The arrival-time parameters in the frequency base obey corresponding relations.We used the above statistical characteristics in security analysis to obtain the secret key rate[15].