A MATLAB based modeling and simulation package for DPS-QKD

Anuj Sethia and Anindita BanerjeeQuNu Labs Pvt Ltd., MG Road, Bangalore, India

(Dated: July 16, 2021)

Quantum key distribution (QKD) is an ingenious technology utilizing quantum information sci-ence for provable secure communication. However, owing to the technological limitations and devicenon-idealities it is important to analyze the system performance critically and carefully define theimplementation security. With an acceleration in the commercial adoption of QKD, a simulationtoolkit is requisite to evaluate the functional architecture of QKD protocols. We present a sim-ulation framework to model optical and electrical components for implementing a QKD protocol.The present toolkit aims to model and simulate the optical path of the DPS-QKD protocol withits imperfections and eventually characterize the optical path. The detailed device-level modelingand analysis capabilities of the present toolkit based on Simulink and MATLAB have the potentialto provide universal toolkit for practical design and implementation of generalized QKD protocolscompared to earlier works. We report a novel work on the implementation of a QKD protocol onSimulink and MATLAB platform. Further, the absence of any modeling framework for DPS QKDand its simplistic optical schematic made it an obvious choice for the authors. We are hopeful thatthis work will pave way for simulating other QKD protocols from the DPR family.

Secure and secret communication techniques arecrucial since the advent of information theory in the20th century. Countless people have worked towardsbuilding technology for secure communication over thelast century. Classical cryptography utilizes the com-plexity of exponentially hard computational problemsto distribute encryption keys. However, the quantumalgorithms pose a severe threat as these problems canbe solved in a polynomial time on a quantum computer.QKD marks a disruptive breakthrough in this journeyby exploiting principles of quantum mechanics. Itis a technique to generate encryption keys betweentwo parties in a quantum secure way. The first QKDscheme developed in 1984 by Charles Bennett and GillesBrassard has revolutionized the cryptography industry.Thereafter, several other QKD protocols are proposedand each protocol is subject to rigorous and extensivesecurity analysis. Many improvements are proposed ontooriginal protocols to enhance the security and range ofcommunication. A common underlying principle amongvarious prepare-and-measure QKD schemes is that thebit information is encoded on a non-orthogonal quantumstate. The indistinguishability of non-orthogonal statesguarantees the security of the communication. Noeavesdropper can overwhelm the quantum principlespaving way for robust security of QKD protocols.

Modeling and simulation are often used as an effi-cient means to understand complex systems and theirdynamics. By systematically defining and decomposingthe complex behavior of interest, one can constructrepresentative models with varying abstraction levels.

Initial research focused on the analysis and perfor-mance of QKD protocols with idealistic assumptionsand limited optical components [4, 5]. As the technologymatured, modeling the practical components and pro-cesses involved in entire QKD protocol gained attention

[1–3, 6–9]. Recently, significant development is made onmodeling and simulation of QKD networks with varyingtopology [10, 11].

In 2015, Mailloux et al. [3] reported a complete soft-ware package with a modular architecture called qkdXfor polarization-based BB84 protocol. Recently, Chat-terjee et al. [1] presented qkdSim, a software package formodeling and analyzing generic QKD protocols, yet onlyB92 protocol is simulated. Further, a recent work [2]have prepared a universal framework for QKD modeling.Although these simulation frameworks are claimed to begeneralised to simulate any QKD protocol, no progresshas been made in simulating the distributed phase ref-erence (DPR-QKD) and continuous variable (CV-QKD).

In the present work, we have designed and imple-mented a QKD simulator for studying the implementa-tion of differential phase shift (DPS) QKD, which fallsunder the DPR-QKD protocols. We have studied thefeasibility of simulating different building blocks of theoptical path of the protocol. The purpose of our work isto find a practical and high-precision platform to analyzethe optical path of the protocol and to explore whetherthe toolkit can be generalized for all QKD protocols.

DPS protocol is widely accepted for commercialapplication by virtue of it’s simplistic implementa-tion. The present toolkit is built using Simulinkand MATLAB, which are widely used for simulationof optical, electronic and opto-electronic systems.Each component and process involved are custom buildand designed to assimilate future technological advances.

This paper is organised as follows. In section I, webriefly explain the DPS QKD protocol and the underly-ing principles. Section II discusses in detail the model-ing scheme and framework in a generalised fashion. Wehave also listed the assumptions considered in the work.

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In section III, we unfold the modeling technique of in-dividual components and processes. In section IV, wepresent our implementation of DPS QKD in the simu-lation toolkit and in section V we present the diversecapabilities of the toolkit for analysing a QKD protocolfor research objectives. Lastly, we conclude in section VI.

I. DPS-QKD PROTOCOL

Inoue et al. introduced DPS QKD in 2002 [12–15]. It isa prepare-and-measure type of QKD protocol where, Al-ice (transmitter) generates heavily attenuated and phaseencoded, train of pulses and transmits then over a quan-tum channel to Bob (receiver). The security is based onnon-deterministic collapse of a wave function in a quan-tum measurement. The optical schematic of DPS QKDis presented in Fig. 1.

FIG. 1. Optical schematic of DPS QKD protocol.

The protocol is executed as follows:

1. Alice generates the state |ψ〉 which is a train of Ncoherent pulses with θ as the initial phase and θn asencoded phase which takes {0, π}. This quantumstate is represented as:

|ψ〉 =

N−1⊗n=0

∣∣∣αei(θ+θn)⟩. (1)

She heavily attenuates the transmission power suchthat mean photon number per pulse is less thanunity.

2. Bob receives the pulse train and passes it through aDelay Line Interferometer (DLI) with a delay whichis inversely proportional to repetition rate of pulsetrain. The interference between adjacent pulses re-sults in selective detection between the two singlephoton detectors D1(2) for a phase difference of0(π).

3. Bob tags each click and the corresponding detec-tor. He shares the time information with Alice overprior authenticated classical channel.

4. Considering the time information shared by Boband the phase encoding information already avail-able with Alice, she can generate a raw data, par-tially correlated with Bob. This process is the be-ginning of key reconciliation. Synchronization mis-

match with electronic noise and delay are the ma-jor sources of error causing irregularities in shareddata.

5. Bob performs the error estimation procedure withAlice. The discrepancies are due to imperfectionsat various levels in the communication system andcontribute to the quantum bit error rate (QBER).QBER is defined as the ratio of incorrect bits toreceived bits. In conservative security analysis, allthe errors are attributed to an eavesdropper. Theamount of information exposed to Eve is estimatedusing the Shannon’s noiseless coding theorem [16].For an estimated QBER of e, the minimum infor-mation exposed is given by

h(e) = −e log2 e− (1− e) log2(1− e). (2)

Thereafter, both the parties execute error-correction to generate correlated bit string on ei-ther end.

6. Finally, Alice and Bob perform privacy amplifica-tion to obtain final secure key. The error correctedkey is compressed to minimise the information leak-age in the raw quantum transmission and during er-ror correction. The extent of compression dependsupon the amount of information leak and the eaves-dropping strategy.

Numerous enhancements have been discussed [15] onthe original DPS protocol to improve the transmissionrange and security against sophisticated eavesdroppingattacks. These advancement include variable delayscheme [17] (RR DPS-QKD), implementation of decoystates [18], a novel four-level scheme (DQPS QKD)[19],and entanglement based scheme. Extensive securityanalysis of these extended schemes is yet an openproblem. The present toolkit will be valuable tool forcomparative analysis of these modification to originalscheme.

II. SIMULATION SCHEME

In this section, we provide an overview of the simula-tion schemes and the assumptions taken to simplify themodeling scheme. We discuss different types of simu-lation schemes and the problem sets addressed by eachcriterion and best practices on modeling and simulation.This analysis helps us to classify QKD protocol amongthe primary simulation classifications and setting the de-velopment ground.

A. Overview

A notable feature of QKD systems is that billions ofoptical pulses are generated and propagated through

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the system during the simulation of system operation.While each pulse is most accurately represented asa continuous-time waveform, it is computationallyintensive to model a complete QKD system usingcontinuous-time simulation. This forces us to simulatethe process in a discrete-time environment.

More broadly, QKD protocols are classified as dy-namic, time-invariant, nonlinear processes [6]. Thedynamic nature is due to the dependence of output overinput signals. Although the system behavior changeswith time, it is invariant to the universal time; thus,it is considered as time-invariant. Nonlinearity arisesfrom the various complex physical components andprocesses. A QKD protocol architecture is dividedamong five discrete parts: (a) source, (b) preparation,(c) transmission, (d) detection, and (e) post-processing.This particular order of events results in a systematicflow of signals in the discrete-event based environment.

An essential characteristic of a QKD protocol is itsprobabilistic nature, where the outcomes of an eventcan be associated with probability. QKD is inherently arandom process represented in its subsystems like a laser,beam splitter, detector, etc. The probabilistic natureis the foundation of QKD thus, it is crucial to modelthe optical path. For each stochastic process, randomvariables are generated from a (pseudo-random numbergenerator) PRNG to emulate the physical processes [20].

A dynamic, time-invariant, nonlinear, stochastic, anddiscrete-time simulation is the appropriate model char-acteristic of a QKD protocol. These are the imperativerequirement in a simulation framework and is essentialto describe before constructing the simulation model.Simulink is one such package that accommodates allthese features quickly and has an open structure todesign and implement a new system. Simulink is bestsuited for the process with its vast simulation toolkitson communication and signal processing.

Independent modules based upon model charac-teristics are developed corresponding to the source,detection, and transmission components used in theactual setup. The modules are interconnected for theflow of logic and mimic the photons’ path in the actualexperimental setup. The simulation aims to illustratethe QKD process alongside various non-idealities dueto device imperfections. Another critical objective isto simulate hacking attacks on the system, helping usperform a security analysis of practical QKD protocols.

Each module is composed of sub-modules correspond-ing to physical components and processes. Each sub-module is parameterized by certain physical variables af-fecting the dynamics. They are categorized as either userinputs or set parameters. User inputs refer to the user’sindividual choices, whereas the set parameters refer to

the specification of the various components.

B. Assumptions

The simulation model creates a virtual experimentalimplementation of DPS QKD by modeling imperfections,but it is not exhaustive. We have to make certain as-sumptions to simplify the system. We have listed belowsome of them:

• Optical source is assumed to produce monochro-matic coherent light with appropriate spectrallinewidth.

• Optical transmission medium is a single mode fi-bre and the signal propagates only in the allowedfundamental LP01 mode.

• The optical signal is represented by an electric fieldpropagating in z direction and it’s components intwo polarisation axis (x and y) where, x being thefaster and y is the slower polarisation axis.

• Interference of optical signal occurs only at DLI,and is ignored elsewhere.

• Optical loss in between the two components is con-sidered negligible, however, insertion loss of indi-vidual components is included. Also, all the opticalcomponents are assumed to be perfectly aligned.

• Both Alice and Bob subsystem runs on a singleclock, which is the simulation clock. This elimi-nates the clock synchronisation imperfections.

• Presently, no eavesdropping strategy or attack isconsidered for the simulation but this would be in-cluded in future works.

• Sifted keys can be obtained from the simulator,however, the error correction and privacy ampli-fication algorithms can be applied separately.

III. QKD UNIT MODULES

In this section, we have described the modeling schemefor each physical component. We have also tabulated thevariables contributing to the modeling scheme of eachcomponent alongside.

1. Laser

Laser is a complex self-consistent system that iscapable of demonstrating a wide range of dynamics.Semiconductor lasers are most frequently used opticalsource in optical fibre communication technologies dueto their compact size, low power consumption andaffordability. Distributed feedback (DFB) laser areprominently used, credit to their superior performances,

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narrow spectral width and low noise.

The operating dynamics of a DFB semiconductor laseris defined by a set of rate equations for the interactionof photons and charge carriers inside the active cavityregion [21]. They are derived from Maxwell’s equationsusing quantum-mechanical approach for the induced po-larization. These single mode optical laser rate equations(3, 4) are used for the simulation of the frequency chirpand output power waveform. For a single-mode laser, therate equations are given below:

dN(t)

dt=I(t)

eVa− N(t)

τn− g0

N(t)−N0

1 + εCS(t)S(t) (3)

dS(t)

dt=

(Γg0

N(t)−N0

1 + εCS(t)− 1

τp

)S(t) +

βΓN(t)

τn(4)

P (t) =S(t)Vaη0hν

2Γτp(5)

Equ. (3, 4) are coupled non-linear differential equa-tions between the charge carrier density, N(t) and photondensity, S(t). The carrier density N(t) increases due tothe injection current I(t) into the active layer volume Vaand decreases due to stimulated and spontaneous emis-sion of photon density S(t). Similarly, the photon densityS(t) is increased by stimulated and spontaneous emissionS(t) and decreased by internal and mirror losses. Thetime variations of the output optical power are related tothe photon density as shown in equ. (5). The resultantelectric field is calculated in equ. 6 where, the amplitude

A(t) =√

2P (t)εVa

.

E(t) = A(t)ei(ωct+φ(t)) (6)

A laser exhibit two types of noise, relative intensitynoise (RIN) (δP (t)) and phase noise (δφ(t)). In our im-plementation, RIN value is obtained from product datasheet and is assumed to be white gaussian. The phasenoise is related to the spectral line width and inverse ofclock frequency. Gaussian noise of calculated varianceis added to the power and the phase to replicate actualsystem imperfections. The rate equation model along-with emulation of laser noise and temperature depen-dence demonstrates response of a monochromatic laserwith a reasonable fidelity. The laser sub-module outputselectric field signal (Ex(t) and Ey(t)) as a bus signal. InQKD protocols based on phase encoding the linewidthplays a critical role. In DPS QKD, improper selectionof linewidth impacts the system QBER and its stability[22]. It is reported that linewidth should be less than0.35% of the free-spectral-range (FSR) of DLI to achieveQBER less than 0.5%. The toolkit provides the flexibil-ity to model and study the same leading to improvementof optical path performance and minimizing the systemQBER.

Symbol Parameter

Γ Optical confinement factorg0 Gain coefficientN0 Carrier density at transparencyεC Gain compression factorτp Photon lifetimeβ Spontaneous emission coupling factorτn Electronic carrier lifetimee Electronic chargeVa Active region volumeηDFB Differential quantum efficiencyN0 Equilibrium carrier densityh Plank’s constantc Speed of lightλ0 Central wavelengthε Permittivity of active regionI(t) Injection currentσI Injection current standard deviation∆ω Spectral linewidthRIN Relative intensity noiseδλ Temperature variation∆f System bandwidthTL Laser set temperatureφ0 Initial absolute phase

2. In-line Polariser

An in-line polariser pass linearly polarized light whileblocking the orthogonal polarization from an unpolarised(or randomly polarized) light source. The orthogonallypolarized light is attenuated with a desired extinctionratio such that only a single principle polarisation modetraverse through the fibre.

Symbol Parameter

ILPloss Insertion lossPER Polarisation extinction ratio

3. Intensity Modulator (IM)

IM is used to externally modulate the optical signalfrom source to result in a pulsed output. A mach-zehnderinterferometer structure is used to modulate the opticalsignal, both the arms experience a phase shift and arerecombined. The two electrodes are biased with a RFsignal causing a phase shift of φ1(t) and φ2(t) for the twobranches. The output field of the light wave carrier isrepresented by

Eo(t) =1

2Ei

(eiφ1(t) + eiφ2(t)

). (7)

For simplicity, equal and opposite potential (V(t)) is ap-plied to the two electrodes, thus the transfer function is

reduced to Eo(t) = Eicos(πV (t)Vπ

). Vπ is a device param-eter for π phase shift. Transfer function is illustrated inFig. 2.

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FIG. 2. Transfer function of IM.

V1 and V0 in Fig. (2) represent the voltages cor-responding to max and min of the optical pulse asshown in Fig. (2). By adjusting these bounds, a desiredextinction ratio can be achieved.

The spectrum of the modulated signals includes setsof symmetric side-bands arranged around the laser car-rier frequency fo. The side-bands are displaced from thelaser carrier peak frequency at integer multiples of themodulation frequency fm (fo ± nfm with n = 1, 2, ...).The relative heights of the side-bands are a function ofthe modulation depth, which is in turn a function of thepeak-to-peak value of the RF driving voltage. As a resultof these side-bands a multi-model interference appears atthe DLI on Bob’s side (discussed in section IV).

Symbol Parameter

IMloss Insertion lossIMER Desired extinction ratioFWHM Full width half maximumV(π)IM Bias voltage for π phase shiftVDC DC bias input to IMVRF(t) RF input to IM

4. Phase Modulator (PM)

PM modulates the phase of optical carrier signals bypassing it through an opto-electrical wave-guide under aRF voltage. As the refractive index of the opto-electricalmedium changes with voltage applied, any desired phaseshift can be provided to the input signal. An importantparameter to characterise PM is Vπ, the RF bias voltagerequired to give a phase shift of π. The resultant electric

field is given by, Eo(t) = EieiπV (t)Vπ .

Symbol Parameter

V(π)PM Bias voltage for π phase shiftVbias Drift in bias voltagePMloss Insertion loss

5. Variable Optical Attenuator (VOA)

VOA attenuates the optical signal to quantum level,reducing the mean photon number to less than unity.The attenuator provide finer control over the quantumamplitude. This is a critical parameter and its precisetuning is very essential from security of quantum channel.

6. Optical Fiber

Light propagating through an optical fiber mainlyundergoes both attenuation and distortion of signal.Optical fiber has a finite number of guided propagationmodes, with defined spatial structures. V-number is adimensionless parameter widely used to determine thefraction of the optical power in a certain propagationmode. V-value below 2.405 indicates a single modepropagation. For a 1550 nm telecommunication wave-length and SMF-28 fiber, V-number = 2.325 suggestinga single mode propagation. We assume light propagatesonly in LP01 mode with transverse profile approximatedas Gaussian.

Transmission of an optical pulses along a single modeoptical fiber (SMF) is governed by the Non-LinearSchrodinger equation (NLSE), derived from Maxwellequations [23]. The influence of optical fiber can beclassified into (a) linear effects, which are wavelengthdepended and (b) non-linear effects, which are intensity(power) depended.

Major impairments of optical signals transmitted viasingle mode fiber are mainly caused by linear effects -dispersion and attenuation. Attenuation limits power ofoptical signals and represents transmission losses. Dis-persion causes spreading of optical pulses in time domainand phase shifting of signals at the fibre end. Based uponthe cause of dispersion it is classified as:

• Chromatic dispersion (CD): It is the dependenceof group velocity associated with the fundamentalmode on the frequency of signal. It is also known asgroup velocity dispersion (GVD). It can be furtherdistinguished as either material dispersion or wave-guide dispersion.

• Higher-order dispersion: The non zero higherderivatives of the total dispersion curve causespulse broadening at zero-dispersion wavelength.

• Polarization mode dispersion (PMD): The birefrin-gence of optical fiber between two principle polari-sation axis results in signal delay.

Nonlinear effects are crucial for long haul optical signaltransmission. The intensity dependence phenomenon offiber refractive index is known as the Kerr effect andis the cause of fiber nonlinear effects [23]. The powerdependence of refractive index is expressed as:

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η′ = η + γP

Aeff(8)

where, P is the average optical power of the guided mode,γ is the fiber nonlinear coefficient and Aeff is the effectivearea of the fiber. Kerr nonlinearity effects include:

• Self-phase modulation (SPM): Optical intensitycauses a nonlinear a time-dependent phase shiftand pulse acquires a chirp.

• Cross-phase effect (XPM): Non-linear phase shiftsimilar to SPM but caused by the interaction withanother beams present in the fiber.

• Four-wave mixing (FWM): Due to third ordernonlinearity, mixing of different frequency com-ponents propagating together generates additionalfrequency components.

Apart from this, nonlinearity induces inelastic scat-tering of incident photons into lower energy photons,causing a downward shift of frequency. Such effectsfound in a optical fiber are - stimulated Brillouinscattering (SBS) and stimulated Raman scattering(SRS). Effects of SRS and SBS are only noticeablewith a high optical power and thus, can be ignored inour application. On the other hand, FWM and XPMbecomes negligible in SMF due to high local dispersion.Thus, SPM is usually the only dominant nonlinear effectat low transmission power. Since, operating power inQKD is significantly low, it is safe to neglect all othereffects.

The numerical method used to solve the NLSE isknown as the split-step Fourier method (SSFM) [23]. Itaccurately models both linearity and non-linearity of aSMF. In SSFM, the fiber length is divided into smallsegments dz, both linear and non-linear effects within thesmall length dz are small and thus considered mutuallyindependent of each other. So, the NLSE is transformedinto operators formulated as follows:

dA

dz= A(D + N) (9)

D = −α2

+i

2β2

∂2

∂t2+

1

6β3

∂3

∂t3± i∆τ

2

∂

∂t(10)

N = i|A√γ|2 (11)

where, D represents the linear differential operatoraccounting for absorption (α), CD (β2), higher order

dispersion (β3) and PMD (∆τ). N is the non-linearoperator representing SPM. The accuracy of SSFMcan be improved by sandwiching the effect of fibernon-linearity by dispersion effect in each segment dz.

Symbol Parameter

L Fibre lengthα Attenuation per kmβ2 2nd order CD factorβ3 3rd order CD factor∆τ Differential group delayγ Non-linear coefficientNFFT Number of samples for FFTdz Step size for SSFM

7. Delay Line Interferometer (DLI)

DLI, encompasses interference of adjacent pulses todemodulate the information stored in relative phase ofpulses. A typical DLI comprises of two 50:50 beam split-ter (BS) each with two inputs and two outputs as shownin Fig. 3 (BS1 assumes vacuum state as second input).The path-splitting operations of BS are parameterizedby reflectivity (r) and transmissivity (t) which follows

|r|2 + |t|2 = 1.

FIG. 3. Input and output ports of a beam splitter.

(a†0a†1

)→(t irir t

)(a†2a†3

)(12)

(a0

a1

)→(

t −ir−ir t

)(a2

a3

)(13)

Equ. (12) and (13) provide the correlation between thecorresponding state creation and annihilation operators.Employing above equations with displacement operator,the input and output states are given by:

|in〉 = |α〉1 |β〉0 (14)

|ou〉 = eiφ |tβ + irα〉2 |tα+ irβ〉3 (15)

Equ. (15) represents the interference of the adjacentpulses in an ideal case. However, the optical signal iscomprised of multiple side bands due to the externalmodulation. Thus, a multi-modal interference occurs atthe second beam splitter and the resultant power is given

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by:

P (t) =

∣∣∣∣∣∣∑i

E(i)x (t) +

∑j

E(j)y (t)

∣∣∣∣∣∣2

(16)

where, E(i)x (t) and E

(i)y (t) are ith mode of electric field

in x and y polarisation field respectively. The central

frequency is represented by E(0)x (t) and E

(0)y (t). This

imperfect interference at the DLI causes a reduction inthe output visibility defined by

V =Imax − Imin

Imax + Imin(17)

Symbol Parameter

∆t Time delayt Transmissivityr ReflectivityDLIloss Insertion loss

The visibility plays a critical role in system QBER andsystem stability for practically, all phase encoding basedQKD protocols. The QBER is given by

1− V2

(18)

which implies that if visibility is 90% then optical QBERis 5% . The impact of proper implementation of DLIwhether based on PLC or free space DLI unit can bestudied by the simulating the DLI unit module accord-ingly.

8. Single Photon Detector (SPD)

For a prepare-and-measure QKD protocol, measuringthe quantum state is the most decisive step. In DPS,Bob’s setup comprises of two single-photon detectors(SPD) placed at the two outputs of DLI. Since, theinformation is encoded in the pulse train with meanphoton number less than unity, we need single photondetectors for measuring the qubit send by Alice. singlephoton avalanche detector (SPAD) are mostly usedfor this application due to it’s relative simplicity andaffordability as compared to superior superconductingnano-wire single photon detectors (SNSPD).

In this work, we try to model and simulate the SPADincorporating it’s complex behavior depending on dead-time, detection efficiency, dark count rate, after-pulseprobability, temperature and time jitter. Many differ-ent approaches have been made to accurately model aSPD [24–27]. We have used a probabilistic approach tomodel the SPD running in continuous mode. Firstly, themean photon number in each pulse is calculated using the

energy content of each pulse. The detection inefficiencyof SPD is equivalent to the transmission losses. Thus,the MPN arriving at detector equals to ηµ and detectordetects all these photons. We also add background pho-ton (Nb) to the incoming photons. The Poisson distri-bution of photon given by Equ. (19) is used to calculatethe probability of more then one photons in the pulse,Pp = 1− eηµ+Nb .

|α〉 = e−(ηµ+Nb)/2∑n

(ηµ+Nb)n/2

√n!

|n〉 (19)

A detection event in SPAD is characterized by multiplephenomena such as dark count, after pulsing, deadtimeetc. To model the after-pulsing effect as a function ofnumber of interval pulses n, we can use an exponentialdistribution function given by:

Pap(n) = p0e−an (20)

The two parameters p0 and a are set to 0.0317 and0.00115 [24], respectively. Given the dark count probabil-ity (Pd) from detector specification sheet the probabilityof registering a detection event, Pclick can be calculatedas:

Pclick = Pp + Pap + Pd − PpPap

− PapPd − PdPp + PpPapPd. (21)

To replicate the probabilistic nature of photon detec-tion, a random number is generated between 0 and 1.If the random number is less then Pclick, a click is regis-tered. Once, a photon click is registered the detector goesOFF for a certain time-interval, known as detector deadtime (τ) and the Pclick is set to zero for the interval. Af-ter each iteration the after-pulse probability is updated.The click interrupt signal from detector is added with asuitable time jitter, similar to realistic devices.

Symbol Parameter

Nb Background photons per pulseτ DeadtimePd Dark count probabilityp0 After-pulse probability pre factora After-pulse probability exponential

factorη Detector efficiencySPDJit Time jitter

IV. DPS-QKD SIMULATION

In this section, we present the modeling architectureof DPS QKD protocol. We have considered the original

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FIG. 4. Block diagram from Simulink for Alice’s subsystem in DPS QKD

DPS scheme [14] already described in section I with stan-dard single-mode telecommunication fiber as mediumof communication. This exercise provides a distinctoverview of simulation toolkit in evaluating QKD systemarchitecture. Similar to a QKD protocol, the communi-cation system is divided into Alice and Bob, comprisingof optical components discussed in section III. Each com-ponent sub-module is a physical representation of com-ponents and together form the quantum communicationpath.

Alice

Alice subsystem in DPS prepares a quantum statewith secret bit information encoded in phase. Fig. 4represents the Alice subsystem’s modeling architecture.A constant injection current to DFB laser modulegenerates a continuous optical signal as a photon source.Injection current can be altered to replicate a directmodulation scheme also. Inline polariser suppresses anorthogonal polarisation mode and results in a linearlypolarised signal. This reduces the dispersion due topolarisation while transmission.

At the next step, IM transforms the continuous signalinto a pulse train (Fig. 5) where, individual pulse repre-sent a coherent quantum state. IM is driven by VDC andVRF , governing the resultant extinction ratio and pulseprofile. Further, PM encodes random binary bit infor-mation into each coherent pulse precisely, 0(1) bit fromthe Simulink’s Bernoulli random number generates trans-lates into a phase shift of 0 (π). Finally, the power levelis reduced to quantum level using the VOA. For regularmonitoring of the average power and mean photon num-ber we have designed a power meter with live display ofphysical quantities.

Bob

In Fig. 6 we have presented the Bob’s subsystem, it re-ceives the weak coherent pulse train transmitted by Alice

FIG. 5. Modulated electric field signal from IM illustratingphoton wave packets.

FIG. 6. Block diagram from Simulink for Bob’s subsystem inDPS QKD

over an optical fibre. To extract the encoded bit infor-mation he passes it through a DLI. The demodulation ofthe signal is achieved by interference of adjacent pulses.In Fig. 7 we have shown the output waveforms from twoports of DLI as observed from the simulator. EventuallyDLI bifurcates the signal based on relative phase differ-ence. The optical noise like poor extinction ratio at DLI,improper phase encoding and non-idealities during sig-nal propagation results in a poor interference and can

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be quantified by the visibility described earlier. SPDs attwo ports of DLI detects single photons based upon theprobabilistic model.

FIG. 7. Optical signal at the two output ports of DLI.

V. CAPABILITIES

The simulation toolkit aims to better understand thefallout of device imperfection on a QKD protocol. Theperformance of a QKD protocol is generally gauged byparameters for example QBER, secure key rate and keyasymmetry. These parameters are calculated after eachcycle of communication between Alice and Bob. QBERis the most essential parameter, defined as the ratio offalse bit and total shared bits. Factors contributing toQBER include laser line width, extinction ration of IM,fiber losses, dark counts, optical visibility, phase jitterand phase modulation demodulation errors. By carefullycontrolling the imperfections, simulation is used to studyand analyze the underlying relations for the performanceparameters.

Simulation also provide visualisation tools for opticalsignal at various step. Parameter which are difficultto calculate theoretically such as signal to noise ratio,visibility, extinction ratio and polarisation extinctioncan be estimated from the toolkit for enhanced imple-mentations.

Theoretical studies on QKD often miss studying thefrequency spectrum of the transmitted signal. With sev-eral wavelength dependent eavesdropping attacks [28], itis crucial to analyse the spectrum of optical communi-cation. We have used Simulink’s spectrum analysis tool[29] to examine the intricate details of optical spectrum.Although, the optical source is said to produce singlemode optical signal, but due to external modulation andscattering, side bands are generated along with the car-rier frequency as shown in Fig. 8. The spectrum analyser

FIG. 8. Spectrum analysis of optical signal from Laser (yel-low) and IM (blue) showing side-bands in optical signal.

incorporated in the simulation toolkit targets this prob-lem and can be a useful tool for research activities.

VI. CONCLUSION

In this work, we present the framework for MATLABbased DPS QKD simulation toolkit, with a bottom-upapproach for modeling. This comprehensive approach ofmodeling optical and electrical components provides theflexibility to push it as a universal toolkit for all discretevariable based prepare-and-measure and entanglementbased QKD protocols. The model helps us in studyingthe impact of different characteristics of components andits impact on system QBER and system stability. Thepresent toolkit based on Simulink and MATLAB offersa user friendly tool leveraging MATALB’s extensivedevelopment and analysis tools to accurately model thephysical devices and their shortcomings. We report forthe first time modeling of physical components from thefirst principles. We note that earlier works [1, 3] posseslimited capabilities to analyze the optical path fromsource generation to signal detection, particularly, ana-lyzing the effects on signal propagation at intermediarypoints.

Future work involves modeling an extensive libraryof optical components and perform simulation of otherQKD protocols based on particularly, prepare-and-measure method. We also intend to simulate quantumattacks on QKD system. This will help us map the loop-holes of the system with quantum attacks. Thus, it willcreate better understanding of Eve’s strategy in real-lifeand effectiveness of proposed countermeasures. Our ob-jective is to build a universal standalone QKD simulationtoolkit to help enhance designing and implementing forbetter performance of QKD protocols in practice.

10

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