Wavelength Assignment in Hybrid Quantum-Classical Networks

0.4 MB·

by  Sima Bahrani

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Wavelength Assignment in Hybrid
Quantum-Classical Networks
Sima Bahrani1, Mohsen Razavi2, and Jawad A. Salehi1
1Electrical Engineering Department, Sharif University of Technology, Tehran, Iran.
2School of Electronic and Electrical Engineering, University of Leeds, Leeds, LS2 9JT, UK.
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Abstract—Inthispaper,weconsidertheproblemofwavelength their practical issues have been proposed in the literature
1
assignment in dense-wavelength-division-multiplexing (DWDM) [11]–[16]. For instance, one of the main advancements in
0
systemsthatintegratebothquantumandclassicalchannels.This
2 the implementation of QKD is the decoy-state technique,
requires the transmission of weak quantum key distribution
which relaxes the need for an ideal single-photon source in
n (QKD) signals alongside intense classical signals on the same
a fiber, where the former can be masked by the background QKD systems. On the other hand, there has been significant
J noise induced by the latter. Here, we investigate how optimal enhancementinreachandperformanceofpoint-to-pointQKD

8 wavelength assignment can mitigate this problem. We consider links. The successful demonstration of point-to-point QKD
2 differentDWDMstructuresandvarioussourcesofcrosstalkand over307km[17],andmeasurement-device-independentQKD
propose several near-optimal wavelength assignment methods
over 404 km [18], has proven the feasibility of running QKD
] that maximize the total secret key rate of the QKD channels.
h Our numerical results show that the optimum wavelength as- over long optical fibers, although at a very low rate. To cover
p signment pattern is commonly consisted of several interspersed longer distances, at a high key-exchange rate, and to support
- quantum and classical bands. Using our proposed techniques, multiple users, QKD must be implemented over large-scale
t
n the total secret key rate of quantum channels can substantially networks.
a be improved, as compared to conventional assignment methods,
The initial steps for the implementation of QKD in a
u in the noise dominated regimes. Alternatively, we can maximize
q the number of QKD users supported under certain key rate network setting have been carried out successfully [19]–
[ constraints. [22]. The European project SECOQC and the Tokyo QKD

network each demonstrated a small quantum network with
1 IndexTerms—Wavelengthassignment,Quantumkeydistribu-
v tion, Wavelength division multiplexing mesh topology. To extend these examples, the first generation
0 of QKD networks are expected to rely on the key exchange
7 in a trusted-note architecture. That is, in order to overcome
2 I. Introduction distancelimitationsofQKD,asetoftrustednodes,atthecore
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Quantum networks are no longer a physicist’s fantasy of the network, can be used to link the two end nodes. The
0
but the emerging reality of today’s complex communications most recent example of such networks is the developing link
.
1
world. With the first quantum satellite in orbit [1], to be fol- betweenBeijingandShanghaiwith32trustednodesalongthe
0
lowedbythelaunchofthe2000-km-longBeijing-to-Shanghai way.
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1 QKD network [2], quantum technologies have reached a new Regardless of the topology, one major requirement in the
: milestone in supporting multiple users at long distances. This widespreaddevelopmentofQKDnetworksistheirintegration
v
trend is boosted by various national and regional programmes with the existing fiber-optic classical networks. This is not
i
X in the UK, the European Union, the States, and far east limitedtothecurrentdevelopingQKDnetworks,butalsonext
r Asia, that aim at bringing the technology to the doorsteps of generationsofquantumnetworksshouldaddressthisissuedue
a
the end users. While the emerging quantum communications to cost efficiency considerations. In all such hybrid networks,
technologies would offer future-proof security for our data weak quantum signals should travel alongside intense clas-
exchange, they are not going to replace the vast investment sical ones. The latter, in this scenario, would produce some
in high data-rate communications. In fact, any commercially backgroundnoise,e.g.,Ramanscatteringandadjacentchannel
sensible solution for quantum networks would rely on its crosstalk, which will enter the quantum receivers.
integration with classical infrastructure [3]–[7]. In this paper, In order to reduce the background crosstalk in hybrid
we address one of the problems that arise in such integrated quantum-classical setups, some methods have been proposed.
networks.WelookataDWDMsetupinwhichmultipleQKD In [5], [6], filtering methods in frequency and time domains
channels are multiplexed with several data channels [8], [9], have been used to suppress the crosstalk. Another useful
and will investigate how wavelength assignment in such a approach is the control of launch power of data signals such
setup can affect the amount of crosstalk, and consequently that it satisfies the receiver sensitivity [5], [6]. Furthermore,
the performance of QKD channels. it has been shown in [8] that orthogonal frequency division
From the first proposed QKD protocols [10] up until now, multiplexing (OFDM) can effectively reduce the crosstalk
the field has seen considerable development and progress. using its optimal filtering.
Various QKD protocols and different techniques to address In this paper, we propose optimal wavelength assignment
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as an additional method of crosstalk reduction in a dense- data signals are transmitted via different fibers. This structure
wavelength-division-multiplexing (DWDM) link at the core is,forexample,usedin100Gsystems.Inthiscase,weassume
of a hybrid network. The problem of optimal wavelength that the classical and quantum signals are transmitted in the
assignment in integrated quantum-classical DWDM systems same direction on both fiber links. That is, in the forward
with one quantum and several classical channels has been link, QKD encoders are located on Alice’s side, whereas for
investigatedin[7].However,themoregeneralscenario,where the backward link, they are on Bob’s side. This assumption is
multiple quantum and multiple classical signals are to be based on the fact that the Raman noise generated in this case
transmitted, has not been fully studied yet. Considering the is smaller than the case of transmitting quantum and classical
shape of Raman spectrum, a conventional solution for this signals in the opposite directions [5], as we will explain later.
problem is the assignment of higher wavelengths to classical Letusintroducethenotationweusefortheemployedwave-
channels, and the lower wavelengths to quantum ones. In lengths in our hybrid link. The set of available wavelengths
[8], appropriate wavelength assignment under the constraint in the system is denoted by G = {λ ,...,λ }. We denote the
1 D
of having two separate quantum and classical bands has channel spacing of the DWDM system by ∆. Furthermore,
been investigated. However, in [9], it has been shown that the set of wavelengths assigned to forward and backward
optimal wavelength assignment does not necessarily follow classical channels are represented by A = {λ ,...,λ } and
A1 AN
this two-band form. Here, we investigate optimal wavelength B = {λ ,...,λ }, respectively. Note that sets A and B, in
B1 BN
assignment and propose several methods to approach it in both structures, could be overlapping. All classical signals
different DWDM setups. We also show that the optimal are assumed to have equal launch power, denoted by I. This
wavelength assignment can improve the performance of QKD launch power is assumed to be minimized, considering the
links effectively. receiver sensitivity, to meet a target bit error rate (BER). In
In our optimization problem, we consider two particular the case of quantum channels, we introduce two wavelength
scenarios. In the first scenario, our objective is to get the sets:U1 ={λq1,...,λqk}representsthechannelswhosequantum
maximumaggregatekeyrateoutofafixednumberofquantum signals travel from Alice to Bob, and U2 = {λqk+1,...,λqM}
channels in the presence of a number of classical channels. represents the channels whose quantum signals travel from
Thisscenarioisrelevantinthesettingsthatallgeneratedkeys BobtoAlice.Inthedual-fiberstructure,parameterk willthen
are to be consumed by our two end nodes. That is, the main representthenumberofquantumchannelsontheforwardlink.
point of multiplexing several QKD channels is to increase Note that, U and U may also be overlapping. In the full-
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the total key rate. Another foreseeable scenario is that each duplex system, k= M, and U would be an empty set.
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quantum channel represents a different user. In such a case, In this paper, we use the BB84 protocol with time-bin en-
we have to be able to guarantee a minimum key rate for all coding[23] forourQKDchannels; seeFig.2. Thismethodis
users. In such a setting, the optimized solution can maximize mainlysuitableforfiberchannels.Wealsousethedecoy-state
the number of users that can be supported. This may or may versionofefficientBB84[24]toallowforweaklaserpulsesto
not coincide with the total maximum key rate as we show in beusedatQKDencoders.Inthetime-binencoding,thequbits
this paper. areencodedonthephasedifferenceoftwoconsecutivepulses,
In the following, the hybrid quantum-classical DWDM r and s, generated by a Mach-Zehnder interferometer (MZI).
system is described in Sec. II. In Sec. III, we present analysis The encoding phase, φA, is chosen from one of the basis sets
of the secret key generation rate. The proposed wavelength {0,π}, for X basis, and {π/2,3π/2}, for Y basis, randomly. At
assignment methods are described in Sec. IV. We present our the QKD decoder, the decoding phase, φB, of Bob’s MZI is
numerical results in Sec. V, and conclude the paper in Sec. chosen randomly, from the set {0,π/2}. He then interferes the
VI. received r and s pulses by means of his MZI and infers the
transmitted qubit by measuring the output pulses.
The integration of classical and quantum signals on the
II. SystemDescription same fiber results in certain problems that may affect the
Consider a DWDM link in the backbone of a quantum- QKD operation. The main challenge is the background noise
classicalnetworkcarryingseveralclassicalandquantumchan- generated by the classical signals that reaches the quan-
nels. We refer to the two end nodes of the link by Alice and tum receivers. Two main sources of this crosstalk noise are
Bob.Weconsiderageneralscenariowhere M channelsareas- the inelastic interactions in an optical fiber and nonideal
signed to the QKD usage, while N forward classical channels performance of DWDM multiplexers and demultiplexers. In
(from Alice to Bob) and N backward classical channels (from particular, the Raman scattering and the power leakage from
Bob to Alice) transmit classical data. As for the fiber link, we adjacentchannelshavebeenshowntobethedominantsources
considertwocasesoffull-duplexoverasingle-modefiber,and of noise in such hybrid systems [25]. In the following, the
dual-fiber, as shown in Figs. 1(a) and (b), respectively. In the effect of these sources of background noise is described in
first case, we assume that each classical channel is equipped more detail. We assume that our QKD decoders are equipped
with optical circulators to enable the transmission of signals with narrow-band filters (NBF) to reduce such noises.
in both directions on the same wavelength. This structure
describes some of the existing, and probably future high- A. Sources of crosstalk noise
capacity, full-duplex DWDM systems with both classical and 1) Ramannoise: Ramanscatteringoccursduetotheinelas-
quantumlinks.InthesetupofFig.1(b),forwardandbackward tic photon-phonon interactions in an optical fiber. Because of
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Fig. 3: Measured Raman cross section for a laser centered at 1550 nm in a
standardsinglemodefiber[25].
dual-fiber case, if m≤k, then λ =λ , otherwise λ =λ .
fn An fn Bn
Therewouldbenobackwardclassicalchannelinthedual-fiber
case, hence λ is not defined. With this notation, the Raman
bn
noisepowercorrespondingtoλ andλ ,respectively,forthe
fn bn
mth quantum channel, is given by [5], [25]:
IFR = Ie−αLLβ(λ ,λ )∆λ (1)
nm fn qm
and, in the full-duplex case,
(1−e−2αL)
IBR = I β(λ ,λ )∆λ, (2)
Fig.1:HybridDWDMlinkwithmultiplequantumandclassicalchannels:(a) nm 2α bn qm
Afull-duplexDWDMsystemoverasingle-modefiber.Eachclassicalchannel
where β(λ ,λ ) and β(λ ,λ ) are the Raman cross section
is equipped with optical circulators to enable the transmission of signals in fn qm bn qm
both directions on the same wavelength. Quantum signals travel from Alice (per fiber length and bandwidth) for forward Raman (FR)
to Bob. (b) A dual-fiber DWDM system, where forward and backward data and backward Raman (BR) scattering, respectively. For our
signalsaretransmittedviadifferentfibers.Quantumsignalstravelalongthe dual-fiber system, IBR = 0. In the above equations, α, L, and
samedirectionastheclassicalones. nm
∆λ are, respectively, the fiber attenuation coefficient, the fiber
length, and the bandwidth of the NBF in wavelength unit.
Figure 3 shows measured Raman cross section, β(1550nm,λ),
for a pump laser centered at 1550 nm in a standard single
mode fiber [25]. As can be seen, the Raman cross section
is slightly higher for wavelengths longer (Stokes regime) than
Fig.2:Phaseencoded(time-bin)QKD.Aliceencodesherkeybitsbychoosing
aphasevalue,φA,fromoneofthebases{0,π}and{π/2,3π/2}.Eachoptical 1550nmthantheonesbelow(anti-Stokesregime)it.Thathas
pulsepassesthroughtheMZIandproducestwooutputpulseswiththerelative resultedinaperceptionthatperhapsthebestwayofallocating
phaseφA.OntheBobside,anotherMZIisusedtorecombinerandsmodes, wavelengths to quantum and classical signals is to use the
followedbyphotodetection.
higher wavelengths for data channels and the lower ones for
quantum.Werefertothissolutionastheconventionalmethod,
and will investigate how far or close it is to the optimum
its wide spectrum, Raman noise can easily leak into quantum
assignment we find in this paper.
channels.TheRamannoiseco-propagatingwiththedatasignal
TheaveragephotoncountofRamannoisecorrespondingto
is referred to by forward scattering, whereas the Raman noise
λ and λ , at the detectors of the mth quantum receiver is,
traveling in the opposite direction is known as backward fn bn
respectively, given by
scattering.BackwardRamanscatteringisknowntobestronger
than the forward one for typical fiber lengths [5]. pFR = IFRλ T η /(2hc)=Cfλ β(λ ,λ ), (3)
nm nm qm d d qm fn qm
IntheDWDMsystemsshowninFig.1,eachclassicalsignal
and
induces a certain amount of Raman crosstalk noise at the
receiver of each quantum link. Consider the quantum channel pBnmR = InBmRλqmTdηd/(2hc)=Cbλqmβ(λbn,λqm), (4)
with wavelength λ for m = 1,...,M, and let us calculate
qm where Cf and Cb are wavelength-independent parameters,
the amount of Raman noise induced by the nth, n=1,...,N,
given by
data channel. The data channel can include signals traveling
Cf = Ie−αLL∆λT η /(2hc), (5)
in the same direction as the quantum signals in channel m, d d
or the opposite direction. Let us denote the wavelength of the
and
dfourpmleexr bsytruλcftnu,raen,dwtehehlaavtteerλbfny=λbnλ.AFnoarnidnstλabnnce=, fλoBrnt.hIenfuthlle- Cb = I(1−2eα−2αL)∆λTdηd/(2hc). (6)
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In the above equations, h and c are, respectively, the Planck’s the average number of photons for the main signal state, in
constantandthespeedoflightinthevacuum,T isdetectors’ the employed efficient decoy-state protocol, by µ, the secret
d
gate interval, and η denotes their quantum efficiency. Note keyratepertransmittedpulseintheQKDchannel,inthelimit
d
that, in the dual-fiber case, pBR =0. of an infinitely long key, is lower bounded by max[0,P(Y )],
nm 0
2) Adjacent channel crosstalk: The DWDM multiplexers where [26]
anddemultiplexerscanalsointroducesomecrosstalknoisebe-
causeoftheirnonidealoperation.Insufficientchannelisolation P(Y0)= Q1(1−h(e1))− fQµh(Eµ). (11)
cancausesomepowerleakagefromdatasignalscopropagating
with quantum signals. Furthermore, there may be some back Here, h(p) = −plog p − (1 − p)log (1 − p) is the Shannon
2 2
reflection from data signals transmitted in the opposite direc- binary entropy function and f denotes the error correction
tiontoquantumsignals,intothequantumreceivers.Ingeneral, inefficiency.In(11),Qµ,Eµ,Q1,ande1,respectively,represent
the power leakage from a classical channel into the two the overall gain, the quantum BER (QBER), the gain of the
immediately adjacent channels is higher than that of the non- singlephotonstate,andtheerrorrateofthesinglephotonstate.
adjacent ones. Moreover, with the use of appropriate NBFs at The overall gain, Qµ, and the QBER, Eµ, are, respectively,
the quantum receivers, the nonadjacent channel crosstalk can given by
be suppressed effectively. In this paper, we only then consider Q =1−(1−Y )e−ηµ (12)
µ 0
theadjacentchannelcrosstalk.Wedenotetheadjacentchannel
isolation of the DWDM module in dB by γa. Furthermore, and
the average value of the transfer function of the NBF at the
E =(Y /2+e (1−e−ηµ))/Q , (13)
passbandoftheadjacentchannelsisdenotedbythecoefficient µ 0 d µ
g . Then, the power leakage corresponding to λ and mth
a fn whereas the gain and the error rate of the single photon state
quantum channel can be expressed as
are, respectively, as follows:
(cid:40)
InFmC = gaIe−αL100(−γa/10) ||λλffnn −−λλqqmm||>=∆∆ . (7) Q1 =Y1µe−µ (14)
Similarly, in the full-duplex case, the crosstalk power corre-
and
sponding to λ at the mth quantum receiver is given by
bn e =(Y /2+e η)/Y . (15)
(cid:40) 1 0 d 1
IBC = gaI10(−χa/10) |λbn −λqm|=∆ , (8)
nm 0 |λbn −λqm|>∆ Here,Y0 representstheprobabilityofhavingdetectorclicksat
Bob’sendwithouttransmittinganyphotons,andY istheyield
whereχ representsthedirectivityoftheDWDMmultiplexer. 1
a
ofasingle-photonstate.Furthermore,theparametere denotes
The indices “FC” and “BC” represent “Forward Crosstalk” d
the misalignment error. For time-bin encoding, e models the
and“BackwardCrosstalk”,respectively.Inthedual-fibercase, d
IBC = 0. The average photon count corresponding to IFC and stability of the relative phase between r and s pulses. The
nm nm
IBC is, respectively, obtained by parameter η represents the total transmissivity of the link, and
nm
is given by
pFnmC = InFmCλqmTdηd/(2hc), (9) η= 1η 10−αL/10. (16)
d
and 2
pBnmC = InBmCλqmTdηd/(2hc). (10) The coefficient 1/2 represents the loss associated with the
decoder setup in Fig. 2. Denoting the repetition period of the
In our analysis, we neglect any crosstalk noise from quan-
QKD system by T , the secret key generation rate of the mth
tum channels on each other. One possible source of such a s
QKD channel is given by
noise can be the synchronization signals sent by the QKD
system. The rate at which such signals are sent is often very
low and can be neglected. Alternatively, one/some of the Rm =max[0,P(Y0)/Ts], (17)
classical channels can be used for time synchronization as
where
wellasotherclassicaltasksthatQKDsystemsrequire.Finally,
we can also use time-multiplexing techniques to separate the Y0 =1−(1−(pdc+pm))2. (18)
quantumandsynchronizationsignalsonQKDchannels.Inthe
latter case, we assume that the time synchronization signals In the above equation, pdc = γdcTd, where γdc denotes the
are transmitted simultaneously on all QKD channels to avoid dark count rate of a single-photon detector and pm denotes
any crosstalk noise on QKD signals. the crosstalk photon count on the mth quantum channel, given
by
III. KeyRateAnalysis (cid:88)N
p = (pFR+pBR+pFC+pBC). (19)
In this section, the secret key generation rate of the QKD m nm nm nm nm
n=1
linksintheDWDMsystemsofFig.1isanalyzed.Weconsider
the mth QKD channel, as an example, and investigate its As explained in Sec. II, in the DWDM structure of Fig. 1(b),
performance in the presence of classical channels. Denoting pBR and pBC are both zero.
nm nm
5
x 106 TABLEI:NominalvaluesusedforQKDsystemparameters.
16
14 Parameter Value
Averagenumberofphotonspersignalpulse,µ 0.48
Secret key rate (bit/s)1146802 LEaRMsrereoricsTrepaQiilcuvmiolugeseranrernmegdtrcaaueettrmpniekoetnticEpintoriffitinoeouebrncnvffiaiteaibnrcnli,aitlceetiTyetnry,dvc,aγyel,d,dcfTs 1E210-1500.7.0001.31n6pp5sss−1
2
0
0 1 2 3 4 5 6 7
Crosstalk photon count x 10−4 where p denotes the crosstalk photon count that results in
th
Fig. 4: Secret key generation rate versus crosstalk photon count (blue solid R = R and is dependent on the fiber length and the QKD
m th
curve)anditslinearapproximation(reddashedcurve).
system parameters. In principle, the above formulation will
lead to a near-optimal solution.
In the following, we examine this optimization problem
IV. OptimalWavelengthAssignment
for each DWDM structure shown in Fig. 1. Note that if the
Wavelengthassignmentinoursettingcansignificantlyaffect NBFs at the quantum receivers suppress the adjacent channel
the performance of QKD links. From Fig. 3 and equations crosstalk effectively, the Raman noise will be the dominant
(1), (2), (7), and (8), we can infer that the crosstalk noise source of crosstalk noise. Otherwise, the adjacent channel
induced by a classical channel onto a quantum one depends crosstalk should also be taken into account. In the following
on the difference between their corresponding wavelengths. subsections,weusethecriteriain(21)tofindthenear-optimal
Therefore, the key rate of QKD channels is dependent on wavelength assignment in each case.
the location of quantum and classical channels, with respect
to each other, in the wavelength grid. In this section, we
A. Full-duplex system
investigatetheoptimalwavelengthassignmentthatmaximizes
the total key rate of QKD channels, in the DWDM systems In this section, we examine the problem of wavelength
shown in Fig. 1, under a minimum key rate per channel assignment for the DWDM system shown in Fig. 1(a). In this
constraint. To this end, we define an optimization problem case, using (19), the cost function in (21) can be rewritten as
that aims to find the sets U , U , A, and B, such that the total
1 2
keyrateofQKDchannelsismaximized.Thisproblemcanbe (cid:88)N (cid:88)M (cid:88)N (cid:88)M
C =Cb λ β(λ ,λ )+Cf λ β(λ ,λ )
formulated as qm bn qm qm fn qm
n=1m=1 n=1m=1
(cid:88)M (cid:88)N (cid:88)M
max R , s.t. R >R , m=1,...,M, (20) + (pFC+pBC). (22)
A,B,U1,U2⊂Gm=1 m m th n=1m=1 nm nm
where R denotes the key rate of the mth quantum channel 1) Raman-noise-only scenario: In this case, (22) is simpli-
m
given by (17), and R is the minimum required value for fied to
th
Rm. The parameter Rth has been defined to take into account C =CbZ1+CfZ2, (23)
quality-of-service considerations for QKD links. In a multi-
user setup, where a minimum key rate needs to be guaranteed where
for each QKD user, R plays the role of this minimum rate. (cid:88)N (cid:88)M
th Z = λ β(λ ,λ ), (24)
Ifweareonlyinterestedinmaximizingthetotalkeyratewith 1 qm bn qm
n=1m=1
no constraints on individual key rates, we can simply use a
negative value for R in our formulation. In the following and
th
sections, we consider both scenarios. (cid:88)N (cid:88)M
Z = λ β(λ ,λ ). (25)
In order to solve the optimization problem in (20), one 2 qm fn qm
n=1m=1
can simplify it by investigating the dependence of R on
m
p . According to the key rate analysis presented in Sec. III, Since Cb and Cf are wavelength-independent, Z and Z
m 1 2
it can be concluded that R is a descending function of should be minimized. In the following, we show that in the
m
p . In Appendix A, we have shown that this curve can optimum solution, the wavelengths allocated to forward and
m
be approximated with reasonable accuracy by a line. As an backward classical channels are identical.
example, Fig. 4 shows the secret key rate of a single QKD Lemma 1: The expression in (23) is minimized, if A= B.
channel as a function of the total crosstalk photon count, and Proof 1: Suppose in the optimum solution A(cid:44) B. Without
its linar approximation, for the system parameters outlined in loss of generality, assume Z < Z . Then, if one uses the set
1 2
Table I and a fiber length of 45 km. With this approximation, A for the backward channels as well, the resulting value for
the optimization problem in (20) can be expressed as Z wouldbelower,whichimpliesthat A= Bfortheoptimum
2
setting.
(cid:88)M
min p , s.t. p < p , m=1,...,M, (21) Using the above result, our optimization problem reduces
m m th
A,B,U1,U2⊂Gm=1 tothecasewherealldatachannelsarebidirectional.Withthis
6
constraint, the optimization problem can be expressed as 12, d is the sorted version of c in the ascending order, and
index is the vector of corresponding indexes. This matrix-
(cid:88)N (cid:88)M (cid:88)N
based algorithm is also applicable to other scenarios we have
min λ β(λ ,λ ),s.t. λ β(λ ,λ )< X ,
A,U1,U2∈Gn=1m=1 qm fn qm n=1 qm fn qm th consider in our work. In each case, we just need to find the
(26) relevant matrix P and apply Algorithm 1 to it.
where Xth = pth/(Cf +Cb). 2) Raman + Adjacent channel crosstalk scenario: In this
To solve the optimization problem in (26), we propose a case,theoptimumwavelengthpatternisdeterminedbysolving
matrix-based algorithm. We define a D× D matrix, P, with the optimization problem in (22). Note that pFC and pBC
nm nm
elements given by are only present for |λ − λ | = ∆ and |λ − λ | = ∆,
fn qm bn qm
(cid:40) respectively. Hence, we use the results of the previous case
λ β(λ,λ ) i(cid:44) j
P = j i j . (27) and propose a suboptimal wavelength assignment method that
ij ∞ i= j
assumes bidirectional data channels. With this constraint, (22)
reduces to
Algorithm 1 Optimization Algorithm (cid:88)N (cid:88)M (cid:110) (cid:111)
1: if (cid:16)ND(cid:17)≤(cid:16)MD(cid:17) then C = n=1m=1 (Cb+Cf)λqmβ(λfn,λqm)+pBnmC+pFnmC . (28)
2: P2 =P
This problem can also be solved by Algorithm 1. In this case,
3: A=matrixof N−combinationsfor{1...D}
the elements of matrix P are given by
4: else
56:: PA2==mPaTtrixof M−combinationsfor{1...D} Pij =(cid:40) (Cf +Cb)λjβ(λ∞i,λj)+pBijC+pFijC ii(cid:44)= jj , (29)
7: end if
8: t=1000 and Xth = pth.
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
9: for i=1...min D , D do
N M
10: b=A(i,:) B. Dual-fiber system
11: c=(cid:80)j∈bP2(j,1: D) Now let us consider the dual-fiber system in Fig. 1(b). In
12: [d,index]=sort(c)
(cid:16) (cid:17) (cid:16) (cid:17) this scenario, the optimization problem in (21) can be split
13: if D ≤ D then
14: Ns=(cid:80)MjM=1dj itnhteontuwmobperroobflecmlass,siocnaelcfhoarnenaeclhspeemrpfilboyeredisfifibxeerd. ItnotNhibsuctawsee,
15: if s<t and d(M)< Xth then have to decide how many, out of M, quantum channels need
16: t= s tobeallocatedtoeachfiber.Suppose M =1.Then,itdoesnot
17: outputq =index(1: M) matter which fiber we use for the QKD channel, but we need
18: outputc =b to find the optimum wavelength assignment that minimizes
19: end if the crosstalk noise. This way we find λ . Now, if M =2, we
20: else q1
2221:: ls==(cid:80)(cid:80)jNj∈=in1ddexj(1:N)P2(b, j) cawanhndeurtehseebotthothetasQlaKmkeDeywcrhaatavenenliesenlesgxtahpreeacsotsenidgtnhtmoeesbnaetmbheuigtfihobenerr.tthhWeaenotchtahenerkficebaeseper
23: if s<t and max(l)< Xth then adding QKD channels to the game, but it can be seen that the
24: t=s optimum assignment should have k = (cid:98)M/2(cid:99) QKD channels
25: outputq =b on one fiber and M−k channels on the other. Now that we
26: outputc =index(1: N) have a fixed number of QKD channels on each link, we can
27: end if
solvethetwooptimizationproblem,correspondingtoforward
28: end if
and backward links, separately. Using (19), the cost function
29: end for
in (21) for each optimization problem is given by
The elements of P are defined based on the summands in (cid:88)N (cid:88) (cid:88)N (cid:88)
C =Cf λ β(λ ,λ )+ pFC, (30)
(26). Pij represents the Raman cross section corresponding to qm fn qm nm
λ, as the data channel wavelength, and λ , as the quantum n=1m∈S n=1m∈S
i j
channel wavelength, multiplied by λ . Since classical and where S = {1,...,k} for the forward fiber link, and S =
j
quantumchannelshavedifferentwavelengths,wehavechosen {k+1,...,M} for the backward one. Note that if M is an even
P = ∞ for i = j. The optimization problem in (26) can be number, k = M/2. In this case, the two optimization prob-
ij
interpretedasfindingN rowsand M columnsofmatrixPsuch lems are identical and achieve similar wavelength assignment
that the summation of elements at the intersection of these patterns.
rows and columns is minimum (the diagonal elements of P 1) Raman-noise-only scenario: In this case, the cost func-
will automatically be excluded because of their infinitely high tion in (30) reduces to
value), and the constraint in (26) is satisfied. The proposed
(cid:88)N (cid:88)
optimization algorithm is presented in Algorithm 1, which C = λ β(λ ,λ ). (31)
is self explanatory. It should only be mentioned that in line qm fn qm
n=1m∈S
7
TABLEII:NominalvaluesusedfortheDWDMsystemparameters.
Parameter Value
Channelspacing 200GHz
AdjacentchannelisolationofDWDMmodule 30dB
DirectivityofDWDMmodule 50dB
BandwidthofNBF 15,125GHz
Comparing(31)and(26),itisconcludedthatthisoptimization
problem can be solved by Algorithm 1 for M = k (and k+1
foroddvaluesof M),withthematrixPdescribedin(27),and
the threshold X = p /Cf.
th th
2) Raman + Adjacent channel crosstalk scenario: In this
case, the optimization problem in (30) should be solved. Here
again, Algorithm 1, for M = k (and k+1 for odd values of
M), at X = p , can be used with matrix P defined as
th th
(cid:40) Cfλ β(λ,λ )+pFC i(cid:44) j
P = j i j ij . (32)
ij ∞ i= j
V. NumericalResults Fig.5:RateEnhancementfor(a)L=45km(b)L=65kmforthefull-duplex
systematRth=0.
In this section, the proposed wavelength assignment meth-
ods are investigated in more detail. Our example DWDM
system uses the wavelength grid ranging from 1530 nm where R and R are the total key rate obtained by the
pr co
to 1565 nm in the C-band. The nominal values for QKD proposed and the conventional methods, respectively.
systems are listed in Table I, and other system parameters are First, we consider the case of “Raman noise only” for
summerized in Table II. These parameters are chosen based the full-duplex system. We choose R = 0, i.e, all quantum
th
on certain practical considerations. We assume that, in the channels are required to have a positive key rate. The rate
full-duplex DWDM system, the classical channels use on-off enhancement parameter, RE, for different values of M and N,
keying with the data rate of 10 GHz. The launch power of is shown in Figs. 5(a) and (b), for L=45 km and L=65 km,
the data laser is controlled by the receiver sensitivity, which respectively. It can be seen that, the proposed method can
is assumed to be −28 dBm, corresponding to a bit error rate improve the total key rate significantly, especially for large
of10−12.Asforthedual-fiberstructure,weassumethat100G N and small M, and if we are in a region that the system is
coherent systems are used in the data links. The power of sensitive to the amount of the background noise.
the received classical signal in both structures is chosen to In order to further investigate the performance of our
be −25 dBm. We consider different cases of “Raman noise proposed methods, we define another measure, denoted by
only” and “Raman + Adjacent channel crosstalk”, based on N , as the maximum possible number of classical channels
max
the bandwidth of the NBF used, for full-duplex and dual-fiber that can be integrated with M quantum channels such that all
DWDM systems. We consider two cases for the bandwidth of of them have a positive key rate. We compare this parameter
theNBFatthequantumreceivers:15GHz,and125GHz.We fortheproposedandtheconventionalmethods.Ournumerical
assume that in the first case, the adjacent channel crosstalk resultsshowthat,dependingonthefiberlength,N canoften
max
is suppressed effectively so that it can be neglected. As for be improved by one or two channels. This means that by the
the 125 GHz NBF, we assume that a Gaussian shaped filter use of optimal wavelength assignment higher data traffic can
is used, which causes an attenuation of about 16 dB at the be supported.
passband of adjacent channels. We have also considered other cases of “Raman noise
In the following, our numerical results are presented. We only” for dual-fiber systems and “Raman + Adjacent channel
have investigated the performance of proposed wavelength crosstalk”forfull-duplexanddual-fiberDWDMsystems.Our
assignment methods in terms of key rate enhancement and numerical results show that the conclusions drawn in this
optimality. Furthermore, the wavelength patterns obtained by section can be extended to these cases as well.
this methods are examined.
B. Near-optimal Wavelength Pattern
A. Rate Enhancement
Inthissection,thewavelengthassignmentpatternsobtained
In this section, we compare our proposed methods with
by the proposed near-optimal methods are investigated. We
the conventional approach of assigning the lower part of the assume that R < 0, which corresponds to the case that the
th
wavelength grid to quantum and the longer wavelengths to
total key rate is maximized with no constraint on channels
classical channels. We define a rate enhancement measure,
individual key rates. We particularly look at the cases where
denoted by RE, as follows:
substantialgaincanbeachievedbyoptimizingthewavelength
R −R assignment, i.e., when a small number of quantum channels
RE = prR co ×100, (33) are present. Here, we choose M = 1 and M = 3 and
co
8
M = 3 is shown in Fig. 7. As can be seen, the wavelength
assignmentpatternsaredifferentfromthe”Ramannoiseonly”
case.Withthechosensystemparameters,theadjacentchannel
crosstalk can be more than the Raman noise. Hence, the
wavelength assignment method avoids the allocation of a
quantumandaclassicalchanneltoadjacentwavelengths.This
would result, especially when N is large, in the optimum
solution converging to the conventional one as can be seen
for N >16 in Fig. 7. More generally, our results imply that if
the adjacent channel crosstalk is the dominant source of noise
and R < 0, the wavelength assignment pattern converges to
th
the conventional method solution, when the capacity of the
system is nearly complete.
It is interesting to study the dependence of the optimal
wavelengthpatternonthetransmissiondistance,or,effectively,
the channel loss. In our formulation, the key parameters
that are distance dependent are Cf and Cb, which affect
the cost function, as well as R , whose value must satisfy
m
our optimisation constraint R > R . In all cases that there
m th
are no constraints on the key rate, i.e., when R < 0, the
th
latter dependence on the distance does not matter. As for the
former, it turns out that in the Raman-noise-only scenarios,
the cost function can be made independent of the fiber length
by eliminating Cf and Cb as in (26) and (31). Hence, for
Fig. 6: Proposed wavelength assignment patterns in a 200 GHz full-duplex
R < 0, the achieved wavelength assignment patterns in the
systeminthe“Ramannoiseonly”casefor(a) M=1and(b) M=3.Each th
row depicts the optimum location of quantum and classical channels in the Raman-noise-only scenarios, e.g. the result in Fig. 6, would
wavelength grid, where ∗ represents a classical channel and ◦ represents a be independent of transmission distance. In all other cases,
quantumone.
the optimal wavelength pattern can, in principle, depend on
the channel loss. However, in the particular cases we have
considered for our numerical results, e.g. in Fig. 7, we have
examine the wavelength patterns, first, in the Raman-noise verified that up to L = 120 km the obtained wavelength
onlycase.Figures6(a)and(b)depicttheproposedwavelength patterns remain the same.
assignment for, respectively, M = 1 and M = 3 quantum
channels and different values of classical channels, N, for a
200 GHz full-duplex DWDM system. Note that these figures C. Optimality
also provide the proposed wavelength assignment for each
individual fiber in a dual-fiber system for, respectively, M =2
and M = 6 quantum channels. In each figure, each row 18
shows the proposed locations for the quantum, represented 16
bwagpnyirivodtehp“no◦ctshl”aen,esdusacminocpdnbaavletctereblnaratonsinsfodinfcsocaalrllaamtsceshateihccatheahnlontdeNwclsoho,faisnrehennanpedvorlsiestn.sgoenfAnetwtcsteheodsecssabaewnyrpialavy“bre∗ael”tcee,onsqmefgoeutprhnaa,netgtiautbrchmilhdee mber of classical channels11168024
even if we only have one quantum channel. This pattern is, Nu 4
2
in general, consisted of multiple interspersed quantum and
0
classical bands. For example, in Fig. 6(a), we can see that 1530 1535 1540 1545 1550 1555 1560 1565
Wavelength (nm)
the QKD channel is between two classical bands for N ≥ 3.
Fig. 7: Proposed wavelength assignment patterns in a 200 GHz full-duplex
This result can be explained by referring to Fig. 3. According DWDMsysteminthe“Raman+Adjacentchannelcrosstalk”caseforM=3
to this figure, Raman noise is smallest at the nearest regions andL=50km.
onthetwosidesofthepumplaserwavelength.IntheRaman-
noiseonlyscenario,ouroptimumwavelengthassignment,then In this section, we compare the proposed near-optimal
benefits from this low-noise regions to improve performance. method with the optimum approach that maximizes the total
The white space in Fig. 6 would also resemble the Raman key rate. First, we choose R <0. In this case, our numerical
th
spectrum shape. results show that, for low fiber lengths, e.g., L = 45 km,
Next, we investigate the wavelength assignment patterns the proposed methods generally lead to the same wavelength
in the “Raman + Adjacent channel crosstalk” case for the assignment patterns, hence, the same total key rates, that
full-duplex system. The proposed wavelength assignment for the optimum solution offers. There are a few exceptions,
9
however,eveninthosefewcases,thepercentageoftherelative VI. Conclusion
difference of the total key rate is below 10−3%.
In this paper, we considered the problem of wavelength
As the fiber length increases, and for sufficiently large val- assignment in a hybrid quantum-classical network. We ex-
uesof M and N,theproposedandtheoptimumsolutionsmay ploited the reconfigurability of optical networks to improve
lead to different total key rates. The reason is mainly because the performance of QKD links by appropriate wavelength
of the linear approximation we use to convert (maximizing) allocation. We considered two system setups, namely, full-
the key rate criterion into (minimizing) the background noise. duplex and dual-fiber, and investigated the optimum wave-
At large distances, QBER for some channels may be very lengthassignmentmethodthatmaximizesthetotalkeyrateof
large resulting in zero key rates for them. In fact, when QKDchannelsineachcase.Inouranalysis,twomainsources
M is sufficiently large, the optimum solution may include ofcrosstalk,Ramannoiseandadjacentchannelcrosstalkwere
some quantum channels with zero key rates. Our noise-based considered.Weusedlinearapproximationstoproposeefficient
solution will instead try to distribute the noise almost equally near-optimal wavelength assignment methods for these cases.
among all channels, which, in certain cases, would result in Furthermore,varioussimulationsandnumericalinvestigations
lower total key rates than that of the optimum solution. As an were carried out to examine the proposed methods. Our
example, Fig. 8 shows the total key rate for N = 8 classical numerical result showed that the conventional wavelength
channels at L = 62 km for different values of M. As can assignment method of two separate quantum and classical
be seen, when R < 0, for M > 5 the optimum solution bandswouldnotnecessarilybetheoptimumsolution.Instead,
th
remains the same (new QKD channels will have zero key theoptimalwavelengthallocationpatterncouldincludeseveral
rates), while our proposed method achieves a lower total key quantum and classical bands interspersed among each other.
rate (by supporting more users with positive key rates). It We showed that in most cases our proposed wavelength
can be concluded that, for each N and L, there is a specific assignment methods were nearly identical to the optimum
valueof M forwhichthemaximumtotalkeyrateisachieved. method. Furthermore, we showed that our proposed method
Increasing M beyondthatvaluedoesnotincreasethetotalkey could significantly improve the total key rate of the DWDM
rateandonlyquantumchannelswithzerokeyratesareadded. system especially in the noise dominated regimes. We found
This is an important observation if maximizing the total key that for any given number of classical channels, there would
rate is the key objective of the operator. exist an optimum number of quantum channels for which the
total key rate would be maximum.
Next, we consider the case of R = 0, i.e., when we need
th
to guarantee a positive key rate for each quantum channel.
AppendixA
From Fig. 8, it can be seen that, the proposed method has a
LinearApproximationofSecretKeyRate
reasonable accuracy in this case, since the linear approxima-
tion method used is more accurate in this positive-key region. Inthisappendix,weanalysethesecretkeyrateofaquantum
Here,weagainseethatthepriceofsupportingalargernumber channelandderivealinearapproximationtoit.From(11),we
of quantum users could be a lower total key rate. It depends can write
on which criterion, number of users versus total key rate, is P(Y0)= Q1x1− fQµx2, (34)
moreimportantfortheoperatorinordertodecideontheright
where
workingpoint.Ouranalysis,inanycase,enablestheoperators
to plan wisely for their resource allocation. x =1−h(e ),
1 1
x =h(E ). (35)
2 µ
If we denote the minimum value of Y that corresponds to
4.5x 106 zero key rate by Y(z), then we have e 0∈ [e ,e(z)] and E ∈
0 1 d 1 µ
4 [ed,Eµ(z)], where e(1z) and Eµ(z) are obtained from (15) and (13),
respectively, for Y = Y(z). Since e(z) and E(z) are typically
3.5 0 0 1 µ
small, we can have approximations of the curves x and x
1 2
bit/s) 3 in their corresponding range, under the assumption of e(1z) <
Total key rate (12..255 a0rep.5sppreaocnxtdiimveEalµt(yzi,)onx<1s=i0n.a5(e.31W4+)e,bwaasensducmaxn2e=ctohknaEtclµuth+deej.rPeSs(uYubl0ts)itni=gtutUliinnYge0st+haerVsee,,
where
1 Optimal, Rth<0
Near−optimal, Rth<0 a k
Optimal, Rth=0 U = µe−µ+b(1−η)µe−µ− f − fje−ηµ, (36)
0.5
Near−optimal, Rth=0 2 2
0 and
0 2 4 6 8 10 12 14
Number of quantum channels
V =aηe µe−µ+bηµe−µ−kfe (1−e−ηµ)− fj(1−e−ηµ). (37)
d d
Fig.8:Comparisonbetweenthetotalkeyrateoftheoptimalandnear-optimal
methodsforN=8andL=62km. On the other hand, (18) can be approximated to
Y (cid:39)2p +2p , (38)
0 dc m
10
under the assumption of (pdc + pm) (cid:28) 1. Hence, the curve [26] M.Razavi,“Multiple-accessquantumkeydistributionnetworks,”IEEE
P(Y ) as a function of p can be approximated by the line Trans.Commun.,vol.60,no.10,pp.3071–3079,2012.
0 m
P(Y )=2Up +2Up +V.
0 m dc
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