Orthogonal Frequency Division Multiplexed Quantum Key Distribution

0.39 MB·English

by  Sima Bahrani

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Orthogonal Frequency Division Multiplexed
Quantum Key Distribution
Sima Bahrani, Student Member, IEEE, Mohsen Razavi, and Jawad A. Salehi, Fellow, IEEE
Trustednodes
Abstract—We propose orthogonal frequency division multi-
plexing(OFDM),asaspectrallyefficientmultiplexingtechnique, Central Central
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for quantum key distribution (QKD) at the core of trusted- office office
1
node quantum networks. Two main schemes are proposed and Accessnetwork Corenetwork Enduser
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2 analyzed indetail,consideringsystemimperfections,specifically, Fig. 1: Trusted-node architecture for emerging quantum networks. The end
time misalignment issues. It turns out that while multiple usersmaybeconnected tothecorenetworkviapassiveoptical networks. In
v service providerscansharethenetwork infrastructureusingthe ordertogenerateasecretkeybetweentwoendusers,onemustfirstgenerate
o proposed multiplexing techniques, no gain in the total secret a key between any two neighboring nodes along their connecting path. The
N key generation rate is obtained if one uses conventional passive key generated between the enduser andits corresponding central office can
all-optical OFDM decoders. To achieve a linear increase in the thenbeencryptedandsecurelyrelayednodebynodeuntilitreachestheother

3 key rate with the number of channels, an alternative active party.Notethattheinternal links(thicker lines)mustcarryahighertraffic.
2 setupforOFDMdecodingisproposed,whichemploysanoptical
switch in addition to conventional passive circuits. We show

] that by usingour proposed decoder the bandwidthutilization is multiplexing (OFDM) techniques [8]–[13] and partly modify
h
p considerably improved as compared to conventional wavelength their setups in order to obtain spectrally efficient high-rate
division multiplexing techniques.
- OFDM-QKD schemes.
t
n IndexTerms—Quantumkeydistribution,orthogonalfrequency QKD enables secure key exchange without relying on
a division multiplexing, quantum networks computational complexity. This is in contrast with existing
u
techniques for key exchange, e.g., the RSA protocol [14],
q
I. Introduction whose security is at risk with the advancement of technology
[
QUANTUM communications has entered a new phase [15]. In that sense, QKD provides a future-proof method of
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in its development targeting new markets and aiming secure communications. The first proposed QKD protocol
v
7 at widespread use and adoption in different scenarios. With by Bennett and Brassard in 1984 (BB84) [7] relied on the
1 the successful demonstration of SECOQC [1] and Tokyo [2] polarization encoding of single photons. Since then new pro-
5 quantum key distribution (QKD) networks, we are now at a tocolsandencodingschemeshaveemergedandQKDhasseen
5 stage to develop many-user quantum networks [3]–[6]. The fielddemonstrationsalongwithconventionaltelecomchannels
0
reach of conventional QKD links is, nevertheless, limited as [2], [16]–[18]. Recent demonstrations cover distances over
.
1 they rely on low-power signals, e.g., single photons [7]. The 250km[19]andwithnearly50users.ThenextstepforQKD
0 initial solution perceived for the first generation of quantum development will focus on extending the reach of the system
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networksrelies on a trusted set of nodes, in a mesh topology, and the number of users QKD networks can support.
1
: at the core network. Such nodes enable secure key exchange Quantumnetworksarefacingseveralchallengesbeforetheir
v betweenanytworemoteusersviaacascadeofkeyexchanges full implementation. One key requirement is their integra-
i
X between neighboring nodes along the path that connects the tion with existing and future classical optical communication
r two users. In order to support many users at the access networks [6], [20]. This implies the need for new quantum
a nodes, it is necessary to proportionally generate longer secret friendly standards for optical networks. That will include
keys between the internal core nodes of the network. The devising proper mechanisms by which weak quantum signals
analogy in classical telecommunications is the ratio between can be separated from classical channels [18], [21]. Multiple-
the end-user data rates and the high traffic of data at the access techniques are also needed to enable interference-free
backbone of the network. One simple idea to achieve higher access to different quantum users [3], [22]. Eventually, QKD
keyratesisto usemultiplexingtechniquesto generatekeysin systems must improve their performance in terms of rate-
parallel. In this paper, we employ one of the most advanced versus-distance behavior and cost.
classical multiplexing techniques to come up with orthogonal One feasible approach to long-distance QKD is based on
frequencydivisionmultiplexedQKD(OFDM-QKD)schemes. trusted-node quantum networks. With current technology, we
We look at existing all-optical orthogonal frequency division are able to generate secret keysat a rate on the orderof Mb/s
at50 km of distance [18]. By cascadingseveralof such links,
ThisresearchwassupportedinpartbytheEuropeanCommunity’sSeventh as shown in Fig. 1, and trusting all intermediate nodes, one,
FrameworkProgrammeGrantAgreement277110,theUK’sEngineeringand
in principle, can exchange secret keys at any distance by first
PhysicalSciencesResearchCouncilGrantEP/M506953/1,andIranNational
Science Foundation (INSF). generating secret keys between neighboring nodes and then
S. Bahrani and J. A. Salehi are with the Department of Electrical Engi- relayingtheinitialkey,inanencryptedway,totheotherparty.
neering,SharifUniversityofTechnology,Tehran,Iran.M.Razaviiswiththe
The main requirement for this approach is to trust all nodes
SchoolofElectronic andElectrical Engineering, UniversityofLeeds,Leeds,
LS29JT,UK,e-mail: [e-mail]. in between the two end users. While this assumption may
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be acceptable for the first generation of quantum networks, it intensities,whereoneofwhichcorrespondstothemainsignal,
can be removed in future generations by relying on quantum and the rest to decoy states. In practice, it is often sufficient
repeater setups at the core network [23]. to use only two decoy states [27], although, in this paper,
In trusted-node networks, the internal nodes in the core for analytical convenience, we assume infinitely many decoy
network are expected to have a high traffic of key exchange statesareused.Ourproposedsetupsarecompatiblewithother
as they are providing service to a large number of end users. QKD protocols, such as continuous-variable or distributed-
It is important then to generate a large number of secret key phase QKD protocols [28]. The detailed analysis of the latter
bits per allocated wavelength to each quantum channel over systems is, however, beyond the scope of this paper.
these core links. One possible approach is to use non-binary BothopticalOFDMandQKDareadvancedtechnologies.It
signalingstosendmorekeybitspertransmittedquantumstate isinterestingtoseehowdrawbacksinonesystemwouldtrans-
[24]. In additionto this, we shouldthink how most efficiently late into the other. While some of the drawbackswith OFDM
we can use the available bandwidth per allocated wavelength, may directly affectour OFDM-QKD system, there are certain
especially with reference to QKD systems that rely on dense issuesthatarelessofaprobleminaQKDsetup.Forinstance,
wavelength division multiplexing (DWDM) techniques [6]. oneknownOFDM problemin the classicaldomainisits high
Ourproposedsolutionherereliesononeofthemostspectrally peak-to-average power ratio, which makes it susceptible to
efficient methods in classical communications, i.e., OFDM. distortions due to nonlinearity effects [29], [30]. Fortunately,
OFDM relies on the full orthogonality of its subcarriers to for QKD applications, nonlinearity is not necessarily a major
multiplexmultiplechannels.Thisfullorthogonalityisessential issue because the QKD transmitted signals are low power.
in QKD applications [3], [25], in order to minimize the Nevertheless, the common OFDM-related imperfections such
interference from other classical and quantum users. In our as time misalignment [31]–[33], phase noise introduced by
case, each subcarrier represents a QKD channel between two the lasers [30], [34], [35], and frequency offsets between the
core nodes. The total key generation rate between these two transmitter and the receiver carriers (in the case of applying
nodesisthenexpectedtoincreaselinearlywiththenumberof a local oscillator at the receiver) [30], [34] can potentially
subcarriers. Moreover, OFDM is compatible with non-binary influence the orthogonality between subchannels, and subse-
signaling techniques, and that would enable us to take the quently affect the performance of OFDM-QKD systems. In
maximum benefit from the available bandwidth. Finally, by this paper, we specifically consider the degrading effects due
usingamultiplexingtechnique,multipleserviceproviderscan to time misalignmnet, which is the major source of error in
use the capacity of the core network without trusting each the most promising setup we propose here.
other.NotethattheOFDM-QKDcanbemodifiedtobeusedas Finally, it is interesting to note that, while one of the
a multiple-access techniquein multi-user QKD setups. In this key advantages of OFDM in the microwave domain is its
paper, we focus on the multiplexing aspect with the objective reliance on digital signal processing, OFDM-QKD setups
of increasing the rate at the core of QKD networks. may less benefit from this feature. At the receiver side, any
Being an optical system, QKD can be merged best with measurement on the OFDM signal before the QKD decoders
OFDM if all-optical OFDM encoders and decoders are used. could alter the transmitted states and result in errors. That is
Here, we consider two possible implementations for the all- why it is important to have a fully optical setup for OFDM
optical OFDM transmitter. In the first approach, the OFDM decoders. At the transmitter side, an optical OFDM setup
subcarriersaregenerateddirectlybyabankoffrequencyoffset would, in principle, allow multiple users to encode their key
locked laser sources or an optical comb generator [8]–[10]. bits without trusting each other. This cannot necessarily be
After encoding the subcarriers, an optical coupler combines achieved if we first generate the OFDM signal electronically
them to generate the OFDM signal. The second approach and then convert it to an optical signal. That said, there will
uses the optical inverse discrete Fourier transform (OIDFT) be much room for improvementin future work, while, in this
circuit to generate the OFDM signal [10]–[13]. Short pulses work, we assess the possibility of OFDM-QKD systems.
are fed into the OIDFT circuit following the QKD encoding The rest of the paper is organized as follows. In the next
stage.Both theseapproachesrelyonreal-timeopticaldiscrete section, we propose two OFDM-QKD schemes and describe
Fourier transform (ODFT) at their receivers. Conventional theirprinciplesofoperation.InSec.III,theproposedOFDM-
passive implementations of ODFT turn out to be too lossy QKD schemes are analyzed from a quantum mechanical
to be useful for our main objective of increasing the rate. In perspective. The analysis of secret key generation rate is
our work, we show how the ODFT circuit can be modified to presented in Sec. IV, in which we particularly focus on time
be effective for QKD applications. misalignment issues within the OFDM system. We propose
Different QKD protocols can be used in our proposed an optimal gating solution to maximize the key rate. Some
OFDM-QKDsetups.Here,wefocusonthedecoy-statevariant numerical results are then presented in Sec. V. We conclude
of the BB84 protocol [26]. The decoy-state technique allows the paper in Sec. VI.
us to use weak laser pulses, rather than ideal single-photon
sourcesas originallyproposedin [7], and that would simplify
II. SystemDescription
the encodingequipmentofQKD. Inordertoobtainimmunity
against the photon-number splitting attacks, in the decoy- In this section, we describe QKD over all-optical OFDM
state protocol, for every transmitted QKD pulse, the sender links. Figure 2 shows the overall system structure. The QKD
has to randomly choose its intensity from a set of available encodersgenerate the quantum signals, in the form of pulses,
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Fig.2:QKDoveranOFDMlink.TheQKDencodedopticalpulsesaremul-
tiplexed byanall-optical OFDMencoder. Atthereceiver, thecorresponding
OFDMdecoderfollowedbyQKDdecodersareusedtogeneratesecretkeys.
Fig.4:OFDM-QKDusingdirectlygeneratedsubcarriers.Theopticalpulses,
that carry the information about the encoded key bits in each generated by N frequency offset-locked laser diodes, are fed into the QKD
subchannel.Theresultingopticalpulsesarefedsimultaneously encoders. At the receiver, an ODFT circuit is required to separate the
intotheOFDMencodertobemultiplexed.Atthereceiver,the subcarriers.
OFDMdecoderfollowedbyessentialQKDdecodingmodules
canbeusedtocompletetheQKDprotocol.Thekeypartinthe (a) b(cid:214)1(t) y(cid:214)0(t) (b)
OFDMdecoderisanODFTcircuit,whicheffectivelyseparates x(cid:214)(t) T/4 0 y(cid:214)2(t)
the subchannels.
In this paper, we assume that the QKD encoders perform b(cid:214)0(t) T/2 0 y(cid:214)1(t)
b(cid:214) (t) (cid:651)/2 y(cid:214) (t)
phase encoding using decoy-state techniques [26]. Based on 2 T/4 3
the phase-encoded BB84 protocol, Alice chooses her phase
GateInterval
value φ from one of the bases 0,π or π/2,3π/2 . The two
A { } { } Fig. 5: (a) The passive ODFT circuit for N = 4. The circuit consists of
phase values in each basis correspond to bits 0 and 1. As
three MZIs with corresponding delays and phase shifts. (b) Shifted replicas
shownin Fig.3,anopticalpulsesentbyAlice passesthrough of the input OFDM signal for N = 4. The shift values for N = 4 are
aMach-Zehnderinterferometer(MZI).Theoutputistwonon- 0,T/4,T/2,3T/4. The time slot, in which all these copies overlap, is
{ }
extracted bythetimegatingoperation.
overlappingsuccessivepulses,denotedbyr and s, ofduration
T with a relative phase correspondingto the chosenbasis and
the transmitted key bit. The QKD decoding includes Bob’s
it is at the output of the combiner that each subchannel has
selectionofhismeasurementbasisbychoosingthephaseφ
B ∈ the right intensity for its correspondingpulse. This will allow
0,π/2 randomlyin one arm of his MZI and the detection of
{ } us to neglect the losses in the encoder box, as we assume
the output signal. In the following, we describe two OFDM-
in this paper. At the receiver, ODFT is used to demultiplex
QKD setups based on the proposed schemes for all-optical
the subcarriers. To comply with the OFDM orthogonality
OFDM.
condition, in Scheme I, the pulse width is T = 1/∆f, where
∆f is the frequency separation of the subcarriers.
A. Scheme I Toillustratetheprinciplesofthisscheme,considertheclas-
Figure 4 depicts the OFDM-QKD system that relies on sicalcase,wheretheOFDMsignalisgeneratedbycombining
directly generated subcarriers. At the transmitter, a bank of classical subchannels as follows:
frequencyoffsetlockedlaser diodesgeneratethe inputoptical N 1
pulses to N QKD encoders. These pulses are individually x(t)= − akejωkt, 0<t<T, (1)
phase randomized, as required by the decoy-state protocol Xk=0
[36], and then go through a bank of encoders as in Fig. 3. where a is the complexamplitudeof the kth subchannelwith
k
Because the information is encoded in the phase difference, frequency ω =ω +2πk∆f for a nominal channel frequency
k 0
these overall random phases do not change the encoded bits. ω . The ODFT circuit, at the decoder, will then separate
0
The same holds for the possible phase noise of the lasers differentsubcarriersandgeneratethefollowingoutputsignals:
so long as their phase is constant during the transmission of
N 1
each bit.In ourforthcominganalysis, we accountforpossible 1 −
y (t)= x(t nT )ej2πnm/N, m=0,1,...,N 1, (2)
relative phase distortions between r and s pulses in Fig. 3. m N − c −
Xn=0
The outputs of the QKD encodersare then combined to form
where T , T/N. With the assumption of T = 1/∆f, we can
the OFDM signal. If we trust all the elements in the Alice c
conclude from (1) and (2) that
box of Fig. 4, we can adjust the transmitted power such that
N 1 N 1
ym(t)= − akejωkt(N1 − ej2πn(m−k)/N). (3)
Xk=0 Xn=0
Theterminthebracketsisnonzeroonlyifk=m,whichleads
to the mth subcarrier extraction.
Different methods can be used to realize the OFDM de-
Fig.3:PhaseencodedQKD.Aliceencodesherkeybitsbychoosingaphase coding, as required by (2), in the optical domain. In Scheme
value φA ∈{0,π/2,π,3π/2}. Each optical pulse passes through the MZIand I, we assume that the ODFT is implemented by a passive
produces twooutputpulses withtherelative phaseφA.OntheBob’s side, a
similarMZIisusedtorecombinerandsmodes,followedbyphotodetection. structure consisted of N 1 MZIs [8]. Figure 5(a) shows an
−
ODFT circuit for N = 4. This structure imitates the efficient
4
OIDFT.Atthereceiver,theODFToperationin(2)shiftseach
of the pulses within an OFDM symbol and combines them
together to generate
N 1N 1N 1
y (t)= 1 − − − A p(t (n+l)T )ej2π(kl+nm)/N, (5)
m N2 k − c
Xn=0Xk=0Xl=0
for m 0,1,...,N 1 . In a real-time implementation,only at
∈{ − }
n+l = N 1, all relevant input pulses are added together at
Fig. 6:OFDM-QKDusing OIDFTcircuit. Atrain ofshortpulses generated −
by an MLL is split into N paths. The OFDM symbol is generated by which ym(t) reduces to
multiplexing the output pulses of the QKD encoders by the OIDFT circuit.
N 1 N 1
TpuhleseOsFfrDoMm dsiyffmerbeonltcionnpsuitsst.sAotftaheserericeesivoefr,paunlsOesD,FeTachciracusiutpdeermpouslittiipolnexeosf zm(t)= − Akp(t (N 1)Tc)ej2πk(N−1)/N(1 − ej2πn(m−k)/N).
N − − N
thesubcarriers. Xk=0 Xn=0
(6)
Here again, the term in the brackets is zero for k,m, which
method of realizing DFT, known as fast Fourier transform implies that, up to a known overall phase factor, the original
(FFT), by meansof delays, couplers,and phase shifters. Each information in Am can be recovered at the mth output port of
output port of the ODFT circuit is a weighted sum of shifted the ODFT circuit of Fig. 6.
replicas of the input as required by (2). It will then provide For the receiver of Scheme II, we have two options. We
us with a real-time DFT operation once all shifted replicas of can either use the passive OFDM decoder used in Scheme I,
the input overlap, as shown in Fig. 5(b). That would require or, alternatively, the active structure shown in Fig. 7(a). The
a time gating operation [8], [9], which can be implemented main advantage of the latter is to remove the inherent loss in
by electro-absorption modulators (EAM), or simply by time- the passive OFDM decoder. To better explain the loss effect
gating the single-photondetectors used in the QKD decoders. in the passive decoder,considera sequenceof N pulsesatthe
Time misalignment can then be a major source of error in input x(t) of Fig. 5(a), and let us look at the output signals.
such a scheme. The quantum operation of the ODFT circuit In this case, each input pulse has four paths to take, with
is discussed in more detail in Sec. III. different delays and phase shifts, to reach to the output ports
ofFig.5(a).Inotherwords,foreachinputpulse,therewillbe
four output pulses at each of the decoder’s four output ports;
B. Scheme II
see Fig. 7(b). Only one out of these four output pulses has
Figure 6 shows an alternative setup for the OFDM-QKD
the right amount of delay and phase shift to be used for our
system. Here, the output of a pulsed laser source, e.g., a
ODFT operation,and that is whytime gating is required.The
mode-locked laser (MLL), is split into several paths by an
inevitable drawback of this approach is that the other three
opticalsplitter.Thepulsesshouldbeshortenoughtocoverthe
pulses, and the power therein, will remain unused and that
spectrumofallthesubcarriersintheOFDMsymbol.Here,we
will contribute to a maximum total efficiency of 1/N for a
assume that the pulse width is slightly lower than T . Similar
c passive decoder as in Fig. 5(a). To overcome this drawback,
to Scheme I, each shortpulse, after splitting, is fed into QKD
our proposed OFDM decoder in Fig. 7(a) employs an optical
encoders to produce successive pulses r and s. Each of these
switchalongwithproperdelays,insteadofapassivecircuit,to
pulses will then go through an OIDFT circuit generating N
performtheserialtoparallelconversion.AsshowninFig.7(c),
short pulses within an OFDM symbol duration T. The delay
this way there will be no extra pulses to be discarded, and
in the MZI of Fig. 3 is assumed to be greater than T.
the ODFT process can be implemented by a passive N-by-N
The required OIDFT can be implemented by a structure
circuit, of a star topology but with phase shifters along each
similar to the ODFT. For instance, the circuit in Fig. 5(a), for
internal path, with no fundamental overall loss [8], [37].
the special case of N = 4, can be employed for OIDFT as
The abovefeature of the active decoderin Fig. 7(a) makes
well. In the case of OIDFT, the input pulses (denoted by y
it a better choice for high-rate QKD links, as we will see
components) enter from the right hand side of Fig. 5(a) and
in the following sections. The passive decoder schemes can
theoutputwillbethesignallabeledby xˆ(t).Assumingthatthe
still be used for the sake of sharing the channel resources
y pulses are synchronous, in each OFDM symbol, the output
between multiple service providers. They do not, however,
signal xconsistsoffourpulsesapartbymultiplesofT within
c offer any total-rate advantage as compared to a single-carrier
a T-long frame. Each of the latter pulses are a combination
system. Note that the OFDM decoder of Fig. 7(a) is mostly
of all input pulses, as shown in Fig. 6.
compatible with Scheme II, due to its discrete nature, as
More generally, in the classical case, the generated OFDM
comparedto SchemeI. Nevertheless,oneshouldtake noteof
amplitude at any carrier frequency ω is given by
possible challenges of using the active decoder with Scheme
N 1N 1 II. While, by using a common pulsed source, Scheme II is
− −
x(t)= Akp(t lTc)ej2πkl/N, (4) more immune to phase noise or frequency offset problems,
−
Xk=0Xl=0 time misalignment is still a key concern. That is why, in the
where p(t) represents the shape of the initial laser pulse and following sections, we study the impact of such timing errors
A is the complex amplitude of the kth subchannel. Note that inourOFDM-QKDsetups.Secondly,whiletheactivedecoder
k
in Fig. 6, subchannels are separated spatially at the input to removesthelossassociatedwithtimegating,itsopticalswitch
5
N (a) 1 bˆ1(t) = kN=−01bˆ1kejωkt and bˆ2(t) = kN=−01bˆ2kejωkt represent the
Delay vacuumPfluctuations of the unusedPports of the MZIs’ beam
… splitters correspondingto all existing frequency modes of the
1 2 N Optical 2 Passive
Delay system.Foracenterfrequency,ω,thetransformationmatrices
Switch FFT
of the three MZIs in Fig. 5 are given by
… Circuit …
1 1 j 1 0 1 j
N B = , (8)
Delay ω,1 2 j 1 ! 0 e−j(ωT2) ! j 1 !
(b) GateInterval (c) for the MZI on the left,
1 1 j 1 0 1 j
B = , (9)
ω,2 2 j 1 ! 0 e−j(ωT4) ! j 1 !
for the one on top right, and
1 1 j 1 0 1 j
B = , (10)
ω,3 2 j 1 ! 0 je−j(ωT4) ! j 1 !
Fig. 7: (a) Proposed active ODFT circuit. By employing an optical switch
instead ofapower splitter, the loss oftimegating is eliminated. (b)Passive for the one on bottom right of Fig. 5(a). Applying the above
approachtoserial-to-parallelconversionforN=4.Somepulsesaregenerated transformations to mode k, we obtain
and then discarded during the time gating process. (c) Active approach to
serial-to-parallel conversion for N=4followed bytheFFTcircuit [8]. aˆ (t) aˆ (t)
′k = B k , (11)
bˆ′k(t) ! ωk,1 bˆ0k(t) !
willintroducesomeadditionalinsertionloss.Thelatter,within yˆ0,k(t) = B aˆ′k(t) , (12)
our practicalregimeof interest, is shown to be less than 2 dB yˆ1,k(t) ! ωk,2 bˆ1k(t) !
for ultrafast optical switches and will be accounted for in
yˆ (t) bˆ (t)
our numerical analysis [38]. Optical switches may also have 2,k = B ′k . (13)
nonzeroextinctionratios,because ofwhichsome powerleaks yˆ3,k(t) ! ωk,3 bˆ2k(t) !
to other undesired output ports. This is a minor problem for The output operator for output port m in Fig. 5 is then given
thedecoderofFig.7(a),becausetheinputpulsestotheswitch by
are non-overlappingin time. By using time gating and proper 3
yˆ (t)= yˆ (t), m=0,1,2,3. (14)
delay lines, the leaked powerto other portsshould notappear m m,k
in the same time slot that time gating is taking place, hence Xk=0
has negligible effect on system performance. Note that the above operations are all linear. Based on the
superposition principle, we can split each output yˆ (t) to two
m
III. QuantumAnalysis parts. The first part is the output obtained by neglecting the
vaccum operators, and the other part is a linear combination
In this section, we analyze the OFDM-QKD systems pro-
of all vacuum operators. More generally, it can be concluded
posed in Sec. II from a quantum mechanical perspective.
thattheoutputofsuchanODFTcircuit,neglectingthevacuum
We choose the Heisenberg picture for our analysis. In two
operators, can be expressed as a function of xˆ(t), as follows:
steps, we first concentrate on the operation of the system
corresponding to each of the two pulses r and s in Fig. 3, N 1
and then we combine the results to find the output operators Xˆm(t)= 1 − xˆ(t nTc)ej2πnm/N, m=0,...,N 1, (15)
N − −
in QKD decoding modules. Xn=0
which is similar in form to (2) for the classical case. Substi-
A. Scheme I tuting (7) into (15) and applying the orthogonality condition,
∆f = 1, we obtain
In the Heisenberg picture, the output operator of the Alice T
box in Fig. 4 can be expressed as N 1 N 1
N 1 Xˆm(t)= − aˆkej2π(f0+k∆f)t(N1 − ej2πn(m−k)/N). (16)
xˆ(t)= − aˆkejωkt, 0<t<T, (7) Xk=0 Xn=0
Xk=0 Note that the term in the brackets is nonzero only if k = m.
We then obtain
where aˆ is the annihilation operator corresponding to the
k
mode representing the kth subcarrier. For the rest of this sec- N 2N 1
tion, we neglectthe path loss effect, which will be considered yˆm(t)=Aaˆm(t)+ − − βikbˆik(t), m=0,...,N 1, (17)
−
when we calculate the secret key generation rate. We then Xi=0 Xk=0
focusonthereceiversetupassumingthatatitsinputthesignal where A is either 1 or j and β ’s are constant coefficients.
ik
xˆ(t) is received. The operator aˆ (t) is the evolved version of aˆ and is given
m m
Forsimplicity,letusfirstconsiderthespecialcaseofN =4. by aˆm(t)=aˆmejωmt, and similarly for the vacuum operators in
As shown in Fig. 5, the ODFT circuit, in this case, is imple- the above equation. As explainedin Sec. II, the orthogonality
mented by three MZIs. The operators bˆ0(t) = kN=−01bˆ0kejωkt, is onlymet in a regionof width T/N, where all of the shifted
P
6
copiesof theOFDM signaloverlap.The signalcorresponding Another option for the receiver in this scheme is the
tothisoverlappingtimeslotwilleventuallybedetectedbythe structure we proposed in Fig. 7(a). For this active decoder,
photodetectors in the receiver module. (21) ismultipledbyanadditionalfactor √N.Furthermore,no
With the phaseencodingQKD protocol,the two successive vacuum components would appear because the beam splitters
pulses r and s for channel m, represented by aˆ(r) and aˆ(s), in the FFT circuit do not have any unused ports [8].
m m
respectively, will be recombined at the receiver’s MZI in
Fig. 3. The output operator corresponding to the recombined IV. KeyRateAnalysis
pulse d in Fig. 3 for the mth output is then given by
Thissectionpresentsananalysisofthesecretkeygeneration
N 2N 1 rate for the proposed OFDM-QKD schemes. We assume that
dˆm(t)= 2j(ejφ(Bm)aˆ(mr)(t)+aˆ(ms)(t))+ − − β(ikr)bˆ(ikr)(t)+ the efficient decoy-state BB84 protocol is employed in the
Xi=0 Xk=0 QKD setup [39], [40]. The average number of photons per
N 2N 1 QKD channel is given by µ, for the main signal state, and
− − β(s)bˆ(s)(t), T T <t<T. (18)
ik ik − c it is calculated at the output of the Alice box. The secret
Xi=0 Xk=0 key generation rate per transmitted pulse, in the limit of an
infinitelylongkey,islowerboundedby max[0,P(Y )],where
0
B. Scheme II
P(Y )=Q (1 h(e )) fQ h(E ), (22)
0 1 1 µ µ
Tostartouranalysisinthisscheme,wedenotetheannihila- − −
tion operator corresponding to the spatial mode at the output and h(p)= plog p (1 p)log (1 p) is the binary entropy
− 2 − − 2 −
of the kth QKD encoder by aˆ . We assume that an OIDFT function with f being the error correction inefficiency. The
k
circuit similar to the one depicted in Fig. 5(a) for N = 4, yet overall gain, the QBER, the gain of a single photon state and
in the reverse direction, is used at the transmitter. Then, we the error rate of a single photon state are, respectively, given
can obtain the output operator of the Alice box by applying by [3]
the transformation matrix of each MZI. It can be concluded
Q =1 (1 Y )e ηµ,E =(Y /2+e (1 e ηµ))/Q ,
that the output operator of the Alice box is given by µ − − 0 − µ 0 d − − µ
Q =Y µe µ,e =(Y /2+e η)/Y . (23)
1 1 − 1 0 d 1
N 1
1 −
xˆ(t)= cˆ p(t lT ), (19) Equations (22)-(23) provide an estimate to the generated
l c
N −
Xl=0 key rate when infinitely many decoy states are in use and
no eavesdropper is present. In the above equations, Y =
where cˆ = N 1aˆ ej2πkl/N is the lth temporal mode at the 1
l k=−0 k (1 η)Y + η is the yield of a single photon state and Y
output of thePOIDFT circuit. Note that the coefficient 1/N in − 0 0
is the probability of any detector clicks without having any
(19) isnotnecessarilya sourceofloss, so longasthe average
transmitted photons from the corresponding QKD encoder.
number of photons per pulse at the output of the transmitter
Furthermore, e represents the probability of phase stability
meets the requirements of the decoy-state protcol. That is, d
errors, betweenr and s pulses, andthe totaltransmissivity of
we can compensate for the internal loss at the transmitter
the link is given by
by tuning the intensity of the input light. In our following
analysis, this factor 1/N has been neglected. With a passive η=η η η 10 αL/10, (24)
g d ins −
OFDMdecodersimilartothatofSchemeIatthereceiver,the
where η is the quantum efficiency of the photodetectors, α
analysis presented in the previous subsection can be useful d
is the channel loss factor in dB per unit of length and η
here as well, except that here we deal with temporal modes. ins
represents any additional insertion loss in the link. Here, η
Substituting (19) in (15) and simplifying the equations at g
represents the additional loss due to the OFDM decoding
n+l= N 1,we canexpresseachoutputofthe ODFT circuit
− scheme. For instance, in Scheme I, with gate interval of T ,
as c
the parameterη equals 1/N in the ideal case. For Scheme II,
g
N 2N 1
yˆ (t)=[aˆ ej2πm(N 1)/N + − − β bˆ ] and the active decoder of Fig. 7(a), ηg is ideally one.
m m − ik ik We calculate the parameters in (23) by finding the prob-
Xi=0 Xk=0
abilities of interest once the QKD measurements are done.
p(t T +T ), m=0,...,N 1. (20)
c
× − − For instance, the measurement operator for the representative
The coefficient ej2πm(N 1)/N is a constant phase term that can photodetector in Fig. 3 is given by
−
be absorbed in aˆ in the above equation.
m
Finally, the output operator obtained by the recombination Mˆ = dˆm†(t)dˆm(t)dt, (25)
Z
of the pulses r and s by means of the receiver’s MZI, dˆ (t), gateinteval
m
is given by fromwhichonecanobtainkeyrateparameters.Intheanalysis
of the key generation rate, the terms containing vacuum
N 2N 1
dˆm(t)=[2j(ejφ(Bm)aˆ(mr)+aˆ(ms))+ − − β(ikr)bˆ(ikr)+ ssitmatepslifiweislltnhoetccaolcnutrlaibtiuotnest.o the key rate parameters, and that
Xi=0 Xk=0
In order to analyze the secret key generation rate of the
N 2N 1
− − β(s)bˆ(s)] p(t T +T ). (21) proposed schemes in more detail, one should consider the
Xi=0 Xk=0 ik ik × − c influence of imperfections in the system, which may degrade
7
TABLEI:Nominalvalues forsystemparameters
Parameter Value 10−3
Averagenumberofphotonspersignalpulse 0.48
QuantumEfficiency 0.3 10−4
Totalinsertionandpathloss,ηins10−αL/10 10dB
ERrreocreicvoerrredcatrikoncoinuenffitrcaieten,cyγ,dcf 1E-17.2n2s−1 nt10−5
LaserPphualsseesrteapbeiltiittyionerrionrt,erevdal, Ts 201.000p5s on cou10−6 NN==48,, PPxxttaallkk
OFDMsymbolduration, T 100ps hot N=16, Pxtalk
Numberofsubcarriers, N 4,8,16 d p N=4, Pdc
groun10−7 NN==81,6 P, Pdcdc
ack
system performance. As explained before, we specifically B10−8
consider time misalignment issues, which are known to be
critical in all-optical OFDM systems. In OFDM, the time 10−9
alignment of the optical subchannels is critical, due to its
effect on their orthogonality. Furthermore, the time gating 10−100 0.005 0.01 0.015 0.02 0.025 0.03
Average time misalignment to symbol duration ratio
at the receiver should be synchronized with the transmitted
pulses to extract the correct time slot. In our OFDM-QKD
Fig.8: Backgroundphotoncountprobabilitycomponentspdcandpxtalkversus
setups, nonidentical QKD encoders or some errors in time- E τk /T for Scheme II with active OFDM decoder, for different values of
{| |}
gating synchronization may introduce time misalignment. N.
In our work, we have found the key generation rate of
our proposed OFDM-QKD schemes in the presence of time
misalignmentissues.Itturnsoutthattheycausetwoproblems. I. These parameters are chosen in accordance to practical
First, they generate some inter-channel crosstalk, denoted by considerations. With the chosen OFDM symbol duration, the
p , which adds to the background noise, and, second, they subchannelfrequencyseparation,∆f,hastobe10GHz,which
xtalk
slightly reduce the transmissivity factor η . One can reduce has been used in several all-optical OFDM experiments [10],
g
the crosstalk noise by reducing the width of the gate interval, [41]. The pulse width, Tp, in Scheme II should be less than
but, by doing so, ηg would further be reduced, as we have to Tc =T/N. Here,weassumethattheratioδ/Tp inthisscheme
leaveoutsomeofthedesiredsignalcomponentsaswell.That is equal to 0.04.
would imply the existence of an optimal gate width at which In order to see the importance of the time misalignment
the total secret key rate R is maximized, where issue, we first look at its induced cross talk contribution as
OFDM
compared to the dark count component. Figure 8 compares
R =max[NP(Y )/T ,0], (26)
OFDM OFDM s the two elements of the backgroundnoise, i.e., p and p ,
dc xtalk
and versus a normalized measure of misalignment, E τk /T, in
{| |}
Y =1 (1 (p +p ))2. (27) the special case of Scheme II with active decoders. A similar
OFDM dc xtalk
− − overall behavior is observed for other schemes as well. It is
Here, Ts representsthe repetitionperiod of the QKD protocol clear that while the cross talk is negligible for low values of
and pdc =γdcTg,whereγdc andTgarethephotodetectors’dark time misalignment, it becomes the major source of noise in
count rate and the gate interval, respectively. In Appendix A, our OFDM-QKD setups. We next consider the effect of time
we have derived all the required terms for calculating the misalignment on each of the proposed setups.
abovekeyrateasafunctionoftheaveragetimemisalignment
Figure 9 shows the effect of time misalignment on the
E τ , where τ ’s are i.i.d random variables representing the
{| k|} k total secret key generation rate, ROFDM, of Scheme I with
time misalignmentof the kth channel with respect to the gate
and without optimal time gating. In the latter case, the gate
interval, and δ , (T T )/2, where T is the pulse width in
c− p p width is a constant Tc. It can be seen that by optimizing the
SchemeII.Inthefollowing,wepresentsomeofournumerical
gate interval the secret key rate significantly improves. It will
findings for a selected set of parameter values.
not,however,surpassthe performanceof a single-carrierlink,
shownontopofFig.9,runatthesameclockrateastheOFDM
V. NumericalResults system. The main reason for this is the loss factor N due to
In this section, we investigate the performance of the thetimegating,whichresultsinareducedkeyratepercarrier.
proposed OFDM-QKD schemes by considering the following Moreover, Fig. 9 shows that a system with a larger number
cases:SchemeIandSchemeIIwithpassiveOFDMdecoders, of subcarriers, N, is more susceptible to time misalignment
and Scheme II with the active OFDM decoder of Fig. 7(a). errors.Thishastodowiththeinterplaybetween pxtalk andηg,
In order to evaluate the effect of time misalignment on the where the latter turns out to be the dominant factor. In short,
performanceof each case, each subcarrier is assumed to have norateadvantageisobtainedbySchemeI.It,nevertheless,can
a time misalignmentwith uniformdistribution, τ U( a,a), be used as a multiplexing tool for sharing the infrastructure
k
∼ −
where a < T is an arbitrary constant. We then find the between multiple service providers.
c
optimal gate width that maximizes the key rate. The nominal Next, Fig. 10 shows the total secret key rate of Scheme
values used for the system parameters are listed in Table II with passive OFDM decoder versus E τ /T. Here again,
k
{| |}
8
x 107 carrier link. That would demonstrate the prospect of using
4
OFDM techniques at the core of QKD networks.
3.5
x 108
b/s) 3 6
e ( N=4
on rat2.5 5 NN==816
Secret key generati1.25 NNN===4816 eneration rate (b/s)34 NND==W48D,, Moopp (ttiipmmeaar llc gghaaattneenel)
1 NND==W48D,, Moopp (ttiipmmeaar llc gghaaattneenel) ecret key g2
0.5 S
0 0.005 0.01 0.015 0.02 0.025 0.03
Average time misalignment to symbol duration ratio 1
Fig. 9: Secret key generation rate versus E τk /T for Scheme I and its
optimalgateversion,fordifferentvalues ofN{.| |} 0
0 0.005 0.01 0.015 0.02 0.025 0.03
Average time misalignment to symbol duration ratio
x 107 Fig.11: SecretkeygenerationrateversusE τk /T forSchemeIIwithactive
4 OFDM decoder and its optimal gate versio{n|, f|o}r different values of N. The
parameter δ/Tp ischosentobe0.04.
3.5
Inadditiontothetotalkeyrate,wealsolookatthespectral
b/s) 3 efficiency of each scheme, S, defined by the ratio of the
e (
at secret key generation rate and the allocated bandwidth.In the
ation r2.5 case of a DWDM link with channel spacing of 50 GHz, in
ner Fig. 11, S =0.11%. For the OFDM-QKD systems, assuming
e
ey g 2 that the allocated bandwidth is given by N/T, S has a peak
et k N=4 value of 0.36%, which is three times higher than that of the
ecr1.5 N=8
S DWDMsystem.Oncetimemisalignmentkicksin,theOFDM-
N=16
N=4, optimal gate QKD systems with lower values of N are favored as they are
1 N=8, optimal gate
DWDM (per channel) less susceptible to such errors. Overall, it can be seen that
by multiplexing 4–8 OFDM subcarriers, one can outperform
0.5
0 0.005 0.01 0.015 0.02 0.025 0.03
Average time misalignment to symbol duration ratio DWDM-QKD systems both in terms of the total key rate and
the spectral efficiency in practical regimes of operation.
Fig.10: SecretkeygenerationrateversusE τk /TforSchemeIIwithpassive
OFDM decoder and its optimal gate versio{|n,|f}or different values of N. The
VI. Conclusions
parameter δ/Tp ischosentobe0.04.
We proposeda spectrallyefficientapproachto multiplexing
QKDchannels,namely,OFDM-QKD.Basedontheprinciples
applying the optimal gate interval results in an enhancement of all-optical OFDM in classical communications, several
in the secret key rate. Yet, no improvement, as compared to OFDM-QKD schemes were considered. These schemes were
the DWDM-QKD system, is observed in the overall key rate analyzedindetail, intermsoftheirsecretkeygenerationrate,
by increasing N, which is mainly due to the inherent loss in considering time-alignment imperfections, which are critical
the passive structure of the OFDM decoder. in all-optical OFDM systems. It was shown that such time
The secret key generation rate of Scheme II with the pro- misalignment issues would introduce a crosstalk noise with a
posed active OFDM decoder is depicted in Fig. 11. It can be degradingeffectonthekeyrate,similar tothatofbackground
seen that by multiplexingmore subchannelsin the system the noise. We showed that by reducing the gate interval to an
secret key rate increases. This increase is initially linear with optimal value this problem could be alleviated to a large
the number of subchannels, but once the time misalignment extent. Most importantly,we showed that the existing passive
kicksinthekeyratealsocorrespondinglydrops. Nevertheless, structures for the OFDM decoder would provide no gain in
italwaysstays abovethatof thesingle DWDM-QKDchannel theirmultiplexing,intermsofthetotalachievablekeyrate.We
depictedinthebottomofthefigure.InfairnesstotheDWDM proposedanactiveOFDMdecoder,which,byusinganoptical
system,wehaveaccountedfor2dBofadditionalinsertionloss switch, followed by proper delays and a passive FFT circuit,
fortheopticalswitchintheOFDM-QKDsystem.TheDWDM could eliminate the inherent loss in passive decoders. We
curve uses 100-ps-long pulses. Under these conditions, for remark that, in the case of active decoders, ultrafast switches
N = 16, and at a normalized average time misalignment of with a transition time on the order of picoseconds may be
0.02,we aredoingmorethanfourtimesbetterthanthe single required. This is due to the short time separation of pulses
9
within an OFDM symbol [38], [42]. This may add to the sourceshaveindependentphases.Inthiscase,thephasediffer-
cost and complexity of the system. Nevertheless, we showed encecorrespondingtoanycrossterm hasuniformdistribution
that,usingourproposedactivedecoders,wecouldoutperform on the interval [ π,π]. These terms are then eliminated due
−
thealternativeDWDM-QKDsystemsintermsofthetotalkey to their zero expectedvalues. At this point, we generalizeour
rateandspectralefficiency.ThisimpliesthatOFDM-QKDcan result to include the case T τ <0. Hence, equation (32)
c k
− ≤
provideahigh-ratespectrallyefficientmethodofkeyexchange can be rewritten as
at the core of trusted-node QKD networks. 2µτ
AppendixA E(φ(Ak)−φkB){hgˆ†kgˆki}= N2|Tk|. (33)
In the last step, the partial crosstalks due to each subcarrier
OFDM-QKDwithmisalignmenterrors
are added to obtain the total crosstalk background count on
In this appendix,we analyze the operationof our proposed the mth subcarrier, denoted by p(m) , as follows:
xtalk
OFDM-QKD setups in the presence of time misalignment.
2µ
p(m) =η E τ , (34)
xtalk ′N2T {| k|}
A. Scheme I Xk,m
We start our analysis by assuming that each subcarrier has whereη isthetransmissivityofthelink,excludingthelossof
′
a time misalignment τ , 0 < τ < T , with respect to the time gating. Under the assumption that τ ’s are i.i.d random
time gating interval. Wkithout l|oks|s of gcenerality, we assume variables, p(m) is independentof the sub|cka|rrierindex.So, we
xtalk
that 0 τ < T (both cases of τ > 0 and τ < 0 have the can express the crosstalk background count per subcarrier as
k c k k
≤
same effect). Figure 12(a) shows the shifted copies of the kth 2µ(N 1)E τ
subcarrier pulse in the presence of time misalignment τk. As pxtalk =η′ −N2T {| k|}. (35)
can be seen in the figure, the shifted copies does not overlap
Such time misalignments change the loss factor in the time
completely in the gate interval, which leads to different
gating operation, η , due to the additional loss occurring in
summation results in two distinct time intervals, as follows: g
the interval (T T ,T T +τ ) in (30). η is then given by
c c m g
− −
t (T T ,T T +τ ) Xˆ (t)= N1aˆk(t) k,m (28) ηg = 1 (E{|τk|})(1 (N−1)2). (36)
∈ − c − c k ⇒ m ( NN−1aˆm(t) k=m N − T − N
0 k,m Now, let us reduce the gate interval by b from each side to
t∈(T −Tc+τk,T)⇒Xˆm(t)=( aˆm(t) k=m . (29) reducethecrosstalkeffect.ItcanbeconcludedfromFig.12(a)
that p =E A , where A is obtained by
xtalk
As a consequence, equation (18) is modified to { }
dˆm(t)= 2j{NN−1(ejφ(Bm)aˆ(mr)(t)+aˆ(ms)(t))|t∈(T−Tc,T−Tc+τm)+ A=( η′2µN(N2T−1)0(|τk|−b) ||ττkk||≤≥bb (37)
1 (ejφ(Bm)aˆ(mr)(t)+aˆ(ms)(t))|t∈(T−Tc+τm,T)+ and pxtalk can2bµe(Nexpr1e)ssed as
N Xk,m(ejφ(Bk)aˆ(kr)(t)+aˆ(ks)(t))|t∈(T−Tc,T−Tc+τk)}, (30) pxtalk =η′ N2T− p(|τk|≥b)(E{|τk|||τk|≥b}−b). (38)
Furtheremore, η =E B , where
g
with the third term representing the inter-subcarrier crosstalk { }
onthemthsubcarrier.Here,weneglectedthevacuumoperators B= N1 − 2Tb − T1(1−(NN−1)2)((|τk|−b) |τk|≥b
due to their elimination once we apply the measurement ( 0 0 τk b
operator. The background count due to this crosstalk may ≤| |≤ (39)
influence the performance of the system, as we will show in and
the following. 1 2b 1 N 1
Defining gˆk(t) , N1(ejφ(Bk)aˆ(kr)(t) + aˆ(ks)(t)) with initial state ηg = N − T −T(1−( N− )2)p(|τk|≥b)(E{|τk|||τk|≥b}−(4b0)).
α rαejφ(Ak) s, we can write
| i | i
2µ τ
hgˆ†kgˆki= N2(1+cos(φ(Ak)−φkB))Tk. (31)
B. Scheme II
Here, µ = α2 and φ(k) is the relative phase produced by
| | A As explained in Sec. II, two receiver structures can be
the QKD encoder of the kth subcarrier. We then calculate the
applied for this scheme: the passive OFDM decoder used
expected value of (31) as a function of (φ(k) φ(k)), which
A − B in Scheme I, and the active OFDM decoder, which exploits
results in
an optical switch. In this subsection, we derive the crosstalk
1 background count for both receiver structures.
E(φ(Ak)−φkB){hgˆ†kgˆki}= 2E(φ(Ak)−φ(Bk)){hgˆ†kgˆki|basis={0,π}}+ First,weconsiderSchemeIIwithapassiveOFDMdecoder.
12E(φ(Ak)−φ(Bk)){hgˆ†kgˆki|basis={π/2,3π/2}}= 2Nµ2τTk. (32) tWoethasesutimmeeagtaimtinegmiinsatelirgvnaml feonrtτeka,c0h<tr|iτbku|t<aryT.c,Fwigiuthrere1sp2(ebc)t
Note that cross terms between any two different subcarriers depictsthereplicasofthepulseseriescorrespondingtothekth
also appear in the dˆm†dˆm . Here, we assume that the laser tributary.We denotethewidthofeachpulsebyTp. Following
h i
10
where u(.)is the step function.Hence, η can be expressedas
g
1 2b (b+δ) N 1
η = ( (1 ( − )2))p(τ b+δ)+
g k
N −{ NT − NT − N | |≥
p p
1 N 1
(1 ( − )2)p(τ b+δ)(E τ τ b+δ )+
k k k
NT − N | |≥ {| ||| |≥ }
p
(b δ) 1
− p(b δ τ b+δ))+ (p(τ b δ)
k k
NT | − |≤| |≤ NT | |≥| − |
p p
(E τ τ b δ ) p(τ b+δ)(E τ τ b+δ ))
k k k k k
{| ||| |≥| − |} − | |≥ {| ||| |≥ }
2(b δ)
− p(τ <b δ)u(b δ) (48)
k
− NT | | − − }
Fig. 12: Shifted copies of the signal corresponding to kth tributary in the p
presence ofthetime misalignment τk,for(a)SchemeI, (b)SchemeIIwith Next, we discuss the time misalignmentissue in Scheme II
passiveOFDMdecoder, (c)SchemeIIwithactive OFDMdecoder.
with an active OFDM decoder. Figure 12(c) shows the pulse
seriesofthekth tributaryinthepresenceoftimemisalignment
τ , 0 < τ < T , with respect to the switching time. Due
thesamestepsasintheprevioussubsection,weconcludethat k | k| c
to the elimination of the loss incurred by passive serial to
the output operator, dˆ (t), for 0<τ <δ is given by
m k parallelconversion,equations(41) and(42) are modifiedby a
dˆm(t)= 2j(ejφ(Bm)aˆ(mr)(t)+aˆ(mr)(t)), (41) marue,ltripeslipceactitviveeflya,ctgoirve√nNb.yH(4e3n)cea,ntdhe(4p4a)r,ammueltteirpslipexdtablkyaNnd. ηg
and for δ<τ <T we have
k c
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{ }
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[14] R.L.Rivest,A.Shamir,andL.Adleman,“Amethodforobtainingdigital
1 2b 1 (1 (N 1)2)((τ (b+δ)) τ b+δ signaturesandpublic-keycryptosystems,”CommunicationsoftheACM,
N − NTp − NTp − N− | k|− | k|≥ vol.21,no.2,pp.120–126, 1978.
B= N1N1−−2|N(τbkTN−|+pδTb)p−uδ(b−δ) |b−δ0|≤≤||ττkk||≤≤|bb(+−47δδ)| [15] PaF1n.2od4uW.n.dfaaScthtiooornri,sn“goA,”flgCoionrmithPpmurotsecrefeoSdrciinqegunsacneto.ufmtIhEceoEmE3p5CuthtoamtiAopnn.n:Sudoaicls.cSrPeyrtemespsl,oosg1ia9ur9mit4h,mopns.