Compensating the Noise of a Communication Channel via Asymmetric Encoding of Quantum Information

0.49 MB·English

by  Marco Lucamarini

Compensating the Noise of a Communication Channel
via Asymmetric Encoding of Quantum Information
Marco Lucamarini1, Giovanni Di Giuseppe2, David Vitali2, and Paolo Tombesi2
1CNISM UdR, University of Camerino, Via Madonna delle Carceri 9, 62032 Camerino (MC), Italy
2Physics Department, University of Camerino, Via Madonna delle Carceri 9, 62032 Camerino (MC), Italy
(Dated: January 28, 2010)
An asymmetric preparation of the quantum states sent through a noisy channel can enable a new
way to monitor and actively compensate the channel noise. The paradigm of such an asymmetric
treatmentofquantuminformationistheBennett1992protocol,inwhichtheratiobetweenconclu-
siveandinconclusivecountsisindirectconnectionwiththechannelnoise. Usingthisprotocolasa
guidingexample,weshowhowtocorrectthephasedriftofacommunicationchannelwithoutusing
reference pulses, interruptions of the quantum transmission or public data exchanges.
0
1 PACSnumbers: 03.67.Dd,03.65.Hk
0
2
n (QIKnDre)ce[1n]thyaesarrseathcheefidelsducohf QauhaingthumdeKgreeyeDofisttreicbhuntiicoanl 0Z ϕ0
a ϕ
J perfection that the emergence of unexplored directions ϕ 1
in one of its oldest protocols, the Bennett 1992 [2], is θ 0 ε
8 1X 0X
2 somewhat surprising. The Bennett 1992 (B92) proto- ϕ ε
1
h] svctoarlliuciestslyboafnstoehdneoorlonthgoiocngalolynbatiwtl,cotooqmuwmahnuitncuhimcaatrseetdaatbseysso,tch|iϕea0tt(cid:105)erdaanntsdhme|iϕtttw1e(cid:105)or, 1Z ϕ1 ϕ0
p
(Alice) to the receiver (Bob) [2]. This simple structure
- FIG. 1: States of the B92 protocol with a characteris-
nt isoneofthemainadvantagesoftheB92protocol, which tic angle θ = π (15◦). Left: channel without phase-drift,
a alsofeaturesunconditionalsecurity[3]andsuitabilityfor ε = 0. Center:12channel with a phase-drift of ε = π (45◦).
u long-distance communications [4, 5]. Right: channel with a phase-drift of ε = − 7 π (−704◦). The
q There is however another peculiarity of the B92 pro- phase-driftrotatesthereferencesystemduri1n8gthetransferof
[
tocol which has not been considered so far and yet it the quantum states from the transmitter to the receiver [7].

1 represents a useful resource in the field of quantum in-
v formation, i.e. the asymmetric distribution of its signal
4 states, |ϕ0(cid:105) and |ϕ1(cid:105). In the left inset of Fig. 1 we de- sketchedinFig.1: intheleftillustrationthesignalstates
3
picted these states as two arrows lying on the equator of reach the receiver without being affected by the noise;
2
the Poincar´e sphere [6] at an angle θ from the horizontal onthecontrary,inthecentralandrightillustrations,the
5
. axis (noiseless case). In this representation, orthogonal states are rotated by an angle ε about the central axis
1 states are associated to antiparallel arrows, to allow for and Bob’s measurement will consequently contain some
0
a one-to-one map between a generic quantum state and errors. We call ε the misalignment angle.
0
1 a single point of the Poincar´e sphere [6]. A symmetric Although the phase drift model can appear naive, it
: distribution of the states would be if |ϕ0(cid:105) and |ϕ1(cid:105) were applies to a vast number of QKD setups, precisely to all
v
lying in opposite directions respect to the origin of the those in which the information is encoded in the rela-
i
X axes, antiparallel to each other. But then they would tive phase of a photon passing through an unbalanced
r be orthogonal, and this can never happen in the B92 interferometer [1, 2]. To show how relevant it is and to
a protocol. Hence the distribution of the signal states is facilitate a comparison between our proposal and exist-
necessarily asymmetric for this protocol. ing solutions, we give in the next a brief overview of the
It turns out that this kind of asymmetry has its own main techniques used to compensate the phase drift.
advantages if adequately exploited. In the B92 protocol Onepopularsolutionistomultiplexinthesamechan-
Bob performs a measurement of the incoming states and nel the quantum signals, e.g. single-photon pulses or
divides the results into two main groups, labeled as con- attenuated laser pulses, and the classical ones, e.g. in-
clusive and inconclusive. The ratio of these two sets of tense light-pulses [8, 9]. In this way a technique already
data is directly related to the noise of the channel thus employed in classical communications is adapted to the
allowing Bob to quantify and correct it. quantum realm, but carries a few drawbacks though.
In the following, we apply this method to correct the First, it has been recently shown that nonlinear effects
phasedrift ofacommunicationchannel. Themethodcan due to the propagation of intense light pulses in optical
beeasilygeneralizedtoincludeothersourcesofnoise,e.g. fiberscangeneratenoiseinthesensitivesingle-photonde-
that due to the birefringence of an optical fiber affecting tectors, thus limiting in practice the maximum distance
thepolarizationofalightpulse. Thephasedriftmodelis of a fiber-based QKD [10]. Second, the co-existence in
2
the same channel of dim and bright pulses makes it hard matrix ρ prepared by Alice is then:
to control the intensity of the incoming light, and this
can be exploited by an eavesdropper to hack the QKD ρ = (|ϕ0(cid:105)(cid:104)ϕ0|+|ϕ1(cid:105)(cid:104)ϕ1|)/2
system [11]. Finally this technique requires additional = β2|0 (cid:105)(cid:104)0 |+α2|1 (cid:105)(cid:104)1 |. (3)
x x x x
hardware to be implemented.
It can be easily verified that the above density ma-
A similar analysis holds for those systems which use
trix is asymmetric because it is not proportional to
a two-way configuration to enable a passive compensa-
the identity operator in the 2-dimension Hilbert space,
tion of the phase-drift [12]. Indeed the pulse sent in
I=|0 (cid:105)(cid:104)0 |+|1 (cid:105)(cid:104)1 |. Thisisaconsequenceofthestrict
the forward direction is necessarily intense, thus causing x x x x
nonorthogonality of the B92 protocol states. To decode
Rayleigh backscattering and opening the door to an in-
theinformation,Bobmeasurestheincomingstatesinthe
terrogation attack [13]. Moreover, this technique cannot
basis B = {|ϕ (cid:105),|ϕ (cid:105)}, k = {0,1}. Upon obtaining the
be employed with a single-photon source, even if some k k k
state|ϕ (cid:105),BobdecodesAlice’sbitasj =k⊕1(thesym-
attempts have been done in this direction [14]. k
bol ⊕ means “addition modulo 2”) and labels the result
There also exist solutions which employ quantum sig-
as conclusive; on the contrary, upon obtaining the state
nalsonly. Inonecase[15]thequantumtransmissionisin-
|ϕ (cid:105), Bob is not able to decode Alice’s bit deterministi-
terruptedandasequenceofquantumsignalswithafixed k
cally, and simply labels the result as inconclusive. For
phase value acting as a reference is sent by the transmit-
example, if Bob detects the state |ϕ (cid:105), then he can say
ter along the channel until the receiver announces that 0
with certainty that the prepared state was |ϕ (cid:105), because
the alignment has been completed . In another case [16] 1
|ϕ (cid:105) is orthogonal to the detected state. The same is
thequantumsignalswithafixedphasevalueareinserted 0
true for the detection of |ϕ (cid:105), which indicates that |ϕ (cid:105)
inthecommunicationusingadifferentwavelength,likein 1 0
was prepared. These are examples of conclusive results.
the classical frequency-multiplexing technique. Both of
On the contrary, if Bob detects the state |ϕ (cid:105), he will
these solutions have some disadvantages. In the former, 0
notbeabletodeterministicallyinferAlice’spreparation,
the interruption of the quantum transmission represents
because that state has a nonzero probability to come ei-
an idle cycle from which no secure bit can be distilled;
ther from |ϕ (cid:105) or from |ϕ (cid:105). Hence this is an example of
moreover this technique is not adaptive, i.e. the system 0 1
inconclusive result.
cannotadapttheinterruptionfrequencytotherealnoise
The heart of our technique is that the asymmetric
present on the channel. In the latter, the presence of
density matrix of Eq.(3) produces different amounts of
anextrawavelengthimpliesthatsingle-photondetectors
conclusive and inconclusive results in Bob’s measure. In
and electronics must deal with two wavelengths rather
particular the ratio between inconclusive and conclusive
than one, thus increasing the complexity of the setup
counts is a function of the angle θ, known to the users,
and reducing the key generation rate due to the fewer
and of the noise. Bob can estimate it during the quan-
detector’s windows available for the signal states.
tum transmission thus obtaining, in real time, informa-
Our solution represents an alternative way to control
tion about the noise, useful to eventually compensate it.
the channel noise at the quantum level without any of
To establish the connection with the noise, let us con-
the above drawbacks. The necessary resources for the
sideracommunicationchannelaffectedbythephase-drift
control are borrowed from the asymmetry of the same
model of noise (Fig. 1). In this case the density ma-
signals used for the very QKD process. So there is no
trix seen by Bob is no more the one given by Eq.(3) but
need to multiplex quantum signal with suitably tailored
rather a new matrix ρ composed by the noise-affected
(cid:101)
bright or reference pulses, or to interrupt the quantum
quantum states |ϕ (cid:105) and |ϕ (cid:105). These states can be ob-
(cid:101)0 (cid:101)1
communication at all.
tained by rotating those of Eq.(1) through the operator
To explain in detail how the compensation technique U =exp(cid:0)iεY(cid:1), with Y the usual Pauli operator and ε
ε 2
works it is useful to recall the B92 protocol encoding- the misalignment angle:
decodingmechanism[2]. Letuswriteexplicitlythequan-
tum states of the protocol: |ϕ (cid:105) = cos[(θ+ε)/2]|0 (cid:105)+sin[(θ+ε)/2]|1 (cid:105), (4)
(cid:101)0 x x
|ϕ (cid:105) = cos[(θ−ε)/2]|0 (cid:105)−sin[(θ−ε)/2]|1 (cid:105). (5)
(cid:101)1 x x
|ϕ (cid:105) = β|0 (cid:105)+(−1)jα|1 (cid:105), (1)
j x x
From the density matrix ρ is then possible to calculate
|ϕ (cid:105) = α|0 (cid:105)−(−1)jβ|1 (cid:105). (2) (cid:101)
j x x the various probabilities associated to Bob’s measure-
ment. To make our description more realistic we intro-
In the above equation |ϕj(cid:105) are the signal states of the duce the quantity η, which is the probability to detect
protocoland|ϕj(cid:105)arethestatesorthogonaltothem,with a single-photon state, i.e. a state different from a vac-
j = {0,1}; |0x(cid:105) an√d |1x(cid:105) are the eigenstates of the Pauli uum or a multi-photon state. It can be thought as a
operator X; β = 1−α2 = cos(θ/2) and θ belongs to sort of total transmission of the QKD setup, including
the open interval (0,π/2). the transmission of the communication channel, η , the
C
To start the communication, Alice chooses at random efficiency of Bob’s detectors, η , and the probability of
B
the value of the bit j, encodes it in the corresponding doubleclicksinBob’sdetectors[3]. Withthisinmind,we
state|ϕ (cid:105)andtransmitsittoBob. Theresultingdensity writedowntheprobabilitythatBobgetsaninconclusive
j
3
outcome, Pkinc = η(cid:104)ϕk|ρ(cid:101)|ϕk(cid:105), or a conclusive outcome, (cid:233)(cid:233)(cid:233)(cid:233)(cid:233)(cid:233)(cid:233)(cid:233) (cid:233)(cid:233)(cid:233)(cid:233)(cid:233)(cid:233)(cid:233)(cid:233)
Pkcon =η(cid:104)ϕk|ρ(cid:101)|ϕk(cid:105), when he measures in the basis Bk: 4 (cid:233)(cid:233) (cid:233)(cid:233) (cid:233)(cid:233) (cid:233)(cid:233)
(cid:233) (cid:233) (cid:233) (cid:233)
Pkinc = η{2+cosε+cos[2θ−(−1)kε]}/4 (6) ons (cid:233)(cid:233) (cid:233)(cid:233) (cid:233)(cid:233) (cid:233)(cid:233)
Pcon = η{2−cosε−cos[2θ−(−1)kε]}/4. (7) cti 3 (cid:233)(cid:233) (cid:233)(cid:233) (cid:233)(cid:233) (cid:233)(cid:233)
k n (cid:233) (cid:233)(cid:233) (cid:233)
The ratio Rk of the two above probabilities is indepen- olfu 2 (cid:233)(cid:233)(cid:233) (cid:233)(cid:233)(cid:233)(cid:233)(cid:233)(cid:233) (cid:233)(cid:233)(cid:233)
dentofηandisacrucialquantity,calledcontrolfunction: ntr (cid:233)(cid:233)(cid:233) (cid:233)(cid:233)(cid:233) (cid:233)(cid:233)(cid:233) (cid:233)(cid:233)(cid:233)
Rk(θ,ε)= 22+−ccoossεε−+ccooss[[22θθ−−((−−11))kkεε]]. (8) Co 1(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)(cid:227)(cid:227)(cid:233)(cid:233)
0
A few examples of Rk are plotted in Fig. 2; the pa- (cid:45)Π (cid:45)Π 0 Π Π
rameters used to draw the curves are k = {0,1} and 2 2
θ = {5π,π,4π}. Among these curves, some are better Misalignmentangle
18 3 9
than others to drive the noise-compensation mechanism.
IftheabsolutevalueofRk istoosmall,likeforthecurves FIG. 2: Examples of control functions Rk(θ,ε) versus the
withθ = 4π,thesystemislessresponsivei.e. bigchanges misalignmentangleε. TheR0 curves(R1 curves)reachtheir
9 maximumintheright-part(left-part)ofthefigure. Thefigure
ofεcausesmallchangesofR . Inthiscasetheriskisthat
k
showsthatagivenmisalignmentanglecausesacorrespondent
themisalignmentanglebecomestoolargebeforeBobbe-
valueinthecontrolfunctions. Bobcanexploitthiscorrespon-
comes aware of it and applies the correction mechanism.
dence to measure the control functions and from that ascer-
On the contrary, the higher the R the fewer the con-
k tain the value of the misalignment angle.
clusive counts registered by Bob. This results in larger The R (π,ε) are drawn with solid lines; the R (5π,ε) with
flcoumctpueantsioantisoninmtheecheasntiimsma,tiaosniotfhRakp,ptehnussfowroersxeanminpglethtoe egmenpttsytkociRr3cle(sπ;,tεh)einRtkh(e49πze,rεo)-nwoiitsheepmoipnttyasrqeuinardeksa.s1h8Tedhelintaesn.-
the curves featuring θ = 5π in Fig. 2. So the best op- k 3
18
tion is to choose an intermediate value of θ that at the
same time provides Bob with a good statistics and a re-
one based on the relative-phase degree of freedom, in
sponsive control. By consequence we choose θ = π (60◦)
3 a one-way configuration. It is constituted by two iden-
andplotinFig.2thecorrespondingcontrolfunctionsto-
tical interferometers placed in Alice’s and Bob’s sta-
gether with their tangents in the zero-noise point. Such
tions [2]. Each of the interferometers features two un-
a value of θ is also interesting because it allows to merge
equal paths which provide the traveling pulses with a
in a single protocol the present technique and that de-
different amount of optical phase. The phase difference
scribed in [5], that is an efficient long-distance version of
can be easily modulated by the users, who are then able
the B92 protocol featuring an optimal θ of about 0.3π
to encode information in this way. In particular, the
(precisely, 55.4◦ [5]).
output ports of the receiver’s interferometer are usually
Whenthenoiseissmall,thecontrolfunctionsofEq.(8)
connected with two detectors that determine the conclu-
can be well approximated by their tangents in the zero-
siveness or inconclusiveness of a certain result.
noise point; furthermore they are monotone, so only one
control function, either R or R , is sufficient to pro- Tocorrectthenoise,Bobexecuteshismeasurementfor
0 1
vide a reliable estimation of ε. This makes the feedback awhile,registeringalltheresultsinacomputermemory.
response very fast and we refer to this situation as to Then, when a sufficient number of occurrences is avail-
a “fast feedback”. On the contrary, when the noise is able, Bob estimates the control functions of Eq.(8) and
largeandfallsoutsidethemonotonicityrangeofthecon- obtains a value of the misalignment angle ε. The last
trol functions, Bob must use both the control functions thingBobhastodoistorecalibratehisphase-modulator
to estimate unambiguously the misalignment angle ε in with the obtained value to re-establish the correct align-
the open interval (−π,π). This procedure is intrinsically ment of his apparatus with Alice’s. The read-and-write
slower than the previous one, so we term it “slow feed- procedure just described is performed by Bob in-course-
back”. From a practical point of view, both the options of-action,andcanbeeasilyimplementedbyanelectronic
arehelpful. Forexample,atthebeginningofacommuni- feedback loop in which the estimated value of Rk(θ,ε) is
cation the users’ apparatuses are completely misaligned the input, the fixed value of Rk(θ,0) is the setpoint, and
andεcanbequitelarge;Bobwillthenusetheslowfeed- a function of these two quantities is the output.
backtogetafirstestimationofεandcompensateit;after We have carried out numerical simulations of such an
that,whentheboxesarenearlyaligned,Bobwillusethe active feedback loop in the case of θ = π and the re-
3
fast feedback to improve the alignment and maintain it sults are reported in Fig. 3. In the upper part of the
during the remaining quantum transmission. figure we considered a linear increase in time of the mis-
At this point let us mention a possible experimen- alignment angle, with a rate equal to 0.05 rad/sec. This
tal implementation of the mechanism used to correct value is easily attainable with some care in the shield-
the noise. The QKD layout we are interested in is the ing process of the users’ interferometers against external
4

0.5 ationiswellbelowtheamountofphase-driftaccumulated
every second on the channel.
0 .0
InthelowerpartofFig.3wehaveconsideredaphase-
d) 2 drift in the form of a step-function of amplitude equal to
a 2 rad, to study the response of the slow feedback loop
r
( in the case of a large phase-drift. In this case Bob ex-

es 1 ploits both R (π,ε) and R (π,ε) to estimate the mis-
l 0 3 1 3
g alignment angle. After acquiring 103 events he evaluates
n
a 0 the mean values of R and R , let us term them R∗ and
0 1 0
Phase -1 mRfin1∗iza.ellsyTthuheseensqhutheaenntvuiatmylue|eRrioc0fa(εlπ3l∗y,εtfio)n−cdoRsm0∗tp|he+ensp|aRoti1en(ttπ3hε,e∗εp)th−haaRste1∗-md|.riiHnftie-.
The empty circles of the figure again represent the value
-2 ofthemisalignmentangleafterthefeedbackhasbeenap-
plied. It can be seen that in correspondence of the noise
0 2 4 6 8 10
stepstheangleseenbyBobundergoesstrongjumps,but
Time (sec) the system recovers immediately after the jump. If one
ignores the jumps, the mean misalignment angle is:
FIG. 3: Noise patterns and feedback responses versus time. εslow =(−0.002±0.098) rad. (10)
fb
Top: linear noise with a rate of 0.05 rad/sec (filled squares)
and response after fast feedback is applied with a statistics ComparedtoEq.(9),theaveragevalueisnearlyunbiased,
of 5×103 events per point (empty circles). The mean angle following the unbiased behavior of the noise, while the
after feedback is εfast = (0.023±0.031) rad. Bottom: step standard deviation is more than three times bigger due
fb
noisewithamplitude2rad(filledsquares)andresponseafter tothereducedstatisticsadoptedforthiskindoffeedback.
slow feedback is applied with a statistics of 103 events per Itcanbenicetonotethatthestep-noisepatternofFig.3
point (empty circles). Other parameters used are (see text): could be employed by the users to create an additional
θ= π3, f =2 MHz, µ=0.5, ηB =10%, ηC =0.1. communication channel between them. More explicitly,
Alice could purposely feed into her phase-modulator a
large phase value in order to send some sort of message
thermal fluctuations. Phase drift values reported in the toBob,likee.g. astringofbitsusefultoidentifyacertain
literaturearewellbelowourthreshold,e.g. 0.033rad/sec part of the quantum transmission.
in[15]or0.0086rad/secin[16]. Wehavealsoassumeda As a last point of our work we want to establish
trigger rate f = 2 MHz, an average photon number per whethertheresultsofEqs.(9)and(10)aregoodenough
pulse µ = 0.5, a detector efficiency ηB = 10%, a chan- for a practical implementation of the B92 protocol. In
nel transmittance ηC = 0.1, equal to about 50 km of a particular we calculate the maximum misalignment an-
standardopticalfiberinthethirdTelecomwindow. Mul- gle for which the secure gain of the B92 protocol is still
tipliedtogether,thesevaluesprovideBobwithanumber positive. The secure gain of a protocol is the ratio be-
ofnon-vacuumeventspersecondoftheorderof104. Let tween the number of secure bits distilled and the num-
usnotethatthisrepresentsawideunderestimationofthe ber of qubits prepared by Alice. For the B92 protocol
cthuerrpenlottQteKdDcuprveerfsorismtahnecerses(usleteoef.ga.n[1a7v]e)r.aEgeacohfp5o×int10o3f it is defined as G = Λcon[1−h(ΛΛcboint)−h(ΛΛcpohn)], with
h the Shannon entropy, Λ the conclusive-count rate,
con
acquisitions, obtainedinpractice byapplyingafeedback
Λ the bit-error rate and Λ the upper bound to the
bit ph
kick every half a second. Following the approach in [16],
phase-errorrateobtainedfromΛ andΛ throughan
bit con
we have empirically verified the optimality of this value:
optimization algorithm [3, 5].
iflarger, thephase-driftwouldbecometoobiginthebe-
Thebit-errorrateisgivenbytheprobabilitythatBob
tween of two consecutive feedback kicks; if smaller, the
finds the state |ϕ (cid:105) (|ϕ (cid:105)) when Alice prepares the or-
statistical error pertaining to Bob’s measurement would 0 1
thogonal state |ϕ (cid:105) (|ϕ (cid:105)). For the present discussion it
0 1
increase considerably. The empty circles in the upper
suffices to assume that the phase-drift is the only source
part of Fig. 3 show the response of the fast feedback
oferrors;inrealitytherearealsothedetectordarkcounts
loop, i.e. the one in which a linear approximation of the
and the unavoidable experimental imperfections. By us-
control functions is adopted. In particular we used the
ing the noisy states of Eqs. (4) and (5), and assuming a
first-order expansion of R (π,ε) in ε = 0 to drive the
0 3 preparation of the bit value 0 (the same holds for the bit
feedback. With feedback, the mean misalignment angle
value 1), we obtain the following bit-error rate:
of the communication channel reduces to:
Λ = η(|(cid:104)ϕ |ϕ (cid:105)|2)/2
εfast =(0.023±0.031) rad. (9) bit (cid:101)0 0
fb = η(1−cosε)/4. (11)
The slight bias towards positive values is due to the Thisexpressionisindependentofθandcorrectlyvanishes
monotoneincreaseofthephase-drift. Thestandarddevi- whenεtendstozero. Thecoefficientηtakesintoaccount
5
the vacuum or multi-photon counts by Bob, while the Alice. It suffices that Bob gets an estimate of ε from
factor 1/2 is the probability to guess the right basis to his data and substitutes it into the given equations [18].
detect the error. This is a further peculiarity of the B92 protocol that
In a similar way, the conclusive-count rate is given by remained unnoticed so far. It can considerably increase
the probability that Bob obtains a conclusive result, and the practicality of the protocol by reducing the classical
can be evaluated from the mean value of Pcon and Pcon communication necessary to distill the final key.
0 1
in Eq. (7):
In conclusion, we introduced a novel scheme to detect
Λ = (Pcon+Pinc)/2 and correct the phase-drift of a communication channel
con 0 1
= η(1−cosεcos2θ)/2. (12) based on some characteristics of the B92 protocol not
considered so far. The scheme features a few remarkable
Thistime,thereisadependanceonθ. Afterfixingθ = π properties: itisentirelyquantum,itcanbeexecutedreal-
and using Eqs. (11) and (12), we have numerically foun3d time without interrupting the communication, it allows
a positive gain for the B92 protocol until toestimatethebit-errorratewithoutabidirectionalcom-
munication, it creates additional communication chan-
|ε|<0.27646 rad. (13) nels for the users. This highly increases the practicality
of the B92 protocol, often considered unsuitable for real-
Thevalueofεgivenbythefastfeedback,Eq.(9),ismuch world implementations. Furthermore, the fully quantum
smaller than the above threshold. Hence our proposal natureoftheschemeononesidereducesthenoisedueto
appearsfeasible,especiallyifoneconsidersthatwemade thepropagationofhigh-intensitylightpulsesinanonlin-
conservative assumptions on the QKD parameters and ear medium, on the other makes it conceivable the con-
that less conservative assumptions would lead to better structionofnetworksanddevicesworkingentirelyatthe
values in Eqs. (9) and (10). quantum level, thus preventing several hacking strate-
Let us point out that Eqs. (11) and (12) establish a gies available to Eve. The asymmetry-based correction
direct dependance of the bit-error rate and conclusive- mechanismcanbeextendedtothepolarizationdegreeof
countrateonthemisalignmentangleε. ThisallowsBob freedom and can play a role in the entanglement distri-
to estimate Λ and Λ without communicating with bution problem.
bit con
[1] N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, Rev. [11] V. Makarov, New J. Phys. 11, 065003 (2009).
Mod. Phys. 74, 145 (2002). [12] H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A.
[2] C. H. Bennett, Phys. Rev. Lett. 68, 3121 (1992). Muller, and W. Tittel, Electron. Lett. 33, 586 (1997).
[3] K. Tamaki, M. Koashi, and N. Imoto, Phys. Rev. Lett. [13] A. Vakhitov, V. Makarov, and D. R. Hjelme, J. Mod.
90,167904(2003);K.TamakiandN.Lu¨tkenhaus,Phys. Opt. 48, 2023 (2001).
Rev. A 69, 032316 (2004). [14] M. Lucamarini and S. Mancini, Phys. Rev. Lett. 94,
[4] M. Koashi, Phys. Rev. Lett. 93, 120501 (2004); K. 140501(2005);A.Cer`e,M.Lucamarini,G.DiGiuseppe,
Tamaki, N. Lu¨tkenhaus, M. Koashi, and J. Batuwantu- and P. Tombesi, ibid. 96, 200501 (2006); R. Kumar, M.
dawe, Phys. Rev. A 80, 032302 (2009); K. Tamaki, ibid. Lucamarini,G.DiGiuseppe,R.Natali,G.Mancini,and
77, 032341 (2008). P. Tombesi, Phys. Rev. A 77, 022304 (2008).
[5] M. Lucamarini, G. Di Giuseppe, and K. Tamaki, Phys. [15] V.Makarov,A.Brylevski,andD.R.Hjelme,Appl.Opt.
Rev. A 80, 032327 (2009). 43, 4385 (2004).
[6] M. Born and E. Wolf, Principles of Optics, 7th (ex- [16] B.B.Elliott,O.Pikalo,J.Schlafer,andG.Troxel,Proc.
panded) edition, Cambridge University Press (1999). SPIE 5105, 26 (2003).
[7] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. [17] Z.L.Yuan,A.R.Dixon,J.F.Dynes,A.W.Sharpe,and
Mod. Phys. 79, 555 (2007). A. J. Shields, Appl. Phys. Lett. 92, 201104 (2008).
[8] C.MarandandP.Townsend,Opt.Lett.20,1695(1995). [18] Since Bob knows the expected number of single-photon
[9] Z.L.YuanandA.J.Shields,Opt.Expr.13,660(2005). events, he can easily appraise η from the single-photon
[10] D. Subacius and A. Zavriyev and A. Trifonov, Appl. events effectively registered by his measuring device.
Phys. Lett. 86, 011103 (2005).