Non-Gaussian postselection and virtual photon subtraction in continuous-variable quantum key distribution

0.56 MB·

by  Zhengyu Li

Non-Gaussian postselection and virtual photon subtraction in continuous-variable
quantum key distribution
Zhengyu Li1,2, Yichen Zhang3, Xiangyu Wang3, Bingjie Xu2, Xiang Peng1†, and Hong Guo1∗
1State Key Laboratory of Advanced Optical Communication Systems and Networks,
Center for Computational Science and Engineering and Center for Quantum Information Technology,
School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
2Science and Technology on Security Communication Laboratory,
Institute of Southwestern Communication, Chengdu 610041, China and
3State Key Laboratory of Information Photonics and Optical Communications,
Beijing University of Posts and Telecommunications, Beijing 100876, China
(Dated: January 13, 2016)
6
Photonsubtractioncanenhancetheperformanceofcontinuous-variablequantumkeydistribution
1
0 (CV QKD). However, the enhancement effect will be reduced by the imperfections of practical
2 devices, especially the limited efficiency of a single-photon detector. In this paper, we propose a
non-Gaussian postselection method to emulate the photon substraction used in coherent-state CV
n
QKDprotocols. Thevirtual photonsubtractionnotonlycanavoidthecomplexityandimperfections
a
of a practical photon-subtraction operation, which extends the secure transmission distance as the
J
idealcasedoes,butalsocanbeadjustedflexiblyaccordingtothechannelparameterstooptimizethe
2
performance. Furthermore, our preliminary tests on the information reconciliation suggest that in
1
thelowsignal-to-noiseratioregime,theperformanceofreconciliatingthepostselectednon-Gaussian

data is better than that of the Gaussian data, which implies the feasibility of implementing this
]
h method practically.
p
- PACSnumbers: 03.67.Dd,03.67.Hk
t
n
a
u I. INTRODUCTION imental techniques, quantum operations were also pro-
q posed to improve the performance of CV QKD, such as
[ the noiseless linear amplification (NLA) operation [20–
Quantum key distribution (QKD) [1, 2] is the most

22]. It can increase the transmission distance roughly
1 applicable technology of quantum information, which
by the equivalent of 20log g dB losses, where g is the
v can allow two users (Alice and Bob) to establish se- 10
9 cure keys remotely through an insecure quantum chan- gain of the NLA. Furthermore, to avoid the difficulty
9 nel controlled by an eavesdropper (Eve). QKD has two of sophisticated physical NLA operations [23–26], non-
7 deterministicvirtual NLAviaGaussianpostselectionhas
main branches, i.e., discrete-variable (DV) QKD and
2 been proposed [27, 28] and experimentally implemented
continuous-variable (CV) QKD [3–5], in which the in-
0
formation is carried by the quadratures (xˆ and pˆ) of the [29].
.
1 Alternatively, a photon-subtraction operation [30–33]
lightfield. CVQKDprotocolsusingGaussianmodulated
0 coherentstates[6–8]notonlyhavebeenprovedtobeun- is shown to be able to significantly improve the trans-
6
mission distance of CV QKD protocols using two-mode
conditionally secure in theory [9–13], but also have the
1
: advantage of being compatible with standard telecom- squeezed vacuum (TMSV) states [34]. By exploiting
v munication technology, which leads to an expectation of the equivalence between the entanglement-based (EB)
Xi better application. However,limited by the practicalex- scheme and the prepare-and-measure (PM) scheme, it
perimental techniques and the non-perfect reconciliation can also be employed in protocols using coherent states.
r
a efficiency, the transmission distances of early CV QKD However, the improvement will be reduced by the im-
setupswerenotsufficiently longfornetworkapplications perfections of devices in a practical photon-subtraction
operation, especially the single-photon detector (SPD),
[14–16]. Thus, research on extending the transmission
distance has attracted much attention in the past few which makes this method unfeasible (see Appendix A).
years. Here, we propose a non-Gaussian postselection
method to emulate the photon subtraction used in
Recent major progressin CV QKD in experiment [17]
coherent-state CV QKD protocols, which is employed
was achieved with 80 km transmission distance using
right before the emission of the coherent states. One
coherent states by taking advantage of the multidimen-
advantage of this virtual photon subtraction is that, it
sionalreconciliationprotocol[18,19]inthelowsignal-to-
can not only remove the complex physical operations,
noise ratio (SNR) regime and the optimization of other
butalsoemulatetheidealphoton-subtractionoperations.
experimentalaspects. Besidestheimprovementofexper-
Another advantageis thatthe postselectioncanbe post-
poned after the parameter estimation, and therefore it
can be adjusted flexibly to optimize the performance.
∗Correspondingauthor: [email protected]. Besides, the postselection filter function does not need
†Correspondingauthor: [email protected]. a cutoff amplitude as does the one in virtual NLA [27–
2
29], because it is bounded. (a)
Furthermore,ourpreliminarytests onthe information
reconciliationofthepostselectednon-Gaussiandatasug-
gest that the multidimensional reconciliation algorithm Het A TMSV B (cid:513)(cid:882)(cid:1767)T B2 (TC,e) B3 Het/Hom
can be directly used for the virtual photon-subtraction B1 BS1
method. Especially in the low signal-to-noise ratio g=xA+ipA BS2 B3Ideal Channel
regime, the performance of reconciliating the postse- Det
lected non-Gaussian data is even better than that of Alice P(cid:513)r(cid:882)a(cid:1767)ctical detector Bob
(cid:2015)
the Gaussian data, which implies the feasibility of im-
plementing this method practically.
This paper is organized as follows: In Sec. II, we in- (b)
troduce some basics of the photon subtraction and pro-
pose the equivalent postselection method of the photon-
A Gaussian B B
subtraction operation in a coherent-state CV QKD pro- Q(g,l,T) Distribution: 2 (TC,e) 3 Het/Hom
g=x +ip Output
tocol. InSec. III, we presentthe performanceofthe vir- A A
coherent state Channel
ttuhaelopphtiomtoanl cshuobitcreacotfiotnhethpraoruagmhenteurmienricvairltusiaml uplahtoitoonn, Alice aB = 2Tlg2 Bob
subtraction, and the tests of information reconciliation
on the postselected non-Gaussian data. In Sec. IV, we
FIG. 1: (Color online) (a) Entanglement-based (EB) scheme
summarize the paper.
of CV QKD with photon subtraction. (b) Prepare-and-
measure (PM) scheme of CV QKD with equivalent post-
selection as virtual photon subtraction. Het: heterodyne
II. PHOTON SUBTRACTION AND ITS detection; Hom: homodyne detection; BS1(2): beams plit-
EQUIVALENT POSTSELECTION IN CV QKD ter; γ: Alice’s measurement result; λ: parameter of TMSV;
Q(γ,λ,T): postselection filter function; T(η): transmittance
Photon subtraction can enhance the entanglement of of BS1(2); TC,ε: channel parameters.
the TMSV state. Various scenarios were proposed, in-
cluding applying photon subtraction on one mode or
A andB will be keptonly when the POVMelement Πˆ
both modes of the TMSV [32], directly or after a pure 2 1
lossychannel[35]. Tomakeourderivationself-contained, clicks. The kept state ρΠˆ1 is denoted as the photon-
AB2
in this section, we first introduce the basics of photon subtracted TMSV state,
subtraction on a TMSV state (noted as the photon-
subtracted TMSV state), then propose its equivalent ρΠˆ1 = trB1(Πˆ1ρAB1B2) . (3)
non-GaussianpostselectioninCVQKDprotocols(noted AB2 tr (Πˆ ρ )
AB1B2 1 AB1B2
as virtual photon subtraction).
where tr () is the partial trace of a multimode quan-
X
·
tum state, and PΠˆ1 =tr (Πˆ ρ ) is the success
A. Photon Subtraction of a TMSV State probability of Πˆ clicks.AB1B2 1 AB1B2
1
Different Πˆ will lead to different types of photon-
1
TheTMSVstateinvolvestwomodesAandB. aˆ,aˆ† subtraction. A general photon subtraction operation is
{ }
and ˆb,ˆb† denotetheannihilationandcreationoperator subtracting k photons, which refers to Πˆ = k k and
of m{odes}A and B, respectively, where [aˆ,aˆ†] = [ˆb,ˆb†] = canbe realizedby aphotonnumberresolv1ing(|PiNhR|)de-
1. A TMSV state can be expressed by tector[36]. Anditisshownin[33]thattheentanglement
will increase as more photons are subtracted. The pho-
∞
TMSV = 1 λ2 λn n,n , (1) ton subtraction can also be extended to the mixture of
| i − | i subtracting different k photons, Πˆ = ∞ c k k and
p nX=0 1 k=0 k| ih |
c 0,1 , among which Πˆ = ∞ Pk k =I 0 0
where λ [0,1), m,n = m n , and n k ∈{ } on k=1| ih | −| ih |
∈ | i | iA ⊗| iB {| i}n∈N corresponds to an on-off detectPor. See Appendix B for
denotes the Fock state.
more details about the above two examples.
The EB scheme of CV QKD with photon subtraction
inside Alice is shown in Fig. 1(a). After generating
the TMSV state, Alice uses a beams plitter (BS1), with
B. Virtual Photon Subtraction in CV QKD via
transmittanceT,tosplitthe modeB intomodesB1 and Postselection
B , getting a tripartite state ρ ,
2 AB1B2
ρ =U [TMSV TMSV 0 0]U† . (2) We suppose Alice uses the photon-subtracted TMSV
AB1B2 BS | ih |⊗| ih | BS state as the source of a CV QKD system, and she will
Then B will be measured by a positive operator-valued perform a heterodyne detection on mode A. As shown
1
measure(POVM)measurement Πˆ ,Πˆ ,andthemodes in Fig. 1(a), the measurement result of a single-photon
0 1
{ }
3
detectorrepresentseitherkeepingthisstate(click)ornot andV = 1+λ2 1 λ2 is the variance ofthe TMSV
−
keeping this state (no click). Alice needs to record this state. (cid:0) (cid:1)(cid:14)(cid:0) (cid:1)
extradataforeachTMSV,andwillrevealittoBobafter In the case where Alice does not use any kind of
Bob measures the mode B . photon-subtraction operation, the output mixed state is
3
According to the extemality of Gaussian quantum
states [9, 10, 37], the secret key rate of the state ρΠAˆ1B3 ρB(G) =Z dxAdpAPxA,pA|αihα|. (7)
isnolessthanaGaussianstateρG whichhasthesame
AB3
covariancematrix, where K(ρΠˆ1 ) K(ρG ). Thus we Compared with the postselected state in Eq. (6), there
AB3 ≥ AB3 aretwodifferences. Firstly,thereisanadditionalweight-
will use ρG to derive the secret key rate. Besides, the
AB3 ing function in Eq. (6),
success probability of Alice’s POVM measurement PΠˆ1
should also be taken into account. Thus, for the reverse PΠˆ1(k xA,pA)
W = | ,
reconciliation, the lower bound of the asymptotic secret PΠˆ1(k)
key rate under collective attack is
whichleadstoafilterfunction,oracceptanceprobability,
K ρG =PΠˆ1 βIG(A:B) SG(E :B) , (4) of each pair of xA,pA ,
AB3 − { }
(cid:0) (cid:1) (cid:0) (cid:1) Q(γ,λ,T)=PΠˆ1(k)W =PΠˆ1(k x ,p ). (8)
where β is the reconciliation efficiency, IG(A:B) is the | A A
mutualinformationbetweenAliceandBob,SG(E :B)is Second, the output coherent state needs to go through
theHolevobound[38]ofthemutualinformationbetween the BS1 with transmittance T, which can be emulated
Bob and Eve. The calculation method of K is shown in viageneratingacoherentstatewithasmallermeanvalue
the Appendix C. √Tα.
Next, we will present the equivalent virtual photon Thus, after exchanging Alice’s heterodyne measure-
subtraction via postselection according to Alice’s mea- ment and the photon-subtraction operation, we get the
surement results, which will benefit the system. First, equivalentvirtual photonsubtractionviapostselectionof
it is not necessary to accomplish the practical photon Alice’s measurement results and scaling the mean value
subtraction which reduces the complexity of the system. of output coherent state by a factor √T. The postselec-
Second,ithasbetterperformancethanthepracticalpho- tion filter function is shown in Eq.(8).
ton subtraction since one can emulate the ideal detector Insummary,the PMschemeofCVQKDusingvirtual
case. photon subtraction, depicted in Fig. 1(b), is as follow:
The heterodyne detection and the POVM measure- Step 1. Alice generates a coherent state α , where
ment Πˆ0,Πˆ1 are commutable since they are con- α = √2Tλγ/2, γ = x + ip , and x ,p | iare cho-
{ } A A A A
ducted on two different modes. Thus, Alice can per- { }
sen randomly from a Gaussian distributed set with zero
form the heterodyne detection on mode A first, and
mean and variance V = V = (V +1)/2. T is the
then the POVM measurement on mode B . It is xA xB
1 transmittanceof BS1andV is the varianceofthe equiv-
known that heterodyne detection on one mode of the
alent TMSV. Then she sends the coherent state to Bob.
TMSV state will project the other mode onto a coher-
Step 2. After receivingthe state,Bobwillperformho-
ent state; thus after BS1, the state of modes B and B ,
1 2 modyne or heterodyne detection, and the measurement
given that Alice’s heterodyne measurement results are
results are denoted by x ,p .
B B
{wxhAer,epAα},=is√2(cid:12)(cid:12)ϕλ((xxAA,pA−)(cid:11)iBpA1B)22.= T(cid:12)(cid:12)√he1−sucTcαes(cid:11)sB1p(cid:12)(cid:12)(cid:12)r√obTaαbEilBit2y, laecccStetpeeptneod3u,.gahnSddteaprtesav.1eaAlaslnictdhee2wdilaelcreidseirocenipdseeatowtehBdicouhbn.dtiaTlthatehweayiclclceobple--
of subtracting k photons, g(cid:14)iven Alice’s heterodyne mea-
tanceprobabilityforeachdataisQ(γ,λ,T)asinEq. (8).
surement results, will be the function of x ,p ,
{ A A} Then Alice and Bob use the accepted data to finish the
postprocessingsteps,includingparameterestimation,in-
PΠˆ1(k x ,p )= k √1 Tα 2
| A A − formation reconciliation and privacy amplification.
=exp (1−T)λ2 (cid:12)(cid:12)x(cid:10)2 (cid:12)(cid:12)+p2 (1(cid:11)−(cid:12)(cid:12)T)λ2 x2 +p2 k k! Since Alice reveals her decision of whether or not she
h− 2 (cid:0) A A(cid:1)i·h 2 (cid:0) A A(cid:1)i (cid:30) accepts each data after Bob’s measurement, the dis-
(5) carded states can be seen as the decoy states, which are
Thenthe mixed state outputfromAlice’s stationwillbe usedintheformernon-Gaussianprotocol[39]toenhance
the security.
ρ(k) = dx dp PΠˆ1(k|xA,pA)P √Tα √Tα , We note that for the practicalimplementation, the se-
B2 Z A A PΠˆ1(k) xA,pA(cid:12) ED (cid:12) cretkeyrate,givenbyEq. (4),shouldbemodifiedtotake
(cid:12) (cid:12)
(cid:12) (cid:12) finite-size effects into account [40], which indicates that
weighting function
the acceptance probability will influence the finite-size
| {z } (6)
analysis. However,thiswillnotaffecttheeffectivenessof
where PxA,pA = π(V1+1)exp(cid:16)−x2AV++p12A(cid:17) is the Gaussian our method; therefore, we only consider the asymptotic
distribution of Alice’s heterodyne measurement results, rate here for simplicity.
4
III. PERFORMANCE OF THE PROTOCOLS 101
Original Protocol
1−photon subtraction
In this section,we firstpresentthe performanceof the se)100 23−−pphhoottoonn ssuubbttrraaccttiioonn
protocolsusingphotonsubtractionintermsofsecretkey pul10−1 4−photon subtraction
rualatetioannsd. tTolheeranbwleeedxicsecsusssnotihsee,ttrharnosumgihttnanucmeeroifcaAlliscime’s- ate (bit/10−2
BS1, and the reconciliation efficiency of the postselected key r10−3
non-Gaussian data. As described in Sec. II B, we will et
use Eq.(4) as the asymptotic secret key rate. Since it Secr10−4
only involves the Gaussian state, the covariance matrix
of ρ(k) will be sufficient to get the rate, which can be 10−5
AB3
gottenaccordingtoAlice’sandBob’saccepteddatawhen
10−6
implementing this protocol practically. Here in the sim- 0 50 100 150 200 250
Transmission distance (km)
ulationwe assumethat the channelcan be characterized
(a)
by two parameters: the channel transmittance T and
C 1
excess noise ε, which means if the covariance matrix of 1−photon subtraction
ρ(k) is 0.9 23−−pphhoottoonn ssuubbttrraaccttiioonn
AB2 4−photon subtraction
0.8
V I φ σ
γ(k) = A AB Z , (9)
AB2 (cid:18)φABσZ VBI (cid:19) 0.7
where I= diag(1,1) and σ = diag(1,-1), then, after the T 0.6
Z
channel transmission,
0.5
V I √T φ σ
γ(k) = A C AB Z , (10) 0.4
AB3 (cid:18)√TCφABσZ TC(VB +χ)I(cid:19)
0.3
0 50 100 150 200 250
where χ = (1 T )/T +ε. The explicit form of γ(k) Transmission distance (km)
can be found i−n ACppenCdix D. And for the rest of theApBa2- (b)
per, we assume Bob uses homodyne detection. We note
FIG.2: (Coloronline)(a)Themaximalsecretkeyrateateach
thatEve’soptimalattackforthis non-Gaussianprotocol
transmission distance,whenchangingthetransmittanceT of
is still an open question.
Alice’s BS1. (b) The optimal T for the maximal secret key
rate in (a). The uppermost black solid line in (a) represents
thecaseoforiginalprotocol. Otherlinesrepresentone-photon
A. Secret Key Rate and Tolerable Excess Noise subtraction (blue solid line), two-photon subtraction (green
dashedline),three-photonsubtraction(pinkdottedline),and
For the virtual k-photon subtraction, the transmit- four-photon subtraction (red dash-dotted line), respectively.
The simulation parameters are as follows: the variance of
tance T of Alice’s BS1 can be chosen arbitrarily from
TMSV state is V =20, channel loss is a=0.2dB/km, excess
0 to 1, which will result in the change of overall accep-
noise is ε=0.01, and reconciliation efficiency is β=0.95.
tanceprobabilityPΠˆ1(k)andalsothe covariancematrix
ofγ(k) . Thus,foreachtransmissiondistance, the secret
AB2
key rate varies with different T, and there should exist
isthe limitedacceptanceprobability,whichisbelow0.25
anoptimalchoiceofT foreachdistance tomaximizethe
under the parameters we used here (see Fig. 6).
secretkeyrate. Figure2(a)showsthemaximalsecretkey
rate at each distance for all possible T. And Fig. 2(b) Fig. 2(a) also shows that the one-photon subtraction
shows the optimal choice of T for each distance, specif- has the longest transmission distance compared to the
ically, only for distances with secret key rate more than other cases. The main reason is that, when subtracting
10−6, respectively. more photons, the non-Gaussianity is higher [33], which
The black solid line in Fig. 2(a) represents the case means the Gaussian state that has the same covariance
of the original protocol, which is outperformed by the matrix, ρG , is more noisy. Thus, when employing the
AB2
protocols of using photon subtraction at long-distance extremality of Gaussian quantum states to do the secu-
range,especiallythecaseofusingone-photonsubtraction rity analysis, it gets a worse result. More specifically, we
(blue solid line). This implies one of the advantages of rewrite the covariance matrix γ(k) as follows:
AB2
usingphotonsubtraction,thatis,expandingthemaximal
transmission distance. However, for the short-distance
range, even for the optimal choice of T, the secret key γ(k) = VAI ηA(VA2−1)σZ , (11)
rate is still worse than the original protocol. One reason AB2 (cid:18) η (V2 1)σ pη (V +χ )I (cid:19)
A A− Z A A A
p
5
nit) 100 Original Protocol noise at each distance for all possible T, and Fig. 3(b)
se u 12−−pphhoottoonn ssuubbttrraaccttiioonn tsohoFwigs.t2h(eao),ptthimeacalscehooficoeriogfinTalfporroetaocchold(ibstlaacnkces.olSidimliinlae)r
ot noi 10−1 34−−pphhoottoonn ssuubbttrraaccttiioonn is also outperformed by other cases at long transmission
h distancerange,whichimpliesanotheradvantageofusing
s
e ( photon subtraction that increases the maximal tolerable
s
oi 10−2 excessnoise for distantusers. It is alsoshownthat when
n
s the channel is less noisy, for instance, ε 0.005, all four
ces photon-subtraction operations will expa∼nd the maximal
x
e e 10−3 transmissiondistancetomorethan200km. Ontheother
bl hand, one could notice that the optimal choice of T for
a
er themaximaltolerableexcessnoiseshowsadifferentform
ol
T 10−4 from the one for the maximal secret key rate. That is
0 50 100 150 200 250 300
Transmission distance (km) because the tolerable excess noise is not affected by the
(a) overall acceptance probability, while the secret key rate
1 is.
1−photon subtraction
2−photon subtraction
0.9 3−photon subtraction
4−photon subtraction
B. The Transmittance of Alice’s Beamsplitter
0.8
0.7 In Step 1 of the PM scheme using virtual photonsub-
traction described in Sec. II, it requires Alice to know
T
0.6 the value of T in advance. For a relatively stable sys-
tem and environment, Alice can use the data of the last
0.5
run to approximately estimate the optimal T for this
run. However, when the system or environment changes
0.4
rapidly, the above method may not result in a suitable
estimation. In this case, one can linearly scale Alice’s
0 50 100 150 200 250 300
Transmission distance (km) heterodyne measurement results first and then accom-
(b) plish the postselection. After the linearly scaling, Al-
ice’s data follows a new Gaussian distribution P′
XA,PA
FIG.3: (Coloronline)(a)Themaximaltolerableexcessnoise with a different variance V′, which can be regarded as
at each transmission distance, when changing the transmit-
the heterodyne measurement results of a new equivalent
tance T of Alice’s BS1. (b) The optimal T for the maximal
TMSV. We assume T is the estimated value accord-
tolerableexcessnoisein(a). Theuppermostblacksolidlinein 0
ing to the data of the last run, and η is the real opti-
(a)representsthecaseoforiginalprotocol. Otherlinesrepre-
mal choice of this run. Let X = Gx , P = Gp ,
sentone-photonsubtraction(bluesolidline),two-photonsub- A A A A
traction (green dashed line), three-photon subtraction (pink and αA = √2λ′(XA iPA) 2 = T0/η α, where
− ·
dotted line), and four-photon subtraction (red dash-dotted G = √T0λ √ηλ′ and λ′2 =(cid:14)(V′ 1p)/(V′+1). Then
−
line), respectively. The simulation parameters are V = 20, the state Al(cid:14)ice initially sends out can be rewritten as
a=0.2dB/km, ε=0.01, and β=0.95.
ρ(k) = dx dp P √T α √T α
where V =2V˜ 1, χ =(1 η )/η , and =B2 dXRAdPAAP′XAA,xPAA,pA√(cid:12)(cid:12)ηαA0 h(cid:11)√hηαA0| |, (13)
A − A − A A R (cid:12) (cid:11)
whereP′ =P G2(cid:12),withx andp substituted
k+1 XA,PA xA,pA A A
ηA =λ2Tk+λ2T. (12) byXA/GandPA/G,ist(cid:14)henew Gaussiandistributionof
thescaleddata. ConsideringtheexplicitformofP′ ,
XA,PA
This means, in the viewpoint of calculating the secret the variance V′ fulfills that V′ 1=T /η (V 1).
0
− · −
key rate, the k-photon-subtracted TMSV can be seen After this linearly scaling step, Alice could do the
as a source with an extra loss on the B mode before equivalent postselection according to the new data
2
being emitted into the channel. And η decreasesas the X ,P . By traversing all possible value of η, one could
A A A
photon-subtraction number k increases, especially when find the optimal value of it to optimize the secret key
theoptimalT islowerforhigherkatlong-distancerange, rate at a certain distance. In this way, the key param-
which implies that subtracting more photons will result eter of virtual photon subtraction, the transmittance of
in more loss. Therefore, subtracting one photon shows Alice’s BS1, can be adjusted after Alice and Bob esti-
better performance under the conditions of Fig. 2, i.e., mate the channel parameters, which makes this method
the initial variance of TMSV is 20. more flexible.
Tolerable excess noise represents another aspect of a Another problem about the choice of T is that if the
protocol. Figure3(a)showsthemaximaltolerableexcess secret key rate varies significantly with T around its op-
6
100 SNR β Type S/T AIN
Secret Key Rate (SKR) Optimal SKR (Ropt) R=0.1 0.1626 92.02% Gaussian 39/40 103
m)250 5900%%RRoopptt((LL)) 5900%%RRoopptt((UU)) 10−1 0.1613 92.71% NoGn-aGusasuiassnian 4303//4400 18324
ce (k200 10−2 0.1600 93.40% NoGn-aGusasuiassnian 2400//4400 115012
n
sta Non-Gaussian 34/40 130
di150 10−3 R=0.02 0.0301 93.37% Gaussian 47/48 111
n
o Non-Gaussian 47/48 101
si
mis100 10−4 0.0296 94.97% Gaussian 37/48 190
s Non-Gaussian 37/48 174
n
a 0.0293 95.94% Gaussian 18/48 157
Tr 50 10−5
Non-Gaussian 33/48 178
0 10−6 TABLE I: Performance comparison of the multidimensional
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SKR
reconciliation method between Gaussian and non-Gaussian
T
data. R: the rate of sparse check matrix; SNR: signal-to-
FIG. 4: (Color online) Secret key rate vs transmission dis- noise ratio; β: reconciliation efficiency; Type: the type of
tance and transmittance T of Alice’s BS1. The middle black tested data; S/T: the number of successfully decoded data
solidlinerepresentstheoptimalT ofeachdistance,whenthe blocks/the number of total tested data blocks; AIN: average
secret key rate approaches its optimal value (Ropt). The left iteration numberwhen thedecoding process succeeds.
red dash-dotted (dashed) line represents the lower bound of
T for each distance, when its secret key rate is 90% (50%)
of its optimum at that distance. The right blue dash-dotted
(dashed) line represents the upper bound of T for each dis-
tance, when its secret key rate is 90% (50%) of its optimum
at that distance. The simulation parameters are: V = 20, Gaussian noise channel (BIAWGNC).
a=0.2dB/km, ε=0.01, and β=0.95.
The two sparse check matrices used here are of two
different rates: one is 0.1 and another is 0.02. For
timal value, then we need to estimate it accurately to each signal-to-noise ratio (SNR), we have tested more
maintain a relatively high performance, which requires than 40 data blocks. Each data block contains 220 bits
complicated implementations in a practicalsystem. For- (1Mbits). Therefore,thenumberofsuccessfullydecoded
tunately, as shown in Fig. 4, the secret key rate varies data blocks and the average iteration number when the
slowly with T at each distance around its optimal value decodingprocesssucceedswillrepresenttheperformance
(black solid line). Specifically, the secret key rate can of the reconciliation method.
maintain higher than 90% of its optimal value (R ), if
opt
the estimated T is in the areabetween the left red dash- TableIshowsourtestresultsinwhich,forthe0.1rate
dotted line and the right blue dash-dotted line. If one matrix, non-Gaussian cases show a better performance
only requiresthat the secret key rate is higher than one- than the Gaussian cases, on both the numbers of suc-
half of its optimum, then the choice of T is much more cessfully decoded blocks and the average iteration num-
flexible, i.e., within the area between the left red dashed ber, given the same SNR. For the 0.02 rate matrix, the
line and the right dashed line. non-Gaussiancases show similar numbers of successfully
decoded data blocks but less average iteration numbers
compared to the Gaussian cases, except the last row. In
the last row, the successful decoding probability of the
C. Reconciliation Efficiency of non-Gaussian Data
Gaussiancasedropssignificantly,whilethenon-Gaussian
case only drops a little, provided there is more average
In all of the above discussions, we assume the recon- iteration.
ciliation efficiency β is 0.95, which is a pretty good ef-
ficiency approached in the Gaussian case [17]. There- In short, from our test, when considering a high rec-
fore, another thing one may be concerned about is, for onciliation efficiency 0.96, the 0.02 rate check matrix
≈
the non-Gaussiandata generated by virtual photon sub- shows a relatively high successful decoding probability,
traction, whether or not the information reconciliation i.e., more than 60%. Besides, the two sparse check ma-
still remains a relatively high efficiency as for the Gaus- trices were initially designed for Gaussian data, not spe-
sian data. We have carried out a preliminary test on cially designed for the photon-subtraction case. Thus,
the performance of the multidimensional reconciliation this result suggests that one may directly use the multi-
method, proposed in [18], on both using one-photon dimensional reconciliation codes for photon subtraction.
subtraction and not using any photon-subtraction cases If the check matrix is specially designed for the non-
(non-Gaussian and Gaussian cases, respectively) with Gaussian data, the reconciliation efficiency may be even
simulateddata,assumingthebinaryinputadditivewhite higher.
7
IV. CONCLUSION 101
Original Protocol
e)100 1−photon subtraction,ηSPD=1
scuoblIsnt,rwatchhtiiicoshncpamanpetebhre,oadcwcioenmcpporlhiosephroeesdnetdb-syttanhtoeen-CGvViarutuQsasKilaDnpphpoorstototsnoe--- bit/puls10−1 11O−−Npp−hhOoottFooFnn dssuuebbtettrrcaaticcottniioo,nn,,ηηηSSSPPPDDD=== 000...851
lection accordingto Alice’s data. It cannot only remove e (10−2
at
tidheealcoompeprlaetxiopnhsywsihciaclhooppetriamtiizoenss,thbeutpearlfsoormemanucleatoef tthhee ey r10−3
k
CV QKD system. The main parameter, i.e., the trans- et
mittanceofAlice’sBS1,ofthispostselectionmethodcan ecr10−4
S
be adjusted flexiblyaccordingtothe channelparameters 10−5
tooptimizethesecretkeyrateortolerableexcessnoiseat
a certain distance. The numericalsimulation shows that 10−6
0 50 100 150 200 250
by choosing the optimal transmittance of Alice’s BS1, Transmission distance (km)
theuseofvirtualphotonsubtractionwilloutperformthe
original protocol at long-distance regime. FIG. 5: (Color online) The detection efficiency of SPD will
Furthermore,ourpreliminarytestsabouttheinforma- influencetheperformanceoftheschemeusingphotonsubtrac-
tionreconciliationsuggestthatbyusingthemultidimen- tion. Thelinesfromtoptobottomareasfollows: theoriginal
sionalreconciliationalgorithm,theperformanceofrecon- protocol without photon subtraction (black solid line), with
ciliating the postselected non-Gaussiandata is even bet- one-photon subtraction under unit DE (blue solid line), 0.8
DE (green dashed line), 0.5 DE (pink dotted line), and with
ter than that of the Gaussiandata. Specifically, for each
on-off detector under 0.1 DE (red dash-dotted line), respec-
SNR, either the successfully decoded blocks are higher
tively. The simulation parameters are the variance V = 20,
or the average iteration numbers are lower which saves
channel loss a = 0.2dB/km, excess noise ε = 0.01, the
decoding time. In our tests, the two sparse check ma-
transmittance of BS1 T = 0.8, and reconciliation efficiency
trices were initially designed for Gaussian data, not for β=0.95.
the non-Gaussian case, which suggests that one can di-
rectly use the multidimensional reconciliation method
here. This implies the feasibility of implementing this we only consider the effect of the limited detection effi-
virtual photon-subtraction method practically. ciency. As depicted in Fig. 1(a), an extra beam splitter
(BS2)withtransmittance1 η isputinfrontofanideal
d
−
detectortomodelthepracticaldetector’sfinitedetection
Acknowledgement efficiency. Figure 5 shows how the SPD’s nonunit detec-
tion efficiency will reduce the performance by numerical
simulation. When the detection efficiency descends to
WewouldliketothankR.G.Patro´nandF.Grosshans
0.8 (greendashed line), although it still outperforms the
forthehelpfuldiscussions. Thisworkissupportedbythe
originalprotocol, the maximal transmission distance de-
National Science Fund for Distinguished Young Scholars
creases significantly. If the detection efficiency descends
of China (Grant No. 61225003), the State Key Project
to 0.5 (pink dotted line), although it is achievable using
ofNationalNaturalScience FoundationofChina (Grant
superconductingtransition-edgesensorsattelecomwave-
No. 61531003), National Natural Science Foundation of
length [36], the maximal transmission distance is worse
China (Grant No. 61501414), and the National Hi-Tech
than the original protocol. And it is even worse if one
Research and Development (863) Program.
uses the commercial on-off SPD with only 0.1 detection
efficiency based on APD (red dash-dotted line).
Appendix A: Influence of the imperfect
single-photon detector Appendix B: Subtracting k photons
The perfect one-photon subtraction will require an Here we use the same notation as depicted in Sec. II.
ideal PNR detector. However, a practical PNR detec- After the BS1, the state is ρ = ψ ψ , where
tor has imperfections, such as finite detection efficiency AB1B2 | ih |
(DE) and dark count, which will reduce the maximal
transmissiondistance. Becausetheaveragephotonnum- ψ =UBS TMSV 0
| i |∞ i⊗| i
ber for a TMSV state used in CV QKD is usually sev- =√1 λ2 λn(U n,0 ) n
eraltens,whichmeansthatthe numberofthe legitimate − BS| i ⊗| iA
count of a well-demonstrated PNR is much greater than =√1 λ2nP∞=0 n λn ClTn−l(1 T)l n,l,n l
the dark count, the finite detection efficiency (ηd) is the − nP=0lP=0 q n − | − iAB1B2
most significant factor. It is similar for the on-off detec- (B1)
tor based on the avalanched photodiode (APD). Thus, and Cl is the combinatorial number.
n
8
0.30
The final state is a mixed state, such that
1−photon subtraction
0.25 2−photon subtraction ∞ PΠˆ1(k)
3−photon subtraction ρon−off = ρ(k) . (B6)
bility 0.20 4−photon subtraction AB2 Xk=1PΠˆ1(on) AB2
a
b
o
Pr Appendix C: Calculation of the Secret Key Rate
s 0.15
s
e
c
Suc 0.10 Suppose the final state ρGAB is a Gaussian state with
covariance matrix
0.05 V I φσ
γG = 1 Z , (C1)
AB (cid:18)φσZ V2I (cid:19)
0
0 0.2 0.4 0.6 0.8 1 where I is diag(1,1), and σ is diag(1,-1), and Alice al-
T Z
ways uses heterodyne detection. The secret key rate of
reverse reconciliation is
FIG.6: (Coloronline) Thesuccessprobabilityofsubtracting
k photons of a TMSV with different transmittances T of Al-
KHom =βIHom(A:B) SHom(E :B), (C2)
ice’s BS. The lines from top to bottom represent one-photon
−
subtraction (blue solid line), two-photon subtraction (green
dashedline),three-photonsubtraction(pinkdottedline),and where the superscript Hom means Bob using homodyne
four-photon subtraction (red dash-dotted line), respectively. detection, and β is the reconciliation efficiency.
The variance of TMSV is V =20, and Alice uses ideal SPD. Therefore,
1 V
IHom(A:B)= log A , (C3)
The success probability of Πˆ1 = |kihk| clicks on mode 2 2VAH|oBm
B is
1
where V =(V +1)/2, V =V , and
PΠˆ1(k)=tr Πˆ ρ A 1 B 2
AB1B2 1 AB1B2
==(cid:0)11−λλ22(cid:1)nP∞=1−kTλ2nkC(cid:16)∞nkTnλ−2kT(1n−(cid:17)CTk)k . (B2) By assuVmAH|ioBnmg=EVvAe−ca2φnV2Bp=uriVfy12+th1e−w2φhV2o2l.e sys(tCem4),
− T n
(cid:0) (cid:1)(cid:0) (cid:1) nP=k(cid:0) (cid:1) S(E :B) = S(E) S(E B) = S(AB) S(AB). And
= 1−λ2 λ2(1−T) k S(AB)isafunctio−nofthe| symplecticeig−envalu|esλ of
1−Tλ2 1−Tλ2 1,2
h i γG , which is
AB
And its relationship with the transmittance of Alice’s
BS1 is shown in Fig. 6. S(AB)=G[(λ1 1)/2]+G[(λ2 1)/2], (C5)
− −
Then the k-photon subtracted state is
where
k ρ k
ρ(k) = h | AB1B2| i = ζ(k) ζ(k) ,
AB2 PΠˆ1(k) (cid:12) ED (cid:12) G(x)=(x+1)log2(x+1)−xlog2x, (C6)
(cid:12) (cid:12)
(cid:12) (cid:12)
where and
(cid:12)(cid:12)ζ(k)E= hPkΠˆ|1ψ(ik) =nX∞=kqp(nk)|n,n−kiAB2, (B3) λ21,2 = 12h∆±p∆2−4D2i, (C7)
(cid:12) q where we have used the notations
and
∆=V2+V2 2φ2,
λ2nCkTn D =V1V 2φ2−. (C8)
p(nk) = ∞ n . (B4) 1 2−
(λ2T)nCk
n And SHom(AB) = G[(λ 1)/2] is a function of the
nP=k symplectic eig|envalue λ o3f−the covariance matrix γb of
3 A
As a comparison, for the on-off detector, the success theAmodeafterBob’shomodynedetection,whereλ =
3
probability is
V V φ2 V . Thus, the secret key rate when Bob
1 1 2
q −
PΠˆ1(on)=1 PΠˆ1(k =0) using(cid:0)homody(cid:14)ne d(cid:1)etection is
=1−(cid:0)1−λ2−(cid:1)nP∞=0(cid:0)λ2T(cid:1)n = (11−−λT2)Tλ2 . (B5) KHom =IHom(A:B)−(cid:2)S(AB)−SHom(A|B)(cid:3). (C9)
9
AppendixsDu:bCtroavcateridanTcMe mSVatsrtixatoef k-photon asunbdstPitxu′,tpe′disbyPxAx,′p,Ap′in. Eq. (6) in which {xA,pA} are
{ }
After simplifying Eq. (D2) by integrating the variable
Supposeγ(k) representsthecovariancematrixofρ(k) , x,
AB2
and it has the following formula,
xˆ2 =2V˜ 1,
γ(k) =(cid:18)hxˆA(cid:10)xˆxˆ2AB(cid:11)iIσZ hxˆAxˆxˆ2BBiIσZ (cid:19). (D1) (cid:10)hxˆxˆA2BAxˆ(cid:11)B=i2=T2λ−√2VT˜λ+V˜1,, (D4)
(cid:10) (cid:11) (cid:10) (cid:11)
Supposex′,p′ arethe heterodynemeasurementresults where V˜ = x′2 W Px′,p′dx′dp′, and further calcula-
of mode A, and x is the homodyne measurement result tion shows R · ·
of mode B. Then,
k+1
V˜ = . (D5)
xˆ2 =2 x′2P (x′,p′,x)dx′dp′dx 1, 1 Tλ2
(cid:10)xˆAAxˆ(cid:11)B =·√R2 x′xP (x′,p′,x)dx′dp−′dx, (D2) −
hxˆ2 =i x2P·(Rx′,p′,x)dx′dp′dx,
B
(cid:10) (cid:11) R
where
2
P(x′,p′,x)=W Px′,p′ x √Tα , (D3)
· ·(cid:12)D (cid:12) E(cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
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