Quantum key distribution relied on trusted information center

12 Pages·0.14 MB·English

by  Guihua Zeng

Quantum key distribution relied on trusted information center
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1Guihua Zeng, 2Zhongyang Wang, and 1Xinmei Wang
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1National Key Lab. on ISDN, XiDian University, Xi’an 710071, China
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a 2Shanghai Institute of Optics and Fine mechanics, Academia Sinica,
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P.O.Box 800-211, Shanghai 201800, China
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Abstract
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0 Quantumcorrelationbetweentwoparticlesandamongthreeparticlesshownonclassicpropertiesthatcanbeused
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0 forprovidingsecuretransmissionofinformation. Inthispaper,weproposetwoquantumkeydistributionschemesfor
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quantumcryptographicnetwork,whichusethecorrelation propertiesoftwoandthreeparticles. Oneisimplemented
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/ by the Greenberger-Horne-Zeilinger state, and another is implemented by the Bell states. These schemes need a
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trusted information center like that in the classic cryptography. The optimal efficiency of theproposed protocols are
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n higher than that in theprevious schemes.
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u PACS:03.67.Dd, 03.65.Bz, 03.67.-a
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X I. Introduction particular, quantum key distribution became especially
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a importantduetotechnologicaladvanceswhichallowtheir
Since the first finding that quantum effects may pro- implementation in laboratory. The first quantum key dis-
tect privacy informationtransmitted in anopen quantum tributionprototype,workingoveradistanceof32centime-
channel by S.Wiesner [1], and then by C.H.Bennett and tersin1989,wasimplementedbymeansoflasertransmit-
G.Brassard [2], a remarkable surge of interest in the in- ting in free space [9]. Soon, experimental demonstrations
ternational scientific and industrial community has pro- by optical fibber were set up [10]. Now the transmission
pelled quantum cryptography into mainstream computer distance is extended to more than 30Km in the fiber [11],
science andphysics. Furthermore,quantumcryptography and more than 205m in the free space [12].
is becoming increasingly practical at a fast pace. Quan-
tumcryptographyisafieldthatcombinesquantumtheory Quantum key distribution is defined as a procedure
withinformationtheory. Thegoalofthisfieldistousethe allowing legitimate two (multi-) users of communication
law of physics to provide secure information exchange, in channel to establish exact two (multi-) copies, one copy
contrast to classical methods based on (unproven) com- for each user, of a random and secret sequence of bits.
plexity assumption. Current investigations of quantum Quantum key distribution employs quantum phenomena
cryptography involve three aspects: quantum key distri- suchastheHeisenberguncertaintyprincipleandthequan-
parties, who share no secret information initially, to com- protocol, called time-reserved EPR protocol, in which
municate over an open channel and to establish between users prestore quantum states in a trusted center, where
themselvesasharedsecretsequenceofbits. Thepresented their quantum states are preserved using quantum mem-
QKDprotocolsareprovablysecureagainsteavesdropping ories. The main procedures are as follows: users store
attack, in that, as a matter of fundamental principle, the quantumstatesinquantummemories,keptinatransmis-
secret data can not be compromised unknowingly to the sion center. Upon request from two users, the center uses
legitimate users of the channel. Several quantum key dis- two-bitgatestoprojecttheproductstateoftwononcorre-
tribution protocols have been proposed, all these proto- latedparticles(one fromeachuser)ontoafully entangled
cols can be classed into two kinds. i) The point-to-point state. As a result, the two users can share a secret bit,
(two parties) quantum key distribution (pQKD). Three which is unknown even to the center. The time-reserved
main protocols of these are the BB84 protocol [3], B92 EPRprotocolwasproposedtobeusedinaquantumcryp-
protocol [4] and EPR protocol [5-7]. ii) The networking tographic networking (QCN). The implementation of the
quantumkeydistribution(nQKD),e.g.,thetime-reserved time-reserved EPR scheme needs four particle for obtain-
EPR protocol [8]. The physical implementation may be ing one qubit. One may ask that whether the nQKD
refer to Townsend’s works [13,14]. scheme may be implemented by three or two particle or
not? InthispaperweproposeseveralQKDprotocolsthat
For the pQKD scheme, the first quantum key distribu-
use three or two particles to obtain one qubit. These pro-
tion scheme, i.e., the Bennett Brassard (BB84) scheme,
tocols can also be used in networking QKD.
waspresentedadecadeago. Itisimplementedbythefour
states , , , , where any of the two states In this work we suggest two nQKD schemes. The sug-
{| ↑i | ↓i | րi | ցi}
, andanyofthe twostates , arenon- gested schemes need three parties: the trusted informa-
{|↑i |↓i} {|րi |ցi}
commuted, , may be any orthogonal states of tion center and two users, by conventional called Alice
{| ↑i | ↓i}
two-dimensional Hilbert space. Its security is warranted and Bob, here the center is trusted. One scheme is im-
by the uncertainty principle of quantum mechanics. In plementedbytheGreenberger-Horne-Zeilinger(GHZ)[26]
1992,Bennett devisedanother protocol,i.e., the B92 pro- triplet state, we call this protocol as GHZ-nQKD proto-
tocol,whichisbasedonthetransmissionofnonorthogonal col,inwhichthecenter’sroleistomeasurehis/herparticle
quantum states. This protocol uses any two nonorthogo- (from the GHZ triplet) by the random measurement like
nalstatestoimplementtheQKD.Itssecurityreliesonthe thatinBB84protocol,andtellAliceandBobthemeasure-
no-cloning of unknown two-nonorthogonal states. A fur- ment results. Another is implemented by the Bell states,
ther elegantscheme has been proposedby Ekert,which is we call it as Bell-nQKD protocol. This scheme needs the
implementedbytheEinstein-Podolsky-Rosen(EPR)pairs center to measure the two-particle entanglement system
[24]. It is called the EPR protocol which relies on the vi- by the Bell operators or the linear combination of Bell
olation of the Bell inequalities [25] to provide the secret operatorsbefore Alice’s andBob’smeasurementandsend
security. Considerthetwo-particlecorrelation,Bennettet the users his/her results. By the center’s assistance, the
al presented a modified version, in which the security is userscanobtainthesecretkey,thecheatingcenteraswell
review the two-particle maximally entangled states, the
1
so-called Bell states and the three-particle maximally en- Φ+ = (z+ z + z z+ ), (4)
c a b a b
| i √2 | i | −i | −i | i
tangledstates,the so-calledGHZ triple state. In addition
weinvestigatethecorrelationpropertiesoftheGHZtriplet
1
and the Bell states. In Sec. III, we propose three proto- Φ− c = (z+ a z b z a z+ b), (5)
| i √2 | i | −i −| −i | i
cols, which are implemented by the GHZ triplet. The
where the subscripts c,a,b denote the states for the in-
efficiencies of these protocols are different, but they are
formation center and the two communicators Alice and
more practical, especially the protocol 3. The securities
Bob. These Bell states can be generated from a type-
of these protocols are analyzed. In Sec. IV, we propose
II parametric down-conversion crystal [28]. Define the x
two protocols implemented by the Bell states. The secu-
eigenstates
rities of these protocols are investigated. In Sec. V, we
discusstheapplicationsofourprotocolsinnetworkQKD.
1
x+ = (z+ + z ), (6)
Conclusions are presented in Sec VI. | i √2 | i | −i
II. The Bell states and the GHZ triplet states
1
x = (z+ z ), (7)
First we review the two and three particles entangle- | −i √2 | i−| −i
ment states. In general, N-particle entanglement states the four Bell states can be rewritten as
may be written as [27]
1
Ψ+ = (x+ x+ + x x ), (8)
N N | ic √2 | ia| ib | −ia| −ib
|ψi=Y|uii±Y|ucii, (1)
i=1 i=1
whereu standsforabinaryvariableu z+ , z and
i i ∈{| i | −i} 1
Ψ = (x+ x + x x+ ), (9)
uc =1 u , z+ and z denote the spin eigenstates,or | −ic √2 | ia| −ib | −ia| ib
i − i | i | −i
equivalentlythehorizontalandverticalpolarizationeigen-
states, or equivalently any two-level system. For N = 2 1
Φ+ = (x+ x+ x x ), (10)
c a b a b
| i √2 | i | i −| −i | −i
they reduce to the Bell states and N =3 and N =4 they
represent the GHZ states. For a general N we shall call-
1
ing them cat states. In this paper, we are interested in Φ = (x x+ x+ x ), (11)
− c a b a b
| i √2 | −i | i −| i | −i
the case of N =2 and N =3, i.e., the Bell states and the
As should be noted, for example, the Ψ+ states give
GHZ triplet state. | i
correlated results in both the z and x bases, but the
1. Bell states
Ψ state give correlated results in the z basis, but an-
−
| i
ticorrelated results in the x basis. Summarizing these
Eq.(1) reduces to the Bell states when N =2
correlated or anticorrelated results of the Bell states
Ψ+ c = 1 (z+ a z+ b+ z a z b), (2) {Ψ+,Ψ−,Φ+,Φ−} in the z and x bases, we get the fol-
| i √2 | i | i | −i | −i
lowing table:
Trent |Ψ+i |Ψ−i |Φ+i |Φ−i |φ−ic = √12(|x+ia|z−ib−|x−ia|z+ib)
(15)
Alice x+ x+ x+ x
| i | i | i | −i
Bob |x+i |x−i |x+i |x+i √12(|z+ia|x−ib+|z−ia|x+ib),
Alice x x x x+ We note that the set of states Φ+,Ψ−,φ−,ψ+ have the
| −i | −i | −i | i { }
Bob x x+ x x following correlated or anticorrelatedresults
| −i | i | −i | −i
Alice z+ z+ z+ z+
| i | i | i | i Table II. The correlationof states Φ+,Ψ ,φ ,φ+
− −
{ }
Bob z+ z+ z z
| i | i | −i | −i Trent Φ+ Ψ φ ψ+
− −
Alice z z z z | i | i | i | i
| −i | −i | −i | −i
Alice x+ x+ x+ x+
Bob z z z+ z+ | i | i | i | i
| −i | −i | i | i
Bob x+ x z z+
| i | −i | −i | i
From Table I it is clear that after the center has pro- Alice x x x x
| −i | −i | −i | −i
jected the two-particle entanglement system onto any of Bob x x+ z+ z
| −i | i | i | −i
the four Bell states Ψ+,Ψ ,Φ+,Φ , the state of any of Alice z+ z+ z+ z+
{ − −} | i | i | i | i
two particles do not give determined results. For exam- Bob z z+ x x+
| −i | i | −i | i
ple, ifthe center’smeasurementbasisis Ψ+ , the state of Alice z z z z
| i | −i | −i | −i | −i
any of two particles may be x+ ,or x with the prob- Bob z+ z x+ x
| i | −i | i | −i | i | −i
ability 1, or z+ or z with the probability 1. Even if
2 | i | −i 2 Thistable showsthe states Φ+,Ψ ,φ ,ψ+ alsohave
− −
{ }
Alice has measured her particle and announced her mea-
the correlation properties in x and z direction. If the
surementbasis,anyone,including Bob,cannotknowsAl-
centerprojectsthetwo-particleentanglementsystemonto
ice’s results, because the probability of making error is 1.
2 anyofthefourbases Φ+,Ψ ,φ ,ψ+ ,andsendsrespec-
− −
{ }
However if the center’s bases are public announced, Alice
tivelyAliceandBoboneoftwo-particleentanglement,Al-
knows the Bob’s qubits and vice versa. These properties
ice’sandBob’sparticleshaveyetnotadeterminedresults
maybe usedto distribute the quantumkeybetweenAlice
beforetheirmeasurement. Forexample,ifthecentermea-
and Bob by the assistance of the trusted center.
sure the two-particle system using the base φ , Alice’s
−
BythefourBellstates(Eq.2-Eq.5)onemayobtainother
measurement may be x+ or x if she measures her
| i | −i
correlated or anticorrelated results. Define a line combi-
particleusexbasis. BeforeAlicerevealshermeasurement
nation of Bell states as
bases, anyone can not know Alice’s results, even if Bob.
However,ifAliceandBobknowthestatemeasuredbythe
1
ψ+ = (Ψ + Φ+ ), (12) center and their measurement directions are determined,
c − c c
| i √2 | i | i
they can judge the qubits each other.
1
ψ = (Ψ Φ+ ). (13)
− c − c c
| i √2 | i −| i 2. GHZ triplet states
One may get
Eq.(1) reduces to eight GHZ triplet states for N = 3.
ψ+ = 1 (x+ z+ + x z ) In this paper we use the following state
| ic √2 | ia| ib | −ia| −ib
Suppose the center, Alice and Bob share one particle Table III. The correlation results of the GHZ triplet
each from a three-particle entangled GHZ state, then the states
GHZ state may be represented by Trent x+ x y+ y
| i | −i | i | −i
Alice x+ x+ x+ x+
| i | i | i | i
1 Bob x+ x y y+
ψ = (z+ c z+ a z+ b+ z c z a z b), (17) | i | −i | −i | i
| i √2 | i | i | i | −i | −i | −i
Alice x x x x
| −i | −i | −i | −i
where the first particle is that of the center, the second
Bob x x+ y+ y
| −i | i | i | −i
that of Alice, and the third that of Bob. Define the y
Alice y+ y+ y+ y+
| i | i | i | i
eigenstates
Bob y y+ x x
| −i | i | −i | −i
Alice y y y y
1 | −i | −i | −i | −i
y+ = (z+ +iz ), (18)
| i √2 | i | −i Bob y+ y x+ x
| i | −i | i | −i
ThetableIIIshowsseveralpropertiesoftheGHZtriplet
1
y+ = (z+ iz ), (19) state: i) anyone of the three parties, i.e., the center, Alice
| i √2 | i− | −i
and using the x eigenstates defined in Eq.(6,7), the GHZ or Bob, can determine whether the other two participa-
triplet state can be rewritten as tors’ results are the same or opposite and also that he
(she) will gain no knowledge of what their results actu-
|ψi= 12[(|x+i|x+i+|x−i|x−i)|x+i ally are, if he (she) knows what measurements have been
(20)
made by the other two participators (that is x or y). ii)
+(x+ x + x x+ )x ], From table III it is clear that allows two parties jointly,
| i| −i | −i| i | −i
or
butonlyjointly,todeterminewhichwasthemeasurement
ψ = 1[(y+ y + y y+ )x+
| i 2 | i| −i | −i| i | i
outcome of the third party. So if the measurement direc-
(21)
tions of the three participators are public, the combined
+(x+ x + x x+ )x ],
| i| −i | −i| i | −i
results of any two participators can determine what the
or
ψ = 1[(y+ x + y x )y+ result of the third party’s measurement was.
| i 2 | i| −i | −i| −i | i
(22)
III. GHZ-nQKD protocols
+(y+ x+ + y x )y ],
| i| i | −i| −i | −i
GHZ states has already found a number of uses. They
or
ψ = 1[(x+ y + x y+ )y+ formthebasisofaverystringenttestoflocalrealisticthe-
| i 2 | i| −i | −i| i | i
(23) ories. Itwasalsoproposedthattheycanbeusedforcryp-
+(x+ y+ + x y )y ]. tographic conferencing or for multiparticle generations of
| i| i | −i| −i | −i
The above decomposition demonstrates the correlation superdense coding [27]. In addition, related states can
amongthreeparticles. Forexample,inEq.(20)ifonepar- be used to reduce communication complexity. Recently,
ticle is in the state x+ and the second particle is in the it wasproposedthat they canbe used for quantum secret
| i
state x+ , the third particle must be in the state x+ sharingandquantuminformationsplit[19]. Inthispaper,
| i | i
because of the correlation of the GHZ triplet state. By we use the GHZ state to distribute quantum key between
A. The protocols particles, either in the x or y direction, but the efficiency
is low by this way, because these results measured along
As discussed in Sec. II, the GHZ state has correlation
y direction have no use in this protocol and a half parti-
properties that if only one communicator’s measurement
cles will be discarded, the efficiency is only 12.5%. The
results is announced, the states of other two particles are
center’s measurement collapses the GHZ triplet state to
still not determined, but two communicators’ results can
be a two-particle system. The state of the two-particle
determine the third result. These properties may be used
entanglementisnotdetermined,becausetheymaybeany
in the QKD relying on a third party. Let us now show
of the states
how to implement our quantum key distribution scheme
1
bythe GHZstate. Thereareseveralwayto distributethe Ψ1 ab = (x+ x+ + x x ),
| i √2 | i| i | −i| −i
communicators the key by the center’s assistance.
1
Ψ2 = (x+ x + x x+ ),
Protocol 1 | iab √2 | i| −i | −i| i
1
Ψ3 = (y+ y + y y+ ),
1. The center measures his GHZ particle in the x direc- ab
| i √2 | i| −i | −i| i
tion and obtains the result x+ or x 1
| i | −i Ψ4 = (y+ y+ + y y ).
ab
| i √2 | i| i | −i| −i
2. The center tell Alice and Bob his measurement re-
In step 4, we use the correlation properties of the GHZ
sults.
states to checkthe eavesdropping. Having measuredtheir
particles, Bob randomly chooses a subset of qubits from
3. Alice andBob makerespectively therandom measure-
his qubits and sends this subset to Alice. Alice compares
ment on their GHZ particles, either in the x or y di-
the corresponding results from the center, Bob and Alice.
rection.
If these results are correlation results, which are satisfy
4. Checktheeavesdroppingbyusingthecorrelationprop-
Eqs.(20-23)orthecorrelationofthreestatesisinthetable
erties of the GHZ states.
III, the results are perfect, otherwise it means eavesdrop-
5. Alice and Bob compare their bases. If their measure- ping or disturbed by noise.
mentbasesaresame,AliceandBobkeeptheirresults, In step 5, Alice and Bob compare their bases. Because
otherwise they discard their results. Alice and Bob randomly measure their particles either in
the x or y direction, some of their bases are different and
6. Alice and Bob obtain the final key by using the data
some are same. If their bases are different, Alice’s and
sifting, the error correction and the privacy amplifi-
Bob’s results are no correlation, thus Alice can not know
cation technologies.
Bob’s qubits and vice versa,in this case they need to dis-
In this protocol, we let the center firstly measures his card these results. However if their bases are same, their
particlefromtheGHZtriplet,andonlymeasureitinthex results are correlated, Alice and Bob keep these results.
direction. The center’sresults willbe x+ or x . After So this protocol discards the results that corresponds the
| i | −i
the center has finished the measurement, Alice and Bob different bases.
measure their particles. This protocol only uses the cor- The rawquantum keydistribution is useless in practice
even in the absence of eavesdropping. For these reasons, 7. Alice and Bob gain the final key by using the data
our scheme needs to supplement some classicaltools such sifting, the error correction and privacy amplification
as the privacy amplification, the error correction and the technologies.
data sifting, so we use these technologies in our protocol.
The protocol 2 lets the center measure his/her particle
The implementation of these supplemented classic tools
either in the x or y direction. It is stresses that here
are the same as in the previous documents [9].
the center’s allmeasurementresults are useful. When the
In quantumkey distribution somequbits (henceforth l)
center’s result is the state x+ or x , Alice and Bob
| i | −i
will be wasted because of the loss and the inexactitude of
need to keep the results which have the same bases, but
equipment, so in order to be left with a key of L qubits
if the center’s result is y+ or y , the communicators
the center 1 shouldprepareL >2(L+l). In this casethe | i | −i
′
discardtheresultswhichhavethesamebases. Thereason
efficiency is
is that the results must be correlated or anticorrelated.
L
η1 = <50%. (24)
2(L+l) This step is finshed in the step 5.
Thisefficiencyislargerthanthatofthetime-reservedEPR ItneedstostressthatAlice’sresultsmustbe consistent
protocol, which is with Bob’s results forgetting the rawquantum key,sowe
L havethestep6. BythepropertiesoftheGHZtripletstate,
η′ = <12.5%. (25)
8(L+l)
Alice(Bob)canjudgeBob’s(Alice’s)resultsbycombining
Protocol 2 her (his) and the center’s results. But the table III can
notgivecompletelyasameresultsalthoughAliceandBob
1. The center measures his GHZ particle either in the
can know the qubits each other. For example, when the
x or y direction and obtains any of the four states
center’sandAlice’sresultsarerespectively y+ and x+ ,
x+ , x , y+ , y | i | i
{| i | −i | i | −i}
Bob’s result should be y , obviously, Alice’s and Bob’s
| −i
2. The center tells Alice and Bob his measurement re-
results are different. For obtaining a same key, Alice’s
sults.
(Bob’s) results need to be consistent with Bob’s (Alice’s)
3. Alice and Bob make respectively a random measure- results. ThemethodisthatAlice(Bob)transfersher(his)
ment on their GHZ particles, either in the x or y qubits to binary bits according to Bob’s (Alice’s) results.
direction. Theefficiencyofthisprotocolisthesameastheprotocol
1. After the center announced his results, Alice and Bob
4. Checking the eavesdropping like protocol 1.
have a possibility of 1/2 to obtain the correct results by
5. Alice and Bob compare their bases. If the center’s therandommeasurement. Considerthewastedqubits(l),
result is x+ or x and their measurement bases inordertobeleftwithakeyofLqubitsthecentershould
| i | −i
are same, or if the center’s result is y+ or y and send L >2(L+l), the efficiency
| i | −i ′
their measurement bases are different, Alice and Bob
L
η2 = <50%. (26)
keep their results, otherwise they discard the results. 2(L+l)
Protocol 3
6. Alice makes her results to be consistent with Bob’s
direction. step6. Considerthewastedqubits(l)inthemeasurement
and the loss in the quantum channel, in order to be left
2. Alice and Bob send their measurement bases (x or y)
withakeyofLqubitsthecentershouldsendL >(L+l),
′
to the center, but not the qubit values.
the efficiency is
3. The center randomly measures his particle according
L
to Alice’s and Bob’s measurement bases. If both Alice η3 = (L+l) <100%. (27)
and Bob measure their particle using the same mea-
B. security analysis
surement basis, e.g. x or y direction, the center mea-
sures his particle using the x measurement basis, oth- These presented schemes are secure against eavesdrop-
erwise, the center measures his particle using the y ping. Their securities are warranted by the correlation of
measurement basis. the GHZ triplet. To see these in a sufficient way, we will
consider several possible eavesdropping in the following.
4. The centerannounces his measurementresults, which
is any of the four states x+ , x , y+ , y . 1. The cheating center’s attacks
{| i | −i | i | −i}
5. Check eavesdropping like the protocol 1.
The cheatingcenter is impossible to knowthe quantum
key. From the table III it is clear that if the center knows
6. Alice and Bob judge the quantum state each other ac-
what measurements bases Alice and Bobmade (that is, x
cording to table III. While the center announces his
ory),hecandeterminewhethertheirresultsarethe same
results, both Alice and Bob know the center’s results.
or opposite and also that the center will gain no knowl-
Then the Alice’s (Bob’s) and the center’s results can
edge of what Alice’s and Bob’s actually are, because the
jointly determine what is the Bob’s (Alice’) measure-
cheating center will has the probability of 1/2 of making
ment outcome.
a mistake. If the center makes measurement on the three
7. Alice and Bob obtain a sharing key by using the data
particles of the GHZ triplet, then send these particles to
sifting, the error correction and privacy amplification
AliceandBob,thecenter’smeasurementwillintroduceer-
technologies.
rors in Alice’s and Bob’s results, thus Alice and Bob can
The important point is that Alice and Bob randomly check it like the four-state BB84 protocol. Of course, a
measure their GHZ particles before the center’s measure- cheating center may use the men-in-middle attack [29] to
ment in this protocol, it is different from the protocols obtainthekeyK andK ,whereK representsthekey
ca cb ca
1 and 2, in which the center’s measurement is completed betweenAliceandthecheatingcenter,andK represents
cb
before Alice’s and Bob’s measurement. This change im- the key between Bob and the cheating center. For pre-
proves the efficiency of this protocol, because the center venting this attacks,Alice and Bob may verify their iden-
may measure his particle according to Alice’s and Bob’s tity using the identity verification technology [30]. This
measurement bases. Although there are the situations method needs a sharingkey between Alice andBob, how-
that Alice’s and Bob’s results are different, all their mea- ever, Alice and Bob, in general, have not sharing key, so
surement states are useful. The correlation of the GHZ this case needs the center to be trustworthy like the key
the key distribution center (KDC) which is often used in If Eve can gain Alice’s (Bob’s) qubits, she can obtain the
the classic cryptography, but here the center process the key. However, Eq.(30) shows that Eve can not obtain
qubits not the binary bits. the results. By the similar method, Eve can not obtain
Alice’s (Bob’s) qubits when the GHZ triplet states satisfy
2. Intercept/resend attacks
Eqs.(21-23).
Letusnowconsidertheintercept/resendattackdefined
IV. Bell-nQKD protocols
in [9]. Suppose that the eavesdropper, by convention de-
notedbyEve,hasmanagedtogetaholdofAliceandBob’s
The above schemes are efficient, however they need
key,shetheninterceptsacommunicator’s(e.g. Alice)par-
three particles. Can we implement the network QKD
ticle from the center and send another particle to Alice.
scheme only by using two particles? In this section we
Inthiscase,threeparticlesofthecenter,BobandtheEve
investigate the two-particle schemes. In the following we
constructaGHZtriplet. However,becausetheAlice,Bob
show that one can also use the Bell states to implement
and the center’s particles are not the GHZ triplet, there
the above quantum key distribution procedure.
are no correlated or anticorrelated result, Eve’s intercep-
tionwillintroduceerrorandcanbe detectedbyAlice and A. protocol
Bob when they check the eavesdropping.
In Sec. II, we see that the two particles of the Bell
3. The entanglement attacks states or the linear combination of Bell states have corre-
lationproperties,they aredemonstratedintable I andII.
The entanglement attacks is no use in our protocol. To
ThesepropertiesmaybeusedintheQKDreliedonathird
show that, Let us assume that the eavesdropper has been
party. Let us now show how to implement the quantum
able to entangle an ancilla in state A with the GHZ
| i
key distribution by Bell states.
triplet state that Alice and Bob are using. The state de-
Protocol 4
scribing the state of the GHZ triplet and the ancilla is
1 1. Thecenterpreparesasetoftwo-particleentanglement
Ψ = (z+z+z+ + z z z ) A . (28)
| i √2 | i | − − −i ⊗| i
pairs and projects each pair onto any of the four Bell
By using the x and y eigenstates and Eq.(20), The eaves-
bases.
dropper get
2. The center sends respectively Alice and Bob one of
the two-particles entanglement and his measurement
U|Ψi= 21(|x+x+i⊗|x+i⊗|A1i+|x−x−i⊗|x+i⊗|A2i
results.
+x+x x A3 + x x+ x A43).. Aliceand Bob make respectively the random measure-
| −i⊗| −i⊗| i | − i⊗| −i⊗| i
(29)
ment on their particle, either in the x or z direction.
whereU denotetheunitarytransformation. Byprojecting
theabovestateonto φ =α1 x+x+ +α2 x x +α3 x+ 4. Alice and Bob check the eavesdropping by using the
| i | i | − −i |
x +α4 x x+ , the eavesdropper creates the states correlation of Bell states.
−i | − i
6. Alice and Bob obtain a sharing key by using the data 6. Alice makes her results be consistent with Bob’s re-
sifting, the error correction and the privacy amplifi- sults.
cation technologies.
7. Alice and Bob obtain a sharing key by using the data
This scheme is similar to the time-reservedEPR proto- sifting, the error correction and privacy amplification
col,but there areseveralimportantdissimilarities. i)The technologies.
time-reservedEPRprotocolusesfourparticleandtwopar-
This protocol is similar to the protocol2, but there are
ticles were prestoredin a transmissioncenter,where their
two dissimilarities: 1) The implementation of protocol 2
quantum states are preserved using quantum memories.
uses the GHZ triplet state, which needs three particles to
Our scheme uses two particles and need not the quantum
obtain one qubit. The center’s measurement result is one
memories. ii)Theefficiencyofthetime-reservedEPRpro-
of the state x+ , x , y+ , y . But the protocol 5
tocol is η <12.5%, but the efficiency of our protocol is {| i | −i | i | −i}
′
uses the Bell states which only use two particles, the cen-
L
η4 = <50%. (31) ter’s results is one of the states Φ+ , Ψ , ψ+ , φ .
2(L+l) {| i | −i | i | −i}
2) The methods for checking eavesdropping are different.
iii)Thecenteronlyusesresultsofthesingletstatesandits
Protocol2usesthecorrelationoftheGHZstates,andhere
correlation properties in the time-reserved EPR protocol,
protocol 5 uses the correlation demonstrated in the table
but the center uses all quantum states in our scheme.
II.
WecanalsousethetableIItodesignanQKDprotocol.
Accordingtotheprotocol5,weseetheefficiencyissame
The protocol goes as follows
as protocol 2:
Protocol 5
L
1. Thecenterpreparesasetoftwo-particleentanglement η5 = <50%. (32)
2(L+l)
pairs andprojects eachpair ontoanyofthefourbases
B. security analysis
Φ+,Ψ ,φ ,ψ+, .
− −
{ }
In term of eavesdropping possibilities, protocols 4 and
2. The center sends respectively Alice and Bob one of
5 have same security with the EPR protocol. After the
the two-particles entanglement and his measurement
center has measured the two-particle entanglement sys-
results.
tems by using any Bell operators or the linear combi-
3. Alice andBob makerespectively therandom measure-
nation Bell operators, Alice and Bob’s particles are two-
ment on their particle, either in the x or z direction.
particleentanglementpairs,whichisoneofthefourstates
4. Check the eavesdropping by using the correlation Ψ+ , Ψ− , Φ+ , Φ− , or Φ+ , Ψ− , φ− , ψ+ , . It
{| i | i | i | i } {| i | i | i | i }
demonstrated in Table II. has the same correlation as the EPR pair, this is there-
fore equivalent to the EPR scheme. So the cheating cen-
5. Alice and Bob compare their bases. If the center’s re-
ter as well as the eavesdropper can not eavesdropthe key
sult is one of the Bell states Φ+ , Ψ and their
−
{| i | i} from the protocols 4 and 5 by the currently eavesdrop-
measurement bases are same, or if the center’s result
ping technologies, e.g., the intercept/resend attacks, the
is one of the states ψ+ , φ and their measure-
−
{| i | i} entanglement attacks etc..