The states of W-class as shared resources for perfect teleportation and superdense coding

0.22 MB·English

by  Lvzhou Li

The states of W-class as shared resources for
perfect teleportation and superdense coding
Lvzhou Li, Daowen Qiu
∗
Department of Computer Science, Zhongshan University, Guangzhou 510275, People’s Republic of China
As we know, the states of triqubit systems have two important classes: GHZ-class and W-class.
Inthispaper,thestatesofW-classareconsidered forteleportation andsuperdensecoding,andare
generalizedtomulti-particlesystems. Firstwedescribetwotransformationsofthesharedresources
for teleportation and superdense coding, which allow many new protocols from some known ones
forthat. Asan application of thesetransformations, weobtain asufficient andnecessary condition
for a state of W-class being suitable for perfect teleportation and superdense coding. As another
application,wefindthatstate|Wi123 = 12(|100i123+|010i123+√2|001i123)canbeusedtotransmit
threeclassical bitsbysendingtwoqubits,whichwasconsideredtobeimpossiblebyP.Agrawaland
7 A.Pati[Phys. Rev. Atobepublished]. Wegeneralize thestatesofW-classtomulti-qubitsystems
0 and multi-particle systems with higher dimension. We propose two protocols for teleportation and
0 superdense coding by using W-states of multi-qubit systems that generalize the protocols by using
2 W 123 proposedbyP.AgrawalandA.Pati. Weobtainanoptimalwaytopartition someW-states
|of miulti-qubit systems into two subsystems, such that the entanglement between them achieves
n
maximum value.
a
J
PACSnumbers: 03.67.-a,03.65.Bz
7

1
I. INTRODUCTION coding are two amazing and interesting processes in
v
quantum information theory, where entangled states as
0
3 Du¨ret al[5]pointedoutthatthestatesofthreequbits shared resources play a crucial role [1, 2]. The two pro-
0 can be entangled in two inequivalent ways: GHZ-class cesseshaveacloserelationshipthathasbeeninvestigated
1 byWerner[4]. Inthe originalprotocols[1,2]fortelepor-
and W-class. States in one class can not be converted
0 tation and superdense coding, EPR pairs were consid-
from states in the other class by stochastic local oper-
7 ered as the shared resources. Latter, more candidates
0 ations and classical communication (SLOCC) [3]. With
have been considered. For instance, maximally entan-
/ respect to loss of qubits, the two classes are rather dif-
h gled states of triparticle were considered in [7, 8, 9],
ferent. The states of W-class are robust against loss of
p and maximally entangled states of multi-particle were
qubits (i.e., if we trace out any one qubit, then there is
-
considered in [17, 18, 19]. Specially, non-maximally
t some genuine entanglement between the remaining two
n entangled states have been considered for teleportation
qubits), while GHZ-states are not. The GHZ-class has
a
and superdense coding by many authors. For instance,
u been extensively studied from many aspects, while W-
non-maximallyentangledstateshavebeenconsideredfor
q class may need more effort to clearly characterize it.
: probabilisticteleportation[22,23,24,25]andprobabilis-
v Recently, more attentions have been taken on the
tic superdense coding [29, 30]. In addition, the classical
i states of W-class. They have been considered for
X information capacity of deterministic superdense coding
many important quantum information processing tasks
withnon-maximallyentangledstateshasbeenstudiedby
r [10, 11, 12, 13, 14]. W-states were considered as quan-
a [26,27,28]. Recently,variouskindsofquantumchannels
tum channel for teleportation of entangled pairs in [10].
have been explored for deterministic and unambiguous
Probabilisticteleportationofa qubitstate via a W-state
superdense coding [31, 32, 33].
was studied in [11]. Furthermore, in [12] the authors
discovereda subclassofW-classsuitable forperfect tele- Observingthestatesoftriparticle,interestinglywefind
portation and superdense coding, but they did not give that some states from GHZ-class are suitable for perfect
thesufficientandnecessaryconditionforaW-statebeing teleportationandsuperdensecoding,butsomestatesare
suitable for that. In addition, W-class has been used for not. For instance, there are two states of GHZ-class:
quantum key distribution [13], and in illustrating viola-
tion of local realism [14]. At the same time, there have
beenvariousproposalstoprepareW-states[21]. Consid- 1
GHZ = (000 + 111 ), (1)
123 123 123
ering the importance of W-class, in this paper we focus | i √2 | i | i
onthe states ofW-classfor perfectteleportationandsu- ^ 1
GHZ = (√2000 + 111 ). (2)
perdense coding. | i123 √3 | i123 | i123
As we know, quantum teleportation and superdense
Itiswellknownthatthefirststatecanbeusedforperfect
teleportationandsuperdensecoding. Butthesecondone
∗Electronicaddress: [email protected](D.Qiu). can not. At the same time, similar case is found in W-
2
class. For the following two states of W-class: where φ = |10i+√2|01i is a partial entangled pair. Cal-
| i √3
culating the entanglement between any two subsystems
1
W 123 = (100 123+ 010 123+√2001 123), (3) resulted, we have
| i 2 | i | i | i
1
|Wi123 = √3(|100i123+|010i123+|001i123), (4) E(12)3(|Wi123)=1,E1(23)(|Wi123)=E2(13)(|Wi123)(<101).
the fifrst state can be used for teleportation of a qubit
From the above results we know that: (i) If we parti-
state and for superdense coding of two classical bits by
tion the state in the way: 123,then the two subsystems
one qubit as shownin [12], but the secondstate cannot. |
resulted sharean entangledstate with one ebit of entan-
Inthispaper,wefocusonthestatesofW-class. Firstly
glement. Therefore, if Alice has particle ‘3’ and Bob has
wehaveabriefanalysisontheabovetwostatesfromW-
particles ‘1’ and ‘2’, then Alice can send Bob two classi-
class, to find out what difference lies between them. We
cal bits of information by sending one qubit, or Bob can
can partition the three particles whose state is W 123 teleport a state of qubit to Alice as shown in [12]. (ii)
| i
into two subsystems in three ways: 123, 213, and 123. The state does not have symmetry property, and thus
| | |
Then we have f the other two partitioning ways lead to two subsystems
2 1 whose entanglement is less than one ebit. (iii) The state
ρ1 =ρ2 =ρ3 = 0 0 + 1 1, also has the robustness against loss of qubits.
3| ih | 3| ih |
(5) In summary, from the above considerations, we have
2 1
ρ23 =ρ13 =ρ12 = ψ ψ + 00 00, seenthatexceptfortherobustnessagainstlossofqubits,
3| ih | 3| ih |
the states of W-class have also some other interesting
where ψ = 1 (01 + 10 ) is an EPR pair. If we cal- properties. For instance, state W has symmetry
| i √2 | i | i | i123
culate the entanglement between the two subsystems re- property that may be very useful for some applications,
sulted from the above partitioning ways, we have but the entanglement involved doefs not attain one ebit,
which, on the other hand, may be its limitation. Op-
E (W )=E (W )=E (W )<1. positely, state W , without symmetry property, can
1(23) 123 2(13) 123 (12)3 123 123
| i | i | i | i
(6) be used as a maximally entangled state if we choose an
f f f appropriate way to partition it. Therefore, the states of
Note thatherewe adoptthe definitionofpartialentropy W-classlike W and W maybeworthyoffurther
123 123
entanglement, where the entanglement between subsys- consideration|,aindwewo|uldiliketodothatinthispaper.
temsAandBinvolvedinthepurestate|ΨiAB isdefined In this paper, based onfthe previous works [5, 12], we
as characterize further the states of W-class, and consider
furthertheprotocolsforperfectteleportationandsuper-
E (Ψ )=S(ρ ) (7)
A|B | iAB A dense coding via W-class. In Section II, our main aim is
to find what states of W-class can be used for telepor-
where ρ =tr (Ψ Ψ) and S(ρ ) is the von Neumann
A B A
| ih | tation and superdense coding. Firstly, in Sec. IIA and
entropy [34].
IIB, we describe two transformations of the shared re-
Now the aboveresults showthat: (i) State W has
| i123 sources for teleportation and superdense coding, which
symmetry property that no matter how we partition it
willallowmanynewprotocolsforthatfromsomeknown
into two subsystems, the results are always fthe same.
protocols. In Sec. IIC, as an application, we obtain a
(ii) The entanglement between any two subsystems re-
sufficient and necessary condition for a state of W-class
sultedfromthe abovepartitionis lessthanoneebit, and
being suitable for perfect teleportation and superdense
therefore it is not suitable for perfect teleportation and
coding. In Sec.IID, asanotherapplication,we find that
superdense coding. (iii) If any qubit of the state is lost,
state W can be used to transmit three classical bits
the residual two qubits still share genuine entanglement | i123
bysendingtwoqubits. InSectionIII,ouraimistogener-
betweenthem, whichiscalledthe robustnessagainstloss
alizestates W and W to multi-particlesystems,
of qubits. Contrary to that, the states of GHZ-class do 123 123
| i | i
andcharacterizethem. InSec.IIIA,wegeneralizestates
not have this robustness.
sysFtoermsstabtyet|hWeis1a2m3,ewweacyasntaatlesodpbaerfotirtei,oannidtitnhteontwweohsuavbe- |wWeip12r3opaonsde|tWwoi12p3rototomcoufllstif-oqrubtietlespyostretamtsio.nInaSnedc.suIIpIeBr-,
dense codingfthatgeneralizethe protocolsby usingstate
1 1 1 1 W indicatedin[12]. InSec.IIIC,forthestatesthat
ρ3 = 0 0 + 1 1, ρ12 = ψ ψ + 00 00, (8) | i123
2| ih | 2| ih | 2| ih | 2| ih | generalize W to N-qubit systems, we point out an
123
| i
optimal way to partition them into two subsystems such
and
thatthe twofsubsystemsresultedsharemaximumentan-
3 1 glement, and, when N is an even number, in a certain
ρ =ρ = 0 0 + 1 1,
1 2
4| ih | 4| ih | sense these generalized states can be exploited as maxi-
(9)
3 1 mally entangled states. In Sec. IIID, we generalizestate
ρ =ρ = φ φ + 00 00,
23 13 4| ih | 4| ih | W to multi-particle systems with higher dimension.
123
| i
3
Finally, some concluding remarks are made in Sec. IV. WecanalsotakeaunitaryoperationV onsubsystem
B
B getting a new entangled state in the following way:
′′
ϕ =(I V )ϕ . (14)
II. PROTOCOLS FOR SUPERDENSE CODING AB A B AB
| i ⊗ | i
AND TELEPORTATION Then we have
(Ux I )ϕ′′ =(Ux I )(I V )ϕ
Inthissection,firstlywewilldescribetwotransforma- A⊗ B | iAB A⊗ B A⊗ B | iAB
=(I V )(Ux I )ϕ (15)
tions of the shared resources for superdense coding and A⊗ B A⊗ B | iAB
teleportation. Then as an application of these transfor- =(I V )Φ .
A B x
⊗ | i
mations, we will obtain a sufficient and necessary con-
From the orthogonality of set Φ and the unitarity
dition for a state of W-class being suitable for perfect {| xi}
of V , the states in set (I V )Φ are clearly mu-
teleportation and superdense coding. Also, we will find B { A⊗ B | xi} ′′
tually orthogonal. Therefore, by sharing state ϕ ,
that state W 123 canbe used to transmit three classical | iAB
| i Alice and Bob can also fulfill the same task as by shar-
bits by sending two qubits, which was considered to be
ing ϕ . The only change is that Bob should make
impossible in [12]. | iAB
a projective measurement on AB with projectors P =
x
(IA⊗VB)|ΦxihΦx|(IA⊗VB†).
A. Transformations of the shared resources for It is worthpointing out that transformations(12) and
superdense coding (14) can occur simultaneously. Then we should combine
(13) and (15) getting a new protocol. For simplicity, we
describe them separately. Similar case will occur in the
The standard protocol for superdense coding can be
protocol for teleportation as we will show in Sec. IIB.
described in the following. Suppose that Alice possesses
subsystemAandBobpossessessubsystemBandthetwo
subsystems share an entangled state ϕ . Alice can
| iAB B. Transformations of the shared resources for
apply operators from the set of unitary operators Ux
{ A} teleportation
on her subsystem A, and then send A to Bob. Then
the states of bisystemAB belonging to Bobwill form an
The standard protocol for teleportation can be de-
orthogonalset Φ . That can be described as
x
{| i} scribedasfollows. Atfirst,AliceandBobshareanentan-
(Ux I )ϕ = Φ . (11) gled state ϕ of bisystem AB, of which subsystem A
A⊗ B | iAB | xi | iAB
belongs to Alice and subsystem B belongs to Bob. Also
Becauseoftheorthogonalityofset Φx ,Bobcanmake Alice possesses another system ‘a’ whose state is Ψ .
a projective measurementon system{|ABi}with projectors Now Alice’s task is to send the state Ψ to Bob s|ucih
Px = Φx Φx to perfectly distinguish the set Φx , thatthestateofBob’ssubsystemturns|toithat,withthe
| ih | {| i}
suchthatBobcanknowexactlywhichoperatorAlicehas help of the shared state ϕ and some classical com-
AB
applied. In this process, if the number of the operators munication between them|.iThis task can be fulfilled, if
oinf sinetfo{r|mΦaxtii}onisfNro,mthAelnicBe.obcangetlog2N classicalbits |Ψia|ϕiAB can be rewritten in the form
D
If we take a viewpoint from discrimination between 1
Ψ ϕ = Φ U Ψ (16)
unitary operations [15, 16], then the core of superdense | ia| iAB √D | xiaA x| iB
coding is to find out as many unitary operators as pos- Xx=1
sible, such that they can be perfectly discriminated by where Φx aA is a set of mutually orthogonalstates of
{| i }
staNteow|ϕsiuApBp.ose there is another state ϕ′ given by jooninstusbyssytsetmem‘aAB’.,Aanlidce{Ucaxn} insoawsefutlfioflluhneitrartaysokpberyattowros
AB
| i steps:
′
ϕ =(U I )ϕ (12)
AB A B AB Make a projective measurement on ‘aA’ in a basis
| i ⊗ | i
•
thatincludes Φ ,gettingmeasurementresult
where UA is a unitary operator acting on subsystem A ‘x’ with proba{b|ilxitiyaA1}.
and IB is identity operator on B. Then we have D
Send the result ‘x’ to Bob who applies a unitary
(UAxUA† ⊗IB)|ϕ′iAB ==((UUAxxUA†I⊗)IBϕ)(UA⊗IB)|ϕiAB • oonpesruabtosyrsUtex†monB.B, and thus recovers the state |Ψi
A⊗ B | iAB
=|Φxi. Now we assume that the shared resource is |ϕ′iAB
(13) given by Eq. (12). Then we have
Thus it is shownthat by sharing state |ϕ′iAB, Alice and |Ψia|ϕ′iAB =(Ia⊗UA⊗IB)|Ψia|ϕiAB
Bob can also fulfill the same task as by sharing state 1 D (17)
ϕ , and the only thing changed is that the set of op- = (I U )Φ U Ψ .
| iAB √D a⊗ A | xiaA x| iB
erators applied by Alice turns to {UAxUA†}. xX=1
4
The aboveprocess shows that if Alice can fulfill her task that will be teleported from Alice to Bob. Then there is
bysharingstate ϕ ,thenshecanalsodothatbyshar-
′ | iAB Ψ a GHZ 123
ing state ϕ , with a projective measurement on her | i | i
| iAB 1
system ‘aA’ in a basis that includes {(Ia⊗UA)|ΦxiaA}. = 2[|ψ1+ia12⊗|Ψi3+|ψ1−ia12⊗σ3|Ψi3 (20)
Similarly, Alice and Bob can also share the state
ϕ′′ AB given by Eq. (14). Then we have +|ψ2+ia12⊗σ1|Ψi3+|ψ2−ia12⊗(iσ2)|Ψi3],
| i
where {|ψ1±i,|ψ2±i} are given by Eq. (25) in Sec. IID.
′′ Alice now makes a projective measurement on particles
Ψ ϕ =(I I V )Ψ ϕ
a AB a A B a AB
| i | i 1 ⊗D ⊗ | i | i (18) ‘tah1e2m’ einasaurbemaseisntinrecsluudltisntgo{B|ψob1±wi,h|ψo2±cai}n,raecnodvetrhsetnatseenΨds
= √D |ΦxiaA(VBUx)|ΨiB. atparticle‘3’byapplyingappropriateoperations. The|rei-
xX=1 foretask(i)isfulfilled. Fortask(ii),similarly, GHZ 123
| i
is shared between Alice and Bob, and we let Alice have
′′
This shows that by sharing state ϕ AB, Alice and Bob particle ‘3’ and let Bob have the left particles. Then Al-
| i
can fulfill the same task as by sharing state |ϕiAB, with ice first applies {I,σ1,−iσ2,σ3} on her qubit and then
theonlychangethatBob’soperatorsetturnsto{Ux†VB†}. sends her qubit to Bob. As a result, the possible states
ofthethreequbitsholdenbyBobwillformanorthogonal
set with four states that can be perfectly distinguished
by Bob. Therefore Bob can get two classical bits from
C. Application I: perfect teleportation of qubit Alice.
states and superdense coding of two classical bits via Nowwiththeaboveprotocolsandthetransformations
states of W-class stated in Sec. IIA and IIB, we can obtain a subclass of
W-class useful for perfect teleportation and superdense
coding. We give our result as follows.
Nowtherearetwotasksdescribedbelow. (i)CanAlice
Theorem 1. A state of W-class is suitable for perfect
teleport a state of qubit to Bob by sharing a W-state
teleportationandsuperdensecodingif,andonlyifitcan
betweenthem? (ii)CanAlicesendBobtwoclassicalbits
be convertedfrom state GHZ by sucha unitary op-
ofinformationbysendingonlyonequbit,withthehelpof | i123
erationthat is the tensor product of a two-qubit unitary
a shared W-state? In Ref. [11], the authors showed that
operation and a one-qubit unitary operation.
by sharing state W between Alice and Bob, task (i)
123 Proof. Thesufficiencyfollowsfromtheprotocolsbyus-
| i
can be fulfilled in a probabilistic manner, but not in a
ing GHZ andthetransformationsstatedinSec.IIA
123
perfect fashion. Aflso it is readily seen that the state can and|IIB. Wi e provide Fig. 1 and Fig. 2 to show that.
notbeusedfortask(ii)inaperfectfashion. Theseresults
Next we verify the necessity. From our knowledge
seem to imply that W-states are not suitable for perfect
about teleportation and superdense coding, it follows
teleportation and superdense coding. However, Ref. [12]
that a state ϕ of triqubit that can be used for tasks
123
found that a class of states within W-class can be used | i
(i)and(ii)cannecessarilybepartitionedintotwosubsys-
to do that. Those states are given in the form
tems,sayAandB,whichshareoneebitofentanglement.
Thatis, ϕ canberegardedasamaximallyentangled
AB
1 state of b|isiystem AB. In addition, we know that all the
W = (100 +√neiγ 010
| ni123 √2+2n | i123 | i123 (19) maximally entangled states of bisystem AB can be con-
+√n+1eiδ 001 ). verted from each other by local unitary operations (i.e.,
| i123 in the form U U ). Therefore, the necessity of the
A B
⊗
theorem follows. This completes the proof.
Aprototypestateinthisclassis W 123 givenbyEq.(3). Now we have a brief test that the states given by
| i
In the conclusions of [12], the authors proposed some Eq. (19) used in [12] for perfect teleportation and su-
problems worthy of further consideration, one of which perdense coding are contained in the subclass stated by
is whether there are other classes of W-states useful for us. Firstly we can rewrite Eq. (19) in the following
perfect teleportation and superdense coding. Now with
1
those transformations we stated previously, indeed we W = (φ 0 +eiδ 00 1 ), (21)
n 123 12 3 12 3
candiscoveraclassofW-statesusefulforperfecttelepor- | i √2 | i | i | i | i
tationandsuperdensecodingwhichincludesthesestates where φ = 1 (10 +√neiγ 01 ). Then W can
given by Eq. (19). As a result, it will explain why state | i √1+n | i | i | ni123
be converted from GHZ by
Wn 123 can be used for tasks (i) and (ii). | i123
| i
As we know state |GHZi123 is suitable for tasks (i) |Wni123 =(V12⊗I3)|GHZi123 (22)
and (ii). Here we have a brief review of these protocols where V is a unitary operator acting on particles ‘1’
12
for that. Alice and Bob share state GHZ 123, of which and ’2’ given as
| i
Bob has particle ‘3’, and Alice has particles ‘1’ and ‘2’.
V = φ 00 + 11 01 + φ 10 +eiδ 00 11, (23)
Also Alice has particle ‘a’ with state Ψ = α0 +β 1 12 ⊥
| ih | | ih | | ih | | ih |
| i | i | i
5
a
Alice 1 (cid:0)(cid:0)(cid:18) Px
( 2 x bits by sending two qubits. Then they conjectured that
? theremay existsome otherstatesofW-classthatcanbe
Bob 36 U3x 6 =⇒ utoseddoftohrastuacshwaetwaislkl.shIonwfabcetl,owst.aFteur|tWheir1m23ocrea,nthbeereusaerde
Ψ a GHZ 123 Ψ indeed many W-states can be used for that.
| i | i | i
Firstly we know that state GHZ can be shared
a | i123
Alice 1 (cid:0)(cid:0)(cid:18) Px′ bclyasAsilciacelbaitnsdoBfionbfo,rsmuachtiotnhabtyAselnicdeincgantwsoenqdubBitosb[7t,h1r2e]e.
( 2 x In this case, Alice possesses the first two qubits, and the
?
unitary operators applied by Alice can be in the set
Bob 3 UxV†
6 3 3 6
Ψ a W′ 123 Ψ U12 ={I⊗I,σ1⊗I,(−iσ2)⊗I,σ3⊗I, (24)
| i | i | i I σ ,I ( iσ ),σ σ ,σ ( iσ ) .
1 2 1 1 1 2
⊗ ⊗ − ⊗ ⊗ − }
AfterAlice’soperationonherqubitsandsendingthatto
Bob, the state of the three qubits will be in one of the
FIG.1: Teleportation ofstate Ψ : whenthesharedresource following states
ischangedfrom GHZ 123 to W| ′i123 =(V12 V3)GHZ 123,
| i | i ⊗ | i 1 1
tfrhoemn thPexmteoasuPrxe′m:e(Inat prVo1j2e)cPtoxr(sIamaVd1e†2b)y,Aanlidcethaereopcheraantgoerds |ψ1±i= √2(|000i±|111i), |ψ2±i= √2(|100i±|011i),
applie{d b}yBo{b are cha⊗nged from {⊗U3x} t}o {U3xV3†}. |ψ3±i= √12(|010i±|101i), |ψ4±i= √12(|110i±|001i).
Alice 3 Ux
3 A (25)
AAU
2 = Sincetheaboveeightstatesaremutuallyorthogonal,Bob
Bob 1 ⇒ can make a projective measurement to perfectly distin-
n 6 6 guish them, such that he can get three classical bits of
GHZ 123 Φx information from Alice.
| i | i
Now from the feasible protocol above, we can obtain
Alice 3 UxV† that W 123 is also suitable for transmitting three classi-
3 3 A cal b|itsiby sending two qubits. First of all, one should
AU
notice that
2
Bob 1 (U I )GHZ = W (26)
12 3 123 123
n 6 6 ⊗ | i | i
W′ 123 Φ′x where
| i | i
U = ϕ+ 00 + 11 01 + ϕ 10 + 00 11, (27)
12 −
| ih | | ih | | ih | | ih |
FIG. 2: Superdense coding of two classical bits: when the and ϕ = 1 (10 01 ). U is a joint unitary oper-
c(shVha1a2rneg⊗dedVr3ef)sr|ooGumHrcZeUii13xs23c,httoahnegnUetd3xhVef3r†oopm,earan|GtdoHrtshZaeip1sp2t3laiettdeosbr|yWesAu′lilit1ce2ed3aar=et abtyomr|aact±triiinxgfoo√rnm2th|ine fitihr±setb|tawsioisq{u|b010i2tis,,|0a1nid,|c1a0ni,a|1ls1oi}b:e given
Bob’endarech{ang}edfro{m Φx} to Φ′x :(V12 I3)Φx . 0 0 0 1
{| i} {| i ⊗ | i}
1 0 1 0
 √2 −√2 . (28)
1 0 1 0
whereφ = 1 (√ne iγ 10 01 ). Now we have √2 √2
| ⊥i √1+n − | i− | i  0 1 0 0
shown that W satisfies the condition given in The-  
| ni123  
orem1. Thus it is naturalto use it for teleportation and
NowweletAliceandBobsharestate W ,ofwhich
123
superdensecodingasdidin[12]. Inaddition,asaspecial | i
the first two qubits belong to Alice and the last qubit
state in Eq. (19), W is of course suitable for that.
123 belongs to Bob. From Eqs. (12), (13) and (26), we soon
| i
get that if Alice chooses operators Ux ∈ U12U1†2 on her
twoqubits andthensendsthatto Bob,the eightorthog-
D. Application II: use state W 123 to transmit onal states given by Eq. (25) will appear at Bob’s end.
| i
three classical bits by sending two qubits
Therefore, Bob can make a projective measurement to
perfectly distinguishwhich operatorhas beenappliedby
In Ref. [12] the authors thought that it may be not Alice, and then gets three classical bits of information
possible to use state W to transmit three classical from Alice. The process is shown in Fig. 3.
123
| i
6
1 Ux
1 A Clearly the state is symmetric in the sense that permu-
Alice( 2 Ux AAU tation of particles does not change state. Therefore,
2 A = if we partition it into two subsystems in such a way:
AU ⇒
Bob 3 (1...i 1,i+1...N)i(simplydenotedby(=i)i),then
6 6 − | 6 |
we get the Schmidt decomposition form:
GHZ 123 Φx
| i | i
N 1 1
1 U1x A |WNi=r N− |WN−1i|0ii+rN|0...00i|1ii. (30)
Alice( 2 U1†2 Ux AAU Thfentwodensity opferatorsassociatedwiththe twosub-
2 A systems are
AU
Bob 3
6 6 1 N 1
ρ = 1 1 + − 0 0,
W 123 Φx i N| ih | N | ih |
| i | i
1 N 1
ρ(6=i) = N|0...0ih0...0|+ N− |WN−1ihWN−1|,
(31)
FIG. 3: Change the channel from GHZ 123 to W 123 for f f
superdensecoding of three classical b|its. i | i where i = 1,2,...,N, and ρ(=i) denotes the density op-
6
erator of WN losing the ith qubit. Calculating the en-
| i
tanglement between the two subsystems, we have
Notethateveryoperatorintheset canbechosenas
12 f
tensor product of two local operatorUs. Therefore, state E (WN )= plog p (1 p)log (1 p) 1,
GHZ 123 can be suitable for the case of two senders (6=i)|i | i − 2 − − 2 − ≤ (32)
| i
andone receiver,where twosenders,spatially separated, f
makeonlylocaloperationsontheirownqubits. Thiscase where we let p = 1. Now from the properties of binary
N
wasgeneralizedtothecaseofmulti-sendervsonereceiver entropy [34], we know that the value decreases while N
in [17, 18, 19]. With respect to state W , although increases,and the equality can be attained when N =2.
123
| i
it can be used to send three classical bits, it may be In fact, when N = 2, the state reduces to the EPR pair
not suitable for the case of two senders and one receiver, 1 (01 + 10 ).
√2 | i | i
because of the non-locality of Ux that derives from the State W 123canbegeneralizedtomulti-qubitsystems
non-locality of U12. Indeed, it was shown in [5] that the in the fo|llowiing form:
statesinW-classcannotbeconvertedfromthe statesin
GHZ-class by local operations. It is also worth pointing N
1
outthat,herewejustproposeaschemeforsendingthree WN = 10...0 + +
| i 2(N 1) | i ··· (33)
classical bits where non-local operators are allowed by
− (cid:16) z }| {
thesender,butwedonotprovethe impossibilityoflocal p0...10 +√N 10...01 .
operators. | i − | i
(cid:17)
NotethatallthestatesofW-classthatsatisfythecon- This state is not fully symmetric. More exactly, permu-
ditiongivenbyTheorem1inSec.IICaresuitableforthe tation of the first N 1 qubits does not change state. If
task stated in this subsection. we partition the stat−e into two subsystems in the way:
(1...i 1,i+1...N)i as before, then when i = N it
− | 6
can be rewritten in the Schmidt decomposition form:
III. CHARACTERIZE W-CLASS OF
MULTI-PARTICLE SYSTEMS 2N 3
|WNi=s2(N −1)|ϕN−1i|0ii
A. Generalize states W 123 and W 123 to − (34)
| i | i 1
multi-qubit systems
+ 0...00 1 ,
i
f 2(N 1)| i| i
−
Consideringthespecialpropertiesofstates W and
| i123 where p
W ,itisinterestingtogeneralizethemtomulti-qubit
123
s|ystiems. Infact,itisnotdifficulttodothat. Asshownin 1 N−1
[5f], state |Wi123 can be easily generalizedto multi-qubit |ϕN−1i=√2N 3 |10...0i+···+|0...10i (35)
systems in the form (N ≥2): − (cid:16) z }| {
f +√N 10...01 .
− | i
N
1 (cid:17)
WN = ( 10...0 + 01...0 + + 0...01 ). Then the entanglement between the two subsystems re-
| i √N | i | i ··· | i sulted is always less than one ebit, and when N = 2 it
z }| { (29)
f attains one ebit.
7
When i=N, the state canbe rewritteninthe formof B. State WN as a shared resource for
| i
maximally entangled state teleportation and superdense coding
1 1
|WNi= √2|WN−1i|0iN + √2|0...00i|1iN. (36) Here we see how state WN is used for teleportation
| i
and superdense coding.
Therefore, if we fpartition WN into two subsystems in
| i
theway: (1...N 1)N,thentheentanglementbetween a. Teleportation Let Alice and Bob share state
− |
the two subsystems is: E (WN ) = 1. Thus WN . Now Alice possesses the first N 1 qubits while
(1...N 1)N
the state can be used as a share−d r|eso|urceifor teleporta- |Bobpiossessesthe lastqubit. Alice also h−as a particle‘a’
tionandsuperdensecoding,whichwillbedetailedinthe in the unknown state Ψ = α0 +β 1 . Then the
a a a
| i | i | i
following subsection. combined input state can be rewritten as
1
Ψ a WN =(α0 a+β 1 a) (WN−1 0 + 0...00 1 )
| i | i | i | i √2 | i| i | i| i
1
= α0 a WN−1 0f+α0 a 0...00 1 +β 1 a WN−1 0 +β 1 a 0...00 1
√2 | i | i| i | i | i| i | i | i| i | i | i| i
(37)
h i
1
= (η+ +fη− )α0 +(ξ+ ξ− )α1 +(ξ+ +fξ− )β 0 +(η+ η− )β 1
2 | i | i | i | i−| i | i | i | i | i | i−| i | i
1h i
= η+ (α0 +β 1 )+ η− (α0 β 1 )+ ξ+ (α1 +β 0 )+ ξ− (β 0 α1 ) ,
2 | i | i | i | i | i− | i | i | i | i | i | i− | i
h i
where η , ξ is a set of orthogonalstates given by Aliceappliestheoperatorsselectedfrom I,σ , iσ ,σ
± ± 1 2 3
{| i | i} { − }
on her qubit. One can readily find that the four states
1 produced are mutually orthogonal. Now Alice can send
|η±i= √2 |0ia|WN−1i±|1ia|0...00i , herqubittoBobwhomakesaprojectivemeasurementon
(cid:16) (cid:17) (38) the N qubits in a basis that includes the four states pro-
1
ξ± = 1 a WfN−1 0 a 0...00 . duced. Since the states produced are orthogonal, Bob
| i √2 | i | i±| i | i
can perfectly distinguish which operator Alice has ap-
(cid:16) (cid:17)
f plied. Therefore, Bob can decode two classical bits of
Now Alice makes a projective measurement in a basis
information from Alice’s encoding.
thatincludesthestates η , ξ ontheparticlespos-
sessedby herself. Then{sh|e±sien|d±sit}he results ofher mea- Note that the two protocols by using state |Wi123
statedin[12]arethespecialcaseoftheabovegeneralized
surement using two classical bits to Bob who can apply
protocols when N =3.
one of the unitary operators I,σ ,σ , iσ to convert
3 1 2
{ − }
the state of his particle to that of particle ‘a’. Now the
teleportation protocol has been completed. This proto-
col consumes one ebit of shared entanglement and two C. The optimal way to partition state WN into
| i
two subsystems
bits of classical communication between Alice and Bob.
f
b. Superdense coding Let Alice andBobshare state
WN . If Bob possesses the first N 1 qubits and Al- As we can see, state WN has a good property—
| i − | i
ice possesses the last qubit, then the coding protocol is symmetrythatisinterestingandusefulforsomeapplica-
described below: tions. Thus this state mayfbe worthy of further consid-
eration. In the previous subsection, we investigated the
1 1 case ofpartitioning it into two subsystems such that one
00: WN I N 1 I WN 1 0 + 0...00 1 ,
| i ⊗ − ⊗ √2| − i| i √2| i| i subsystem possesses one qubit and the other subsystem
−−−−−−−→
1 1 possessestheresidualqubits,andthenwefoundthatthe
01: WN I⊗N−1 σ1 fWN−1 1 + 0...00 0 , entanglementbetweenthetwosubsystemsbecomesclose
| i ⊗ √2| i| i √2| i| i
−−−−−−−−→ to0whenN increases,whichmaybenotagoodnewsfor
1 1
10: WN I⊗N−1 iσ2 fWN−1 1 0...00 0 , our teleportationandsuperdense coding. However,if we
| i ⊗− √2| i| i− √2| i| i makelightofthisstatejustbecauseoftheseresults,then
−−−−−−−−−−→
11: WN I⊗N−1 σ3 1 WNf−1 0 1 0...00 1 . thatmaybethereallybadnews. Inthefollowingwewill
| i ⊗ √2| i| i− √2| i| i findanoptimalpartitioningwayinthesensethattheen-
−−−−−−−−→
(39) tanglement between the two subsystems resulted attains
f
8
its maximum value. At the first blush, it seems impossible to be comparable
Ageneralwayofpartitioningstate WN intotwosub- with the following maximally entangled state
| i
systemsistoletonesubsystemhavex(1 x<N andx
isanintegernumber)qubitsandtheotfher≤subsystemthe GHZ4 = 1 (0000 + 1111 ). (43)
| i √2 | i | i
left N x qubits. Then, due to its symmetry property,
−
nomatterwhatthe xqubitsare,the statecanalwaysbe However, according to what we have obtained before, if
rewritten in the form: we partition it into two subsystems such that every sub-
x N x system has two qubits, then we can rewrite it in this
WN = 1 10...0 + + 0...01 0.−..0 form:
| i √N | i ··· | i | i 1
h(cid:16) z }| { (cid:17) z }| { W4 = (ϕ+ 00 + 00 ϕ+ ) (44)
f x N−x | i √2 | i| i | i| i
+ 0...0 10...0 + + 0...01
| i | i ··· | i where ϕ =f 1 (10 01 ). In this way, the entan-
zx}| { (cid:16) zN−}x| { N x x (cid:17)i glemen|t b±eitwee√n2th|e tiw±o s|ubisystems is one ebit. At the
= Wx 0...0 + − 0...0 WN−x sametime,note thatifwepartition GHZ4 inthe same
N| i| i N | i| i | i
r r way, then the entanglement between two subsystems re-
z }| { z }| { (40)
f f sulted is also one ebit. Therefore, in a certain situation,
where states Wx and WN x are resulted from N in W4 can be used to fulfill the same task as by using
Eq. (29) bei|ng xi and |N −x,irespectively. Then the |GHiZ4 .
| i
entanglement fbetween thfe −two subsystems resulted, say fTeleportation of entangled pairs via three-particle
A and B, is the binary entropy given by states was discussed in [8, 9, 10]. Next we see how state
x x x x W4 is used as a shared resource for teleportation of an
EA|B(|WNi)=−N log2 N −(1− N)log2(1− N). e|ntaingled pair. Detailedly, Bob and Charlie possess the
(41) 3fth and 4th qubits, respectively, and Alice possesses the
f firsttwoqubits. Alicealsopossessesanentangledpairof
Thusfromthepropertiesofbinaryentropy,weknowthat qubits given by
theabsolutevalue x 1 issmaller,thevalueinEq. (41)
isbigger,andthem|Nax−im2u|mvalue1isattainedwhen x = ψ ab =α00 +β 11 , or ψ′ ab =α01 +β 10 . (45)
N | i | i | i | i | i | i
1. Therefore, there are two cases we should consider:
2 Then we have
(i) N is even. Then the maximum value 1 can al-
• ways be attained by letting x = N. That is, in |ψiab|W4i
2 1
this case, a balanced partition always leads to two =(α00 +β 11 ) (ϕ+ 00 + 00 ϕ+ )
subsystems sharing 1 ebit of entanglement. |fi | i √2 | i| i | i| i
(46)
1
• (ii) N is odd. In the case, we let x = ⌊N2⌋, that = 2 |µ+i(α|00i+β|ϕ+i)+|µ−i(α|00i−β|ϕ+i)
ivsa,luxeisxthe1intiesgtehrepsamrtalloefstnuammbonerg aN2ll.pToshseibnlethxe, + ωh+ (αϕ+ +β 00 )+ ω− (αϕ+ β 00 ) ,
|N − 2| | i | i | i | i | i− | i
and thus E (WN ) attains its maximum value i
AB and similarly,
amongallpos|sib|lex.iOfcourse,inthiscase,theen-
tanglementcannfot attain1 ebit forever,but when ψ′ ab W4
| i | i
N is large enough, the value can arbitrarily close 1
to 1. =(α|0f1i+β|10i)√2(|ϕ+i|00i+|00i|ϕ+i)
In summary, for state WN , among all the ways of 1 (47)
| i = π+ (α00 +β ϕ+ )+ π (α00 β ϕ+ )
partitioning it into two subsystems, the optimal way is 2 | i | i | i | −i | i− | i
such one that leads to twofbalanced (or approximately h
+ ̟+ (αϕ+ +β 00 )+ ̟ (αϕ+ β 00 ) ,
balanced) subsystems. Below we provide an example to | i | i | i | −i | i− | i
apply the way. i
where
1
1. An example: state W4 used for teleportation of |µ±i= √2(|00i|ϕ+i±|11i|00i),
| i
entangled pairs 1
f |ω±i= √2(|00i|00i±|11i|ϕ+i),
(48)
As an example, we consider state WN for N = 4. 1
That is the following state | i |π±i= √2(|01i|ϕ+i±10i|00i),
1 f 1
W4 = (1000 + 0100 + 0010 + 0001 ). (42) ̟ = (01 00 10 ϕ+ ).
±
| i √4 | i | i | i | i | i √2 | i| i±| i| i
f
9
The set S = µ , ω , π , ̟ can be extended subsystem A. Alice can now manipulate her subsystem
± ± ± ±
to an orthogon{a|l bias|is oif t|heiH|ilberit}space H2⊗4. Now by unitary operations
Alice canmake a projectivemeasurementina basis that
includesS onthefourqubitspossessedbyher,andsends d 1
−
the results of measurement using three cbits to Bob and U(m,n)= e2πikm/d k k n (51)
| ih ⊕ |
Charlie. Then Bob and Charlie can together choose an k=0
X
appropriate joint unitary operator (non-local operator) where denotes addition modulo d, and m,n =
′ ⊕
on their qubits to recover the state ψ or ψ at 0,...,d 1. After that, Alice sends her subsystem to
ab ab
their qubits. For instance, if the resul|t iof meas|ureiment Bob. Th−en similar to the case in [18], d2 mutually or-
is µ+, then Bob and Charlie make a unitary operation thogonal states will be produced at the disposal of Bob
U = 00 00 + ϕ+ 11 + 11 ϕ+ + ϕ ϕ on their who can make a projective measurement on the N par-
− −
| ih | | ih | | ih | | ih |
two qubits. ticles to distinguish what operation Alice has applied.
Notethatintheaboveprotocol,thetwokindsofstates Therefore, Bob gets 2log d classical bits of information
2
′
ψ and ψ canbeteleportedbyusingthesamepro- from Alice.
ab ab
| i | i
tocol with three bits of classical communication. Here if
we just want to teleport one kind of them, say ψ ,
ab
| i
then two bits of classical communication is enough. In
IV. CONCLUSION
addition, since the operations done by Bob and Char-
lie are non-local, Bob and Charlie can not be spatially
separated, which may be the limitation of this protocol. In this work, we investigated that the states of W-
Onthe otherhand,becauseofthis non-locality,Boband class are used for teleportation and superdense coding,
Charlie must cooperate to recover state ψ′ or ψ , and characterizedW-class in multi-particle systems. We
ab ab
so this protocol may be considered for q|uaintum |seciret describedtwotransformationsofthesharedresourcesfor
sharing [20]. By the way, state W4 can also be used to teleportation and superdense coding, with which we ob-
transmit three classical bits by|sendiing two qubits. tained a sufficient and necessary condition for a state of
W-class being suitable for perfect teleportation and su-
f
perdensecoding,andwefoundthatthe state W can
123
| i
D. Generalize state W 123 to multi-particle be used to transmit three classical bits by sending two
systems with hi|ghier dimension qubits, which was considered to be impossible in [12].
We generalized the states of W-class to multi-qubit sys-
tems and multi-particle systems with higher dimension.
Inthefollowing,weconsiderthestateofmulti-particle
We proposed two protocols for teleportation and super-
systems with d-dimensional particles,such that it gener-
dense coding by using W-states of multi-qubit systems
alizes the state W . First we let
123
| i that generalize the protocols by using W stated in
123
| i
N [12]. We obtained an optimal way to partition some W-
ξN = 1 i0...0 + 0i...0 +... 0...0i (49) states of multi-qubit systems into two subsystems, such
| (i)i √N | i | i | i that the entanglement between them achieves maximum
(cid:0) z }| { (cid:1) value.
for i=1,...,d 1. Then we let
− As we pointed out in Sec. IID, although state W 123
| i
can be used to transmit three classical bits by send-
d 1
|ΩNi= √1d − |ξ(Ni)−1i|i−1i+|0...0i|d−1i . (50) itnwgotswenodqeursbiatns,ditonmearyecbeievenro.tTsuhietnabilneffuotrurteh,eocnaesecaonf
hXi=1 i considerwhetherthere existW-states suitablefor super-
Nowstate ΩN maybetakenasareasonablegeneraliza- dense coding with two senders and one receiver. If the
| i
tionofstate W . Infact,Eq. (50)canberegardedas answerisyes,thenonecanfurthergeneralizethatto the
123
theSchmidt|deciompositionforthe stateofbisystemAB, case of multi-sender and one receiver. Otherwise, one
where subsystem A consists of the first N 1 particles should prove the impossibility.
−
andsubsystem B has the Nthparticle. Thus, the entan- This work is supported in part by the National Natu-
glement between them is log d. Clearly, when d = 2, ral Science Foundation (Nos. 90303024, 60573006), the
2
state ΩN reduces into state WN given by Eq. (29). Higher School Doctoral Subject Foundation of Ministry
Nex|t wie see how state ΩN| isiused for superdense of Education (No. 20050558015), and the Natural Sci-
| i
coding. Suppose that Alice and Bob share state ΩN ence Foundation of Guangdong Province (No. 031541)
| i
suchthatAlicepossessessubsystemBandBobpossesses of China.
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