Generation of GHZ and W states for stationary qubits in spin network via resonance scattering

0.44 MB·English

by  L. Jin

Generation of GHZ and W states for stationary qubits in spin network via resonance
scattering
L. Jin, and Z. Song†
Department of Physics, Nankai University, Tianjin 300071, China
We propose a simple scheme to establish entanglement among stationary qubits based on the
mechanismofresonancescatteringbetweenthemandasingle-spin-flipwavepacketindesignedspin
network. Itisfoundthatthroughthenaturaldynamicalevolutionofanincidentsingle-spin-flipwave
packetinaspinnetworkandthesubsequentmeasurementoftheoutputsingle-spin-flipwavepacket,
multipartite entangled states among n stationary qubits, Greenberger-Horne-Zeilinger (GHZ) and
W states can be generated with success probabilities PGHZ = 2/1 + t−n 2 and PW = t2/n
| | | |
respectively,where t is thetransmission amplitude of thenear-resonance scattering.
9
0 PACSnumbers: 03.67.Bg,75.10.Pq,03.67.-a
0
2
I. INTRODUCTION ate multipartite entanglement among stationary qubits
n
via scattering process. We introduce a scheme that al-
a
J In quantum information science it is a crucial prob- low the generation of the GHZ and W states of station-
ary qubits in spin networks. In the proposed scheme,
5 lem to develop techniques for generating entanglement
among stationary qubits. Entanglement as unique fea- the flying qubit is a Gaussian type single-flip moving
] wavepacketontheferromagneticbackground,whichcan
h ture of quantum mechanics can be used not only to
propagate freely in XY chain. The stationary qubit is
p test fundamental quantum-mechanical principles [1, 2],
consisted of two spins coupled by Ising type interaction
- but to play a central role in applications [3, 4, 5].
t with strength Jz. A single spin flip can be confined in-
n Especially, multipartite entanglement has been recog-
side such two spins by local magnetic field h to form a
a nized as a powerful resource in quantum information
u double-dot(DD)qubit. ThesystemofanXY spinchain
processing and communication. There are two typ-
q with a DD qubit embedded in exhibits a novel feature
ical multipartite entangled states, Greenberger-Horne-
[ under the resonance scattering condition h = Jz, that a
Zeilinger (GHZ) and W states, which are usually re-
single-flip moving wave packet can completely pass over
2 ferred to as maximal entanglement. Numerous protocols
v for the preparation of such states have been proposed a DD qubit and switch it from state |0i to |1i simul-
1 taneously. We show that the scattering between a fly-
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].
6 ing qubit and a DD qubit can induce the entanglement
1 Most of them are scattering-based schemes which uti- between them and the operation on the DD qubit can
0 lize two processes: the natural dynamic process of an
be performed by the measurement of the output flying
. always on system and the final project process carried
1 qubit. It allows simple schemes for generation of mul-
out by a subsequent measurement. Another feature of
0 tipartite entanglement, such as GHZ and W states by
9 suchkindofschemesisthattherearetwokindsofqubits
simply-designed spin networks. We also investigate the
0 involved in: target qubits and flying qubit. The target
influence of near-resonance effects on the success proba-
: qubitsarethemainentitiesthatwillbeentangledbythe
v bilities ofthe schemes. It is found that the success prob-
above two processes, which are usually stationary and
Xi can be realized by atoms, impurities, or other quantum abilities are PGHZ = 2|1+t−n|2 and PW = |t|2/n for
the generation of GHZ and W states, respectively. Here
r devices. The flying qubit is a mediator to establish the
a t is the transmission probability amplitude for a single
entanglement among the target qubits via the interac-
DD qubit and n is the number of the DD qubits.
tion between them, which is usually realized by photon
or mobile electron. This paper is organized as follows. In Sec. II the DD
qubit and spin network are presented. In Sec. III we
In this sense, the type of interaction between station-
investigate the resonance-scatteringprocess between the
ary qubits and the flying qubit as a mediator and the
flying and stationary qubits. Sec. IV and V are devoted
transfer of the flying qubit are crucial for the efficiency
totheapplicationoftheresonancescatteringonschemes
of the entanglement creation. In general, such two pro-
of creating GHZ and W states. Section VI is the sum-
cesses are mutually exclusive. The scattering between
mary and discussion.
stationary and flying qubits can convert information be-
tween them, while it also reduces the fidelity of the fly-
ing qubit, which will affect the efficiency of the entan-
glement, especially for multi-particle system. It is still
II. DOUBLE-DOT QUBIT
a challenge to create entanglement among massive, or
stationary qubits.
In this paper, we consider whether it is possible to Thespinnetworkweconsiderinthispaperisconsisted
use an arrangement of qubits, a spin network, to gener- ofspinsconnectedviatheXXZ interaction. TheHamil-
2
tonian is
J⊥
H = ij(σ+σ−+H.c.)+Jzσzσz (1) (cid:0) (cid:1)
− " 2 i j ij i j#
hXi,ji
(cid:2)(cid:3)(cid:4)
+ h σz,
i i
i y (cid:10)(cid:11) (cid:0)
X
where σ± = σx iσy, and σα (α = x,y,z) are the
i i ± i i
Pauli spin matrices for the spin at site i. The total z- (cid:9) y (cid:6)(cid:7)(cid:8)
component of spin, or the number of spin flips on the
(cid:2)(cid:5)(cid:4)
ferromagnetic background, is conserved as it commutes
with the Hamiltonian. For Jz = 0, it reduces to XY
ij
spin network, which has received a wide study for the FIG.1: (Coloronline)Thebasicbuildingblockoftheproposed
purpose of quantum state transfer and creating entan- scheme. (a) Schematic illustration for a DD qubit embedded
in a spin chain, which consists of two spins with only Ising
glement between distant qubits by using the natural dy-
namics [21]. For Jz = J⊥, the Hamiltonian describes interaction. The local external magnetic field can confine a
ij ij spin flip in the two spins to form a DD qubit with states 0
isotropic Heisenberg model. In the antiferromagnetic | i
and 1 . (b)Schematicillustrationoftheresonancescattering
regime (Jz 0), a ladder geometry spin network, a | i
ij ≺ processbetweenasingle-spin-flipwavepacketandaDDqubit.
gapped system [22], has been employed as a data bus
Under the resonance scattering condition, an incident wave
for the swapping operation and generation of entangle- packet can totally pass through a DD qubit and switch itfrom
ment between two distant stationary qubits. It has been state 0 to state 1 .
| i | i
shown that a moving wave packet can act as a flying
qubit [23, 24, 25] like photon in a fiber. On the other
hand, the analogues of optical device, beam splitter can Thereisaquasi-invariantsubspacewiththediagonalen-
be fabricated in quantum networks of bosonic [26], spin ergy being h and under the condition h J⊥ , which
≫
and ferimonic systems [27]. is spanned by basis
(cid:12) (cid:12)
Inthispaper,weconsideranewtypeofqubit,double- (cid:12) (cid:12)
dotqubit,whichcanbeembeddedinsuchspinnetworks. |ϕ1i = |↑i1|↓i2|↑i3|↓i4,
A DD qubit consists of two ordinary spins at sites d and ϕ = , (4)
| 2i |↓i1|↑i2|↑i3|↓i4
d+1, connected via Ising type interaction in the form
ϕ = .
| 3i |↓i1|↑i2|↓i3|↑i4
Hd =−Jzσdzσdz+1+h σiz. (2) The matrix of the Hamiltonian in this subspace reads
i=d,d+1
X h J⊥
Wnethwenorskuscwhitkhin|dhs|≫of Jtwi⊥joasnpdinhsia=re0,emabspeidndfledipiinsctohnefisnpeind −J2⊥ −hJ2⊥ −hJ2⊥  (5)
withinitanadndfor1ms=aDDqubit.wWithetwhiellnsohtoawtiotnhsa|t0siduc=h  − 2 
a|↓inde|w↑itdy+p1e of qu|biit h|a↑sida|↓niodv+e1lfeature when it interacts with eigenstates |ψii and eigen energies εi (i = 1,2,3)
being
with another spin flip in the spin networks.
1
ψ = (ϕ ϕ ),ε =h; (6)
1 1 3 1
| i √2 | i−| i
III. RESONANT SCATTERING
1
ψ = ϕ √2 ϕ + ϕ ,
2,3 1 2 3
| i 2 | i∓ | i | i
The main building block in the spin network of our (cid:16)1 (cid:17)
scheme is the DD qubit. It acts as a massive or sta- ε2,3 = J⊥+h.
±√2
tionary qubit, like atoms or ions in cavity-QED-based
schemes. To demonstrate the property of a DD qubit in Obviously,intheinvariantsubspace,sucha4-sitesystem
a spin chain, we investigate a small system of 4-site, a acts as a normal 3-site system. Note that the DD qubit
DD qubit connecting to two spins. In order to provide as the center of the 3-site system can be in two different
a clear exposition, we firstly assume a specific coupling states 0 or 1 while another spin flip is at left or right
configuration with h = Jz, which leads to the following site. T|hius a|tiime evolution process can accomplish the
4-site Hamiltonian transformation or
|↑i1|↓i2|↑i3|↓i4 −→ |↓i1|↑i2|↓i3|↑i4
0 1 with 100% success proba-
Hs = −J⊥ σlxσlx+1+σlyσly+1 (3) |b↑iili1ty|.iSdu|↓cih4a−f→eat|u↓ir1e|isidde|↑siir4able for the quantum infor-
l=1,3 mation processing. It is because that, on one hand, a
X (cid:0) (cid:1)
hσzσz +h(σz +σz). spin flip passing over a DD qubit can operate the qubit
− 2 3 2 3
3
state; and on the other hand, the state of a DD qubit
can indicate whether there is a spin flip passing over it.
= α 0 +β 1 , (10)
A similar transformationhas been proposedthroughthe |րid | id | id
π
cavity input-output process in adiabatic limit [28]. A = α φ( ,L) +β Vac ,
|րiin 2 | i
particular merit of the present scheme is that it is based
(cid:12) E
on a natural dynamic process rather than an adiabatic where α andβ arearbit(cid:12)rarycoefficients satisfying α2+
process. β 2 =1. (cid:12) | |
Nowweconsiderthedynamicprocessoftheinteraction | |In practice, the difference between h and Jz will leads
between a moving wave packet and a DD qubit. We
to the reflection of the incident wave packet. Then the
embedaDDqubitintoachainasillustratedinFig. 1(a).
scattering process can be expressed as
It has been shown that a single-spin-flip wave packet in
the form [23, 24]
π π π
φ( ,L) 0 r φ( ,L) 0 +t φ( ,R) 1
(cid:12)φ(π2,Nc)E= √1ΩXj e−α22(j−Nc)2+iπ2jσj+|Vaci (7) w(cid:12)(cid:12)(cid:12)ith2 r 2E+| itd2−→= (cid:12)(cid:12)(cid:12)1. −2The Etr|anidsmiss(cid:12)(cid:12)(cid:12)ion2coeffiEc|(i1ein1dt)
can(cid:12)(cid:12)propagatealongauniformspinchainwithoutspread- T(Jz,|h)|= t|2|isacrucialfactorinthefollowingschemes
| |
forquantuminformationprocessing. Onthe otherhand,
ing approximately, where the vacuum state is fully
the strength of the local field h also results the spread-
ferromagnetic state Vac = . Here Ω =
N1 exp(−α2(j − Nc)|2/2)i is thQe ln=o1r|m↓iallization factor, Wingeopfertfhoermspinnumflieprifcraolmsismtautlaet|i1oindfaonrdthreedsucacettseTri(nJgzp,hro)-.
N is the center of the wave packet at t = 0 and N
Pc cessinorderto investigatesuchphenomenon.Numerical
is the number of sites of the chain. At time t, it will
result for T(Jz,h) with α = 4/15 is plotted in Fig. 2.
evolvesto φ(π/2,N +J⊥t) . Letus firstly assumethat
c It shows that the transmission coefficient is close to 1 if
initiallythequbitisinthestate 0 ,whileawavepacket
(cid:12) (cid:11) | id h Jz 5 J⊥ , which is feasible in practice. It ensures
of type (7(cid:12)) |φ(π/2,Nc ≺d)i ≡ |φ(π/2,L)i is coming th∼atasp≻innetworkwithanembeddedDDqubitinaspin
from the left. Similarly, we define |φ(±π/2,Nc ≻d+2)i chaincanpe(cid:12)(cid:12)rfor(cid:12)(cid:12)mthetransformation(8,9)viaanatural
φ( π/2,R) , φ( π/2,N d) φ( π/2,L) to
≡ | ± i | − c ≺ i ≡ | − i dynamic process rather than an adiabatic process.
denote a transmitted or reflected wavepacketafter scat-
tering. In the strong local field regime, h J⊥ , the
≫
spin flip is firmly confined in the DD qubit. From the
IV. GENERATION OF GHZ STATE
(cid:12) (cid:12)
analysis of the above 4-site system, which is c(cid:12)alle(cid:12)d the
resonant case with h = Jz, the wave packet will pass
Now we focus on the practical application of resonant
freely through the DD qubit. Comparing to the case
scatteringeffect onthe quantuminformationprocessing.
withouttheembeddedDDqubit,theoutputwavepacket
As mentioned above, although the totally transmitted
gets a forward shift with a lattice space, while switches
wavepacketswitchesthe stateofthe DDqubitfrom 0
the DD qubit from |0id to |1id, i.e., to 1 , there is no entanglement between the DD |anidd
| id
the wave packetarisingfrom sucha process. However,if
π π
φ( ,L) 0 φ( ,R) 1 . (8) the incoming wavepacketis not polarizedin z direction,
2 | id −→ 2 | id but in the form
(cid:12) E (cid:12) E
In contrast(cid:12), if the qubit is in(cid:12) state 1 , the scattering
process is (cid:12) (cid:12) | i π
=α φ( ,L) +β Vac , (12)
|րiin 2 | i
(cid:12) E
π π where α and β are rest(cid:12)ricted to be real for simplicity in
φ( ,L) 1 φ( ,L) 1 , (9) (cid:12)
2 | id −→ −2 | id the followingcontext,the entanglementbetweenthe DD
(cid:12) E (cid:12) E qubit and the scattered wave packet can be established.
i.e., the in(cid:12)coming wave pack(cid:12)et is totally reflected and
(cid:12) (cid:12) Actually, the corresponding resonant scattering process
maintains the qubit to be in state 1 . Interestingly,
| id can be expressed as
the states of wave packet and the DD qubit are both
altered through this process, i.e., being shifted with a
π
lattice space. However, such a shift brings about totally 0 α 1 φ( ,R) +β 0 Vac . (13)
|րiin| id −→ | id 2 | id| i
different effects on the DD qubit and the wave packet,
(cid:12) E
respectively: it switches the DD qubit from 0 to 1 , Thereduceddensitymat(cid:12)rixofthefinalstateinthebasis
butdoesnotalterthewavepacketinthesame| midanne|riads {|1id|Vaci, |1id|φ(π/2,R(cid:12) )i, |0id|Vaci, |0id|φ(π/2,R)i}
classical perfectelasticcollision.Itisworthytopointout is
thattheDDqubitandthewavepacketarenotentangled 0
in such resonant case. In the case of non-resonance h = α2 αβ
Jz, or initially the DD qubit and/or the incident wav6 e  αβ β2 , (14)
packet are in a superposition states as 0
 
 
4
(cid:13) videdbybeamsplitter(α′, β′),andcontributetotheout-
put wave packet ψout = φ(π/2, ) . Then the whole
| i | ∞ i
process can be expressed as
(cid:14)(cid:15)(cid:22)
(cid:13) (cid:14)(cid:15)(cid:21) ψin 0 (15)
| id
(cid:14)(cid:15)(cid:12) (cid:12)step1(cid:11)α φ(π,L) 0 +β φ(π) 0
(cid:13)(cid:14)(cid:12) (cid:14)(cid:15)(cid:20) s(cid:12)−te→p2α(cid:12)(cid:12)1 2φ(πE,AR|) id +β(cid:12)(cid:12)0 2φE(Bπ|) id
(cid:13)(cid:12) −→ |(cid:12) id 2 A (cid:12)| id 2 B
(cid:13)(cid:14) (cid:13)(cid:14) (cid:14)(cid:15)(cid:19) step3(α′α 1(cid:12)(cid:12) +β′βE0 ) ψout (cid:12)(cid:12) E
−→ |(cid:12)id | id (cid:12)
(cid:18) (cid:12) (cid:12) (cid:16)(cid:17) + β′2α 1 α′β′β 0 (cid:12)φ( π(cid:11),R)
(cid:14) | id− | id (cid:12) −2 A
(cid:24)(cid:25)(cid:26) (cid:23) +(cid:0)α′2β 0 α′β′α 1 (cid:1)(cid:12)(cid:12)φ( π) E.
(cid:29) | id− | id (cid:12) −2 B
(cid:12) E
(cid:0) (cid:1)(cid:12)
From step 2 to step 3 we have us(cid:12)ed formulas (28, 29)
(cid:27)(cid:30)"
derived in Appendix A. When ψout is measured in the
| i
output lead, the operation
(cid:27)(cid:30)!
% 0 α′α 1 +β′β 0 (16)
(cid:27)(cid:30) | id −→ | id | id
is implemented. The success probability of this op-
(cid:27)(cid:30)(cid:31) eration is (αα′)2 + (ββ′)2. In the optimal case with
α = β = α′ = β′ = 1/√2, we can perform the opera-
(cid:27) tion 0 (1 + 0 )/√2bytheprocessofresonant
&’( (cid:27) (cid:28) (cid:29)(cid:27) (cid:29)(cid:28) | id −→ | id | id
#$ scatteringandsubsequentmeasurementwiththesuccess
probabilityup to 0.5. The measurementof ψout canbe
| i
FIG. 2: (Color online) Profiles of transmission coefficient T implemented by embedding another DD qubit to record
as a function of h and Jz in the unit of J⊥, obtained by the passing of the output wave packet.
numerical simulations. (a) 3-D plot of the function T(h,Jz). Now we consider the case of multiple DD qubits em-
It indicates that the transmission coefficient gets the maxima bedded in A arm, which is schematically illustrated in
undertheresonancescatteringcondition h=Jz. (b)Theplot Fig. 3(b). All the n stationary DD qubits are prepared
of the transmission coefficient under the resonance scattering initially in state 0 ⊗n = n 0 . Similarly, we have
condition. It shows that the transmission coefficient is close | i l=1| il
to 1 in the region h∼Jz ≻˛˛5J⊥˛˛. ψin 0 ⊗n Q (17)
| i
s(cid:12)tep1(cid:11)α φ(π/2,L) 0 ⊗n+β φ(π) 0 ⊗n
twhheicchonhcausrrceonncceurbreetnwceeenCt=hem2|αrβea|.chFeosrtoαt=heβm=ax1im/√um2, s(cid:12)−−te→→p2α||φ(π/2,R)iiAA||1ii⊗n+β(cid:12)(cid:12)(cid:12)φ(2π2)EBB||0ii⊗n
wCamvaex p=ac1k.etA(7fl)yviniag aqu1bi×t (21-2b)eacmanobreYg-ebneearmatesdplfirtotemr s−te→p3 α′α|1i⊗n+β′β|0i⊗n (cid:12)(cid:12)(cid:12)ψout E
[t2w7o].cWoneneccotnesdid1er th2e-bsepaimn nseptlwitoterkrs.wiTthhetsheetgweoomYe-tbreyamof + β′(cid:16)2α 1 ⊗n α′β′β 0 ⊗n(cid:17)(cid:12)(cid:12)φ( π(cid:11),R)
× | i − | i −2 A
splitters are characterized by (α, β) and (α′, β′) respec- (cid:16) (cid:17)(cid:12) π E
+ α′2β 0 ⊗n α′β′α 1 ⊗n (cid:12)φ( ) ,
tively,whichareschematicallyshowninFig. 3(a). There | i − | i (cid:12) −2 B
is a DD qubit embedded in one of the two arms. Now (cid:16) (cid:17)(cid:12) E
(cid:12)
we consider the dynamic process with the initial state with the notation 1 ⊗n = n 1 .(cid:12)
baneiningcoψmining|0wida,vewhpearceketψianlon=g t|φhe(πl/e2ft,−ch∞a)ini,. dIennotthees In the optimal c|asie withQαl==1β| i=l α′ =β′ =1/√2, we
can perform the operation
first st(cid:12)(cid:12)ep, t(cid:11)hrough the b(cid:12)(cid:12)eam(cid:11)splitter (α, β), ψin is di-
vided into two wave packets φ(π/2,L) and φ(π/2) 1
| iA (cid:12)| (cid:11) iB 0 ⊗n 1 ⊗n+ 0 ⊗n (18)
along A and B arms, respectively. In the se(cid:12)cond step, | i −→ √2 | i | i
sub-wave packet φ(π/2,L) passes over DD qubit and (cid:16) (cid:17)
| iA
becomes φ(π/2,R) , while sub-wave packet φ(π/2) by the subsequent measurement with the success proba-
| iA | iB
propagates along B arm and meets φ(π/2,R) at the bility up to 0.5. Then by using natural dynamics and
joint of beam splitter (α′, β′). In th|e third stieAp, wave subsequent measurement, multipartite entangled GHZ
packets φ(π/2,R) and φ(π/2) are reflectedanddi- state can be generated. This provides a simple way of
| iA | iB
5
entangling n stationaryqubits throughscattering with a DDqubitembeddedintheltharm,andswitchesitsstate
flying qubit. In the near-resonance scattering case, the from 0 to 1 simultaneously. Inthe thirdstep, allthe
| il | il
transmission probability amplitude t will effect the suc- sub-wave packets φ(π/2,R) are reflected and divided
| il
cess probability to be at the node of the right beam splitter, and contribute to
theoutputwavepacket ψout = φ(π/2, ) . Thewhole
2 | i | ∞ i
process can be expressed as
P = , (19)
GHZ 1+t−n 2
| | ψin 0 ⊗n (22)
| i
undertheoptimalconditionsα=α′ = 1/(1+tn),β = n
s(cid:12)tep1(cid:11) 1 π n
β′ = tn/(1+tn). Note that the succpess probability is (cid:12)−→ √n φ(2,L) l l=1|0il
reducedexponentially as the number ofqubits increases. Xl=1(cid:12) E Y
p n (cid:12)
step2 1 (cid:12) π n
φ( ,R) 1 0
−→ √n 2 l| il i6=l| ii
V. GENERATION OF W STATE Xl=1(cid:12) E Y
n (cid:12)
step3 1 1(cid:12) n 0 ψout
Now we turn to the scheme of the generation of an- −→ n | il i6=l| ii
other type of multipartite entangledstate, W state. The Xl=1 Y (cid:12) (cid:11)
configuration of the spin network we utilized consists of 1 n n ψout 1 (cid:12) n 0
two1 n-beamsplitterswithoneDDqubitembeddedin −n√n j | il i6=l| ii
× l=1j=1
each parallel arm in the same way, as illustrated in Fig. XX(cid:12) (cid:11) Y
n (cid:12)
3(c). + ψout 1 n 0
We start our analysis by considering n = 2 case with l | il i6=l| ii
l=1
two 1 2-beam splitters being characterized by (α, β) X(cid:12) (cid:11) Y
and (α×′, β′) respectively. Denoting the qubit states of From step 2 to(cid:12)step 3 we have used the formula (35)
|t0w,o1iDBDreqsupbeicttsiveemlyb,etdhdeeddyinnaamrmicspAroacnedssBcaans|b0e,1wiAritatnend douertipvuetdlienadA,ptpheenodpixeraBt.ioWnhen |ψouti is measured in the
as
n
1 n
0 ⊗n 1 0 (23)
ψin 0 0 (20) | i −→ √n l=1| il i6=l| ii
| iA| iB X Y
step1 π π
(cid:12) (cid:11)α φ( ,L) 0 0 +β φ( ,L) 0 0 is implemented with the success probability 1/n. Then
(cid:12)−→ 2 A| iA| iB 2 B| iA| iB byusingnaturaldynamicsandsubsequentmeasurement,
step2α(cid:12)(cid:12)1 φ(πE,R) 0 +β(cid:12)(cid:12)1 φ(πE,R) 0 multipartite entangled W state can be generated. In the
−→ |(cid:12) iA 2 A| iB (cid:12)| iB 2 B| iA near-resonance scattering case, the transmission proba-
(cid:12) E (cid:12) E
step3(αα′ 1(cid:12) 0 +ββ′ 1 0 ) ψ(cid:12)out bility amplitude t reduces the success probability to
−→ | (cid:12)iA| iB | iB| iA (cid:12)
π
+ αβ′2|1iA|0iB−βα′β′|1iB|0iA (cid:12)(cid:12)φ(−(cid:11)2,R) A P = |t|2. (24)
+(cid:0)βα′2 1 0 αα′β′ 1 0 (cid:1)(cid:12)(cid:12)φ( π,R)E . W n
| iB| iA− | iA| iB (cid:12) −2 B
(cid:12) E
(cid:0) (cid:1)(cid:12) Asthecomparisonofthesuccessprobabilitiesofcreat-
In the optimal case with α = β = α′ (cid:12)= β′ = 1/√2, we
ing GHZ andW states ofn qubits with the transmission
can perform the operation
coefficient T, we plot Eqs. (19) and (24) in Fig. 4 for
1 the cases with T = 1.0, 0.9, 0.8, and 0.7; n = 2, 3, ...,
0 0 (1 0 + 0 1 ) (21) 8. It shows that when T is close to 1, the difference be-
| iA| iB −→ √2 | iA| iB | iA| iB
tweenP andP becomes largeas n increases.As T
GHZ W
by the subsequent measurement of the output spin flip decreases from 1, the difference between PGHZ and PW
with the success probability up to 0.5. becomes smaller for fixed n.
Now we extend the above conclusion to n-DD qubits
case. Fora1 n-beamsplitter,weonlyconsiderthesim-
plestcasewit×hidenticalhoppingconstantbeingJ⊥/2√n VI. SUMMARY
between the input lead and each arm, which is schemat-
ically illustrated in Fig. 3(c). An incident wave packet We have shown how a spin network can be used
will experience the following process. In the first step, to generate multipartite entanglement among stationary
ψin is divided into n wave packets through the beam qubits. The key of this scheme is the alternative of mas-
splitter, with φ(π/2,L) (l=1,2,...,n) being the wave siveorstationaryqubitinaspinnetwork,DDqubit. The
(cid:12) (cid:11) | il
p(cid:12)acket along the lth arm. In the second step, every sub- resonance scattering between a DD qubit and a SFWP,
wave packet φ(π/2,L) passes over the corresponding which acts as the alternative of a flying qubit in a spin
| il
6
y )* a , a + y ./0 qubit, is constructed by the element, two neighbor spins
of the spin network, which is applicable to all types of
123 b - b + the scalable multi-qubit systems. Secondly, it is based
on a natural dynamic process rather than an adiabatic
process. There is certainly significant potential for spin
networkstofindapplicationsinsolidstatequantumpro-
y )* a ,, a + y ./0 cessing and communication.
1 We acknowledge the support of the CNSF (grant No.
4
10874091,2006CB921205).
153 b -- b +
6 VII. APPENDIX DYNAMICS OF WAVE
PACKETS IN BEAM SPLITTERS
6
y )* y ./0
In this appendix, we present the exact results for the
7 7 dynamics of wave packets in beam splitters.
1:3
8 8
A. Appendix A: 1 2 beam splitter
9 ×
FIG. 3: Schematic illustrations for the scheme of entangle- Inthe workofRef. [27]the dynamicsofawavepacket
mentgenerationinspinnetwork. (a)Realizationofoperation inthespinnetworksbasedontheXY modelwasstudied.
0 (1 + 0 /√2. (b) Generation of the GHZ state. (c) A 1 2-beam splitter is consisted of three uniform spin
G| ein−e→ratio|niof|thie W state. chain×s with coupling constant J⊥/2. The connections
between three uniform chains are αJ⊥/2 and βJ⊥/2 as
0.7 in Fig. 5. It has been shown that under the condition
T=1.0 α2+β2 =1, an input moving wave packet ψin will be
T=0.9
0.6 TT==00..87 dividedinto twowavepackets|ψαoutiand ψ(cid:12)(cid:12)βout (cid:11)without
0.5 any reflection, which can be expressed as(cid:12) E
(cid:12)
(cid:12)
0.4 ψin α ψout +β ψout . (25)
−→ α β
0.3 (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
Similarly, the(cid:12)inverse proc(cid:12)ess also ho(cid:12)lds, i.e.,
0.2
α ψin +β ψin ψout , (26)
α β −→
0.1
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
0.0 where states ψi(cid:12)n , ψαin ,(cid:12)and ψβin(cid:12) (|ψouti, |ψαouti, and
2 3 4 5 6 7 8 ψβout )repre(cid:12)(cid:12)sent(cid:11)th(cid:12)(cid:12)ewa(cid:11)vepack(cid:12)(cid:12)(cid:12)etscEomingin(out)ofthe
FIG. 4: (Color online) Success probabilities for the genera- (cid:12)nodeEalong the three chains respectively. Contrarily, for
(cid:12)
tion of GHZ and W states of n qubits in the system with (cid:12)two wavepackets alongthe twoarms, whichinterference
transimission coefficient T. destructively at the node, we have
β ψin α ψin β ψout α ψout . (27)
α − β −→ α − β
network, allows a perfect transformation: an incident
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
wave packet can totally pass through a DD qubit and Combin(cid:12)ing the ab(cid:12)ove Eqs. (2(cid:12)6 and 27),(cid:12)we have
switch it from state 0 to state 1 . The resonance scat-
tering condition is in|veistigated a|nialytically and numeri- ψin α ψout +β2 ψout αβ ψout , (28)
α −→ α − β
cally. Itshowsthattheresonancescatteringisfeasiblein
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
practice. This ensures that through the natural dynam- and (cid:12) (cid:12) (cid:12) (cid:12)
ical evolution of an incident single-spin-flip wave packet
in a spin network and the subsequent measurement of ψin β ψout αβ ψout +α2 ψout . (29)
β −→ − α β
the output single-spin-flip wave packet, multipartite en-
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
tangled states among n stationary qubits, GHZ and W Then(cid:12)for a given(cid:12)incident wav(cid:12)e packet al(cid:12)ong any branch
states can be generated. There are two merits in our ofa1 2beamsplitter, theprobabilityamplitudesofall
×
scheme. Firstly, the massive or stationary qubit, DD the output wave packets can be obtained exactly.
7
y TNOP y UNOP y QN-OVP y QNOP
y CDE y WXY
y a=>? y b@AB 1
n
a b
a b 1
y LM y IJK n y ][\
a b
y ;< y RS
y aFGH y a^_‘ y b^_‘ y ZW-XYc y ZWXY
FIG.5: (Coloronline)Schematicillustrationfora 1 2beam FIG. 6: (Color online) Schematic illustration for a 1 n
splitter. Leftpanel: anincidentwavepacket ψin is×splitinto beam splitter. Left panel: an incident wave packet ψin×, is
Rttawwnioogdhsatuψrpbmoa-uwsntaeis.lv:edaipvnaicdikenedctisidne|tψnoαotuthwtiraeaveensdpuab|ψc-wkβoeuattvie|ψwpαiiantchiko,e|uattsloa|nnψigyαouortenifle,e|coψtfiβootuhntei. osr(ejpnflle=ietcoi1tfni,tot.ho.n.e,.nnnR)sauiagrbnhm-dtwsapiaψvsenodeuipltva:ic.dakenedtsiinn|tcψoijdonuetn+it1(wjsau=vbe-1w,pa.a.v.c,eknep)tawc|kψietliht|nsoiu|ψatiljoaounntygi
| i | i
B. Appendix B: 1 n beam splitter
×
n−1
= + , (32)
Now we consider a 1 n beam splitter consists of one c l
H H H
×
chain of length la and n arms of length lb with identi- Xl=1
cal connecting coupling strength J⊥/2√n for each arm, J⊥ la+lb−1
whichisshowninFig. 6. TheHamiltonianofsuchquan- Hc = − 2 c†ici+1+H.c. ,
tum network reads Xi=1 (cid:16) (cid:17)
J⊥ lb−1
= d† d +H.c. ,
Hl − 2 l,i l,i+1
J⊥ la−1 J⊥ n Xi=1 (cid:16) (cid:17)
= a†a a† b (30) (l =1,2,..,n 1).
H − 2 i i+1− 2√n la l,1 −
i=1 l=1
X X Note that the sub-Hamiltonians satisfy
J⊥ n lb−1
b† b +H.c.,
− 2 l,j l,j+1
l=1 j=1
XX [ , ]=0,[ , ]=0, (33)
c l l m
H H H H
where a† and b† are particle operators at site i of chain
i l,j which indicate that the original quantum network, 1
l and site j of the arml. They can be boson or fermion ×
a n beam splitter can be decomposed into n independent
operators. The conclusion for such model is available
chains, one of them is the length of l +l and the rest
a b
for the dynamics of a single flip in the analogue of spin
are all the length of l . Based on the fact that a moving
b
network.
wavepacketcanpropagatefreelyalongthenindependent
Similarly as shown in Ref. [27], we can perform the chains, we have the following processes
following transformations
n
c†i = a†i,(in≤la), (31) ψin −→ √1n ψjout , (34)
c† = 1 b† ,(i l ), (cid:12) (cid:11) Xj=1(cid:12) (cid:11)
i+la √n j,i ≤ b 1 n (cid:12) (cid:12)
Xj=n1 √n ψjin −→ ψout ,
d† = 1 e−i2nπljb† ,(l=1,2,...,n 1), Xj=1(cid:12) (cid:11) (cid:12) (cid:11)
l,i √n j,i − n (cid:12) (cid:12)n
Xj=1 e−i2πnmj ψjin −→ e−i2πnmj ψjout ,
j=1 j=1
X (cid:12) (cid:11) X (cid:12) (cid:11)
wherec† andd† arealsocorrespondingbosonorfermion (cid:12) (m=1,2,..,(cid:12)n 1).
i l,j
−
operators. Under the transformations, Hamiltonian (30)
can be written as Through a straightforward algebra, we obtain a set of
8
expressions Then for a given incident wave packet along any branch
ofa1 nbeamsplitter,theprobabilityamplitudesofall
n ×
1 1 the output wave packets can be obtained exactly.
ψin ψout ψout + ψout ,(35)
l −→ √n − n j l
j=1
(cid:12) (cid:11) (cid:12) (cid:11) X(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (l=(cid:12)1,2,..,n). (cid:12) (cid:12)
[ ] emails: [email protected] 024302 (2003).
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