The effect of spin-orbit interaction on entanglement of two-qubit Heisenberg XYZ systems in an inhomogeneous magnetic field

0.43 MB·English

by  Fardin Kheirandish

The effect of spin-orbit interaction on entanglement of two-qubit Heisenberg XYZ
systems in an inhomogeneous magnetic field
Fardin Kheirandish , S. Javad Akhtarshenas and Hamidreza Mohammadi
∗ † ‡
Department of Physics, University of Isfahan, Hezar Jarib Ave., Isfahan, Iran
The role of spin-orbit interaction on the ground state and thermal entanglement of a Heisenberg
XYZ two-qubit system in the presence of an inhomogeneous magnetic field is investigated. For a
certain value of spin-orbit parameter D, the ground state entanglement tends to vanish suddenly
and when D crosses its critical value Dc, the entanglement undergoes a revival. The maximum
valueof theentanglement occurs in therevival region. In thefinitetemperatures, thereare revival
regions in D−T plane where increasing of temperature first increases the entanglement and then
tends to decrease it and ultimately vanishes for temperatures above a critical temperature. This
8
critical temperature is an increasing function of D, thus the nonzero entanglement can exist for
0
larger temperatures. In addition, the amount of entanglement in the revivalregion dependson the
0
spin-orbit parameter. Also, entanglement teleportation via the quantum channel constructing by
2
above system is investigated and influence of spin-orbit interaction on the fidelity of teleportation

n and entanglement of replica state is studied.
a
J
PACSnumbers: 03.67.Hk,03.65.Ud,75.10.Jm

2
1
I. INTRODUCTION where h(x) = xlog x (1 x)log (1 x) is the bi-
] − 2 − − 2 − 4
h naryentropyfunctionandC(ρ)=max 0,2λ λ
max i
p { − }
i=1
A. Einstein, B. Podolsky and N. Rosen, in their fa-
t- mousEPRparadox,arguedthatingeneraltwoquantum is the concurrence of the state, where λis are poPsitive
n squarerootsoftheeigenvaluesofthenon-Hermitianma-
a system cannot be separated even if they are located far trixR=ρρ˜,andρ˜isdefinedbyρ˜:=(σy σy)ρ (σy σy).
∗
u fromeachother[1]. E.Schr¨odingernamedthis quantum ⊗ ⊗
ThefunctionΞisamonotonicallyincreasingfunctionand
q mechanical property as Entanglement [2]. Today, entan-
rangesfrom0 to1asC goesfrom0 to1,sothatone can
[ glement is a uniquely quantum mechanicalresourcethat
takethe concurrenceasa measureofentanglementinits
1 plays a key role in many of the most interesting applica- own right. In the case that the state of the system is
v tionsofquantumcomputationandquantuminformation pure i.e. ρ = ψ ψ , ψ = a00 +b01 +c10 +d11 ,
7 [3, 4]. Thus a great deal of efforts have been devoted | ih | | i | i | i | i | i
the above formula is simplified to C(ψ )=2 ad bc .
9 to study andcharacterizethe entanglementinthe recent | i | − |
The spin chain is the natural candidates for the real-
8
years . The central task of quantum information theory
ization of entanglement and Heisenberg model is the
1
is to characterize and quantify entanglement of a given
. simplest method for studying and investigating the be-
1 system. A mixed state ρ of a bipartite system is said
havior of the spin chains . Nielsen [8] is the first per-
0 to be separable or classically correlated if it can be ex-
son who studied the entanglement of two-qubit Heisen-
8
pressed as a convex combination of uncorrelated states
0 berg XXX- chain modeled with the Hamiltonian H =
v: ρA and ρB of each subsystems i.e. ρ = i ωiρiA ⊗ρiB Jσ1·σ2+B·(σ1z +σ2z). He showed that entanglement
insuchsystemsexistsonlyforantiferromagnatic(J >0)
i such that ωi 0 and ωi =1, otherwise ρPis entangled
X ≥ i case below a threshold temperature Tc. After Nielsen,
r [4, 5]. Many measuresPof entanglement have been intro- entanglement in the two-qubit XXX, XXZ and XY sys-
a duced and analyzed [3, 6, 7], but the one most relevant temsinthepresenceofhomogeneousandinhomogeneous
to this work is entanglement of formation, which is in- magneticfieldhasbeeninvestigated[9,10,11,12,13,14].
tended to quantify the resources need to create a given The effect of anisotropy due to spin coupling in the
entangled state [6]. For the case of a two-qubit system x,y,z direction has also studied in a number of works
Wootters has shown that entanglement of formation can [15, 16]. In ref [17] Yang et al. have shown that in XYZ
be obtained explicitly as: Heisenberg systems, an inhomogeneous external mag-
netic field can make the larger revival, improve the crit-
ical temperature and enhance the entanglement. Spin-
1+√1+C2 orbit (SO) interaction cause another type of anisotropy
E(ρ)=Ξ[C(ρ)]=h , (1)
[18, 19, 20, 21, 22, 23, 24]. The effect of SO interaction
2
!
inthe thermalentanglementofatwo-qubitXXXsystem
in the absence of magnetic field has been studied in [25].
However,theentanglementforaXYZHeisenbergsystem
under an inhomogeneous magnetic field in the presence
∗fardin−[email protected]
†[email protected] of SO interaction has not been discussed. Therefore, in
‡[email protected] this paper we investigate the influence of SO interaction
2
on the entanglement and entanglement teleportation of maximallyentangledcomponentoftheresource. Thisen-
two-qubit system at thermal equilibrium. ables the usage of any quantum channel as a generalized
On the other hand, among the numerous concepts to depolarizing channel without additional twirling opera-
implement a quantum bit (qubit), approaches based on tion” [41]. Using the property of linearity of teleporta-
semiconductor Quantum Dots (QDs) offer the great ad- tion process [38], J. Lee and M.S. Kim have shown that
vantage that ultimately a miniaturized version of quan- ”quantumteleportationpreservesthenatureofquantum
tum computer is feasible. Indeed, at first D. Loss and correlation in the unknown entangled state if the chan-
D.P. Diviencenzo proposed a quantum computer proto- nel is quantum mechanically correlated” and then they
colbasedonelectronspintrappedinsemiconductorQDs investigate entanglement teleportation via two copies of
in 1998 [26, 27, 28]. Here, the qubit is represented by a werner states [42]. The entanglement teleportation via
single electron in a QD which can initialize, manipulate thermallyentangledstateofatwo-qubitHeisenbergXX-
and read out by extremely sensitive devices. Compared chain and XY-chain has been studied by Ye Yeo et al.
withother systemssuchasquantumopticalsystems[29] [43, 44]. The effect of spin-orbit interaction on entangle-
andnuclearmagneticresonance(NMR)[30,31,32],QDs mentteleportationonatwo-qubitXXX-Heisenbergchain
arearguedto be morescalableandhaslong decoherence in the absence of magnetic field is reported by G. Zhang
time [3]. The above Heisenberg system is suitable for [45].
modelingandcomputingtheentanglementofatwo-qubit Inthispaperweinvestigatetheabilityoftheabovemen-
system representedby two electrons confined in two ver- tioned two-qubit system for the entanglement teleporta-
tically coupled quantum dots (CQDs), respectively. Due tion. Weshowthatspin-orbitinteractionandinhomoge-
to weak lateral confinement electrons can tunnel from neousmagneticfieldhaveeffectiveeffectontheentangle-
one dot to another dot and spin-spin and spin-orbit in- ment of replica state and the fidelity of teleportation. A
teraction between the two qubits exists. In GaAs dou- minimal entanglement of the thermal state in the model
ble QDs, hyperfine interactions have been identified to is requiredto realize efficiententanglementteleportation
dominate spin mixing at small magnetic fields, while SO and we can attain to this minimal entanglement, in the
interactionisnotrelevantinthisregime. HoweverSOin- case of J <0, by introducing SO interaction.
z
teraction and the coupling magnetic fields are expected Thepaperisorganizedasfollows. Insec. 2weintroduce
to be orders of magnitude stronger in InAs compared to the HamiltonianofatwoqubitHeisenbergsystemunder
GaAs [33]. SO interactionin suchnanostructurescanbe inhomogeneous magnetic field with SO interaction and
investigated with the help of quantum optical methods write the thermal density matrix of the system related
[34]. tothisHamiltonianandultimatelycalculatethethermal
Taking the advantage of tunability of SO strength [34, concurrenceofthe system. Insubsection2.1,the ground
35, 36, 37], we show that this type of interaction cause stateentanglementiscalculatedandresultsareplottedin
to enhancement of entanglement in the revival region, figs. 1-2c. Insubsection2.2thefinitetemperatureentan-
increase the volume of revival region and improve the glement of system is computed. The figs. 3-5 illustrate
critical values of other parameters. the obtained results. The entanglement teleportation of
In the following, as an application of the above system, a two-qubit pure state and its fidelity derived in sec. 3
the entanglementteleportationof a two-qubitpure state andresults areplotted infigs. 6-8. Insec. 4 adiscussion
via the above two-qubit system is investigated and av- concludes the paper.
erage fidelity between input and output states is calcu-
lated. C. Bennet et al. have shown that two entangled
spatially separated particles can be used for teleporta- II. MODEL AND HAMILTONIAN
tion [38]. They also argued that states which are less
entangled still could be used for teleportation but they TheHamiltonianofatwo-qubitanisotropicHeisenberg
reduce ”the fidelity of teleportation and/or the range of XYZ-model in the presence of inhomogeneous magnetic
state φ whichcanaccuratelybeteleported”. Afterthen field and spin-orbit interaction is:
| i
S. Popscu by using hidden variable model, have shown
that, teleportation of a quantum state via a pure classi- H = 1(J σxσx +J σyσy+J σzσz +B σ +B σ
cal communicationcannot performedwith fidelity larger 2 x 1 2 y 1 2 z 1 2 1· 1 2· 2
than 32 [39]. Thus, mixed quantum channels which al- + D·(σ1×σ2)+δ σ1·Γ·σ2), (2)
lows to transfer the quantum information with the fi- where σ = (σx,σy,σz) is the vector of Pauli ma-
delity larger than 32 are worthwhile. Horodecki et al. trices, B (jj = 1j,2)jis jthe magnetic field on site j,
have calculated the optimal fidelity of teleportation for j
J (µ = x,y,z) are the real coupling coefficients (the
bipartite state acting on Cd Cd by using the isomor- µ
phism between quantum cha⊗nnels and a class of bipar- cmhaaginneitsica(nFtiM-fe)rfroormJag<net0i)ca(nAdFDMi)sfcoarllJedµD>z0yaalonsdhfinersrkoi--
tite states and twirling operations [40]. G. Bowen and µ
Moriya vector, which is first order in spin-orbit coupling
S.Bosehaveshownthat”standardteleportationwithan
and is proportional to the coupling coefficients (J ) and
arbitrary mixed state resource is equivalent to general µ
Γ is symmetric tensor which is second order in spin-
depolarizing channel with the probabilities given by the
orbit coupling [18, 19, 20, 21]. For simplicity we assume
3
B =B zˆsuchthatB =B+bandB =B b,whereb In the above equations N = 1 and M =
j j 1 2 ± ±
indicatesthe amountofinhomogeneityofma−gneticfield. 1+ (b±ξ)2
The vector D and the parameter δ are dimensionless, in 1 are the normalizaqtion Jc2o+n(sJztaDn)2ts. Here we
system like coupled GaAs quantum dots |D| is of order 1+(B±η)2
Jγ
ofafew percent,while the orderoflasttermis 10−4 and dpefine, ξ := b2+J2+(JzD)2 and η := B2+(Jγ)2,
is negligible. If D = JzDzˆ and ignore the second order for later convenience.
p p
spin-orbit coupling, then the above Hamiltonian can be The state (density matrix) of a system in equilibrium at
expressed as: temperature T is ρ = Z 1exp( H ), where Z is the
− −kBT
partition function of the system and k is the Boltzman
B
constant. Forsimplicitywetakek =1. Inthestandard
B
H = Jγ(σ1+σ2++σ1−σ2−)+(J +iJzD)σ1+σ2−+(J −iJzD)σb1−aσs2+is, the density matrix of the system in the thermal
J B+b B b equilibrium can be written as:
+ zσzσz +( )σz +( − )σz, (3)
2 1 2 2 1 2 2
where J := Jx+Jy, is the mean coupling coefficient
2 µ 0 0 ν
in the XY-plane, γ := JJxx+−JJyy, specifies the amount ρ = 0+ w1 z 0 , (8)
of anisotropy in the XY-plane (partial anisotropy, T  0 z w 0 
∗ 2
−ra1isi≤ngγop≤era1t)orasn.d σ± = 12(σx ±σy) are lowering and  ν 0 0 µ− 
The Hamiltonian (3), in the standard basis
00 , 01 , 10 , 11 ,has the following matrix form:
{| i | i | i | i} where
1J +B 0 0 Jγ
2 z
H = J00γ −J −12J0izJ+zDb J−+21J0izJ−zDb 12Jz00−B . (4) µ± = e−Zβ2Jz (coshβη∓ Bη sinhβη),
  βJz
The spectum of H is easily obtained as w = e 2 (coshβξ bsinhβξ),
1,2 Z ∓ ξ
(9)
H|ψ±i=ε1,2|ψ±i, (5) ν = Jγe−β2Jz sinhβη,
− Zη
H Σ =ε Σ ,
± 3,4 ±
| i | i βJz
where the eigenstates and the corresponding eigenval- z = (J+iJzD)e 2 sinhβξ,
− Zξ
ues are, respectively
|ψ±i=N±(−(J−bi±JξzD)|01i+|10i), (6) and Z =2eβ2Jz(coshβξ+e−βJzcoshβη) .
|Σ±i=M±(−(BJ±γη)|00i+|11i), Ienntwanhgaltefmolelnotwso,fotuhrepaubropvoesetwisot-oquqbuiatnstyifsytetmhevaemrsouusntthoef
parameters of the system, with the main concerning on
and
D . For density matrix in the form (8), one can show,
ε = 1J ξ, bystraightforwardcalculations,thatthe squarerootsof
1,2 −2 z ±
(7) the eigenvalues of matrix R=ρρ˜are:
ε = 1J η.
3,4 2 z ±
λ = √w w z
1,2 1 2
| ±| ||
βJz
e 2
= ξ2+J2+(J D)2sinh2βξ J2+(J D)2sinhβξ ,
z z
ξZ ±
(cid:12)q (cid:12)
(cid:12) p (cid:12) (10)
(cid:12) (cid:12)
(cid:12) (cid:12)
λ3,4 = √µ+µ ν
| −∓ |
βJz
= e− 2 η2+(Jγ)2sinh2βη Jγsinhβη .
ηZ ∓
(cid:12)q (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
C loss of generality we can assume J > 0 and γ > 0, since
0.7 the above formula are invariant under substitution J
→
0.6 J and γ γ. For the special case D = 0, these
− → −
0.5 equations give the same results as ref [17]. The obtained
0.4 results are given in the following subsections.
0.3
0.2
0.1
A. Ground state entanglement and critical
b
0.5 1 1.5 2 2.5 3 parameters
Thebehaviorofthesystematthequantumphasetran-
FIG. 1: The ground state concurrence vs. b for D =0 (solid
line), D = 0.8 (dashed line) and D = 1 (doted line) where sition (QPT) point [49], such as B = Bc, b = bc or
B=0.8,J =1,Jz =−1 and γ =0.5. D = Dc, may be determined from the density matrix
at T = 0, at which the system is in its ground state. If
we consider T 0 (or β ) then the concurrence C
→ → ∞
Now, it is easy to calculate the concurrence. Without can be written analytically as:
Jγ if ξ <η J
η − z
C(T =0)= (cid:12)(cid:12)(cid:12)21√(cid:12)(cid:12)(cid:12)(cid:12)|JJ(cid:12)(cid:12)(cid:12)2ηγ+ξ(|J−zD√)2J2+ξ(JzD)2(cid:12)(cid:12)(cid:12)(cid:12) ififξξ>=ηη−−JJzz (11)
 (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)
Also, this formula can be derived by calculating en- the concurrencedropsandexhibit arevivalphenomenon
tanglement of ground state of H directly: When ǫ < ǫ whenDcrossesitscriticalvalueD . Intherevivalregion
4 2 c
(or ξ < η J ) the ground state of H is Σ and then (larger D) the concurrence reaches its maximum value
z −
− | i
C = Jγ and when ǫ <ǫ (or ξ >η J ) the ground (C =1). Fig. 2aillustratesthegroundstateconcurrence
| η | 2 4 − z
variationversus D for three values of b, by decreasing b,
state of H is |ψ−i and then C =| √J2+ξ(JzD)2 |. On the the critical value of D (Dc ) increases. Fig. 2b shows
otherhand,atcriticalpoint(whereǫ4 =ǫ2orξ <η Jz), ground state concurrence versus D for three values of γ,
−
the ground state of the system is an equally mixture of byincreasingγ,D increases(amountofγdeterminesthe
c
Σ− and ψ− i.e. GS = (Σ− + ψ− )/2 and then valueofentanglementbeforereachingthe criticalpoint).
| i | i | i | i | i
C = 1 Jγ √J2+(JzD)2 . In fig. 2c the ground state concurrence versus D is plot-
2 η − ξ ted for three values of Jz, when Jz increases, Dc also
vTahlueecson(cid:12)(cid:12)(cid:12)(cid:12)o(cid:12)(cid:12)(cid:12)cfuDrr(cid:12)(cid:12)(cid:12)(eDnce=C0a,s0.a8,fu1n)c(cid:12)(cid:12)(cid:12)(cid:12)taiorenpoflobtt(eadtTin=fig0.)f1o.rWthritehe iHnecnrecaes,ews.e can control value of D|c by| adjusting the pa-
increasing b, the concurrence C is initially constant and rameters of the system such as b,γ and Jz. We demand
equaltoC =| Jηγ |=0.53,thendropssuddenlyatcritical to decrease Dc (because for D >Dc concurrence will be
maximized). Therefore we should choose, J = 1,γ
value of b (bc = (η Jz)2 ((JzD)2+J2)). At this | z |
− − small and b as large as possible.
point (T = 0,b = b ), the concurrence becomes a non-
c
p
analyticalfunctionofb andQPToccurs[17]. Forb>b ,
c
concurrence C undergoes a revival before decreasing to
B. Thermal entanglement
zero. The amount of concurrence at revival region de-
pends on D, by increasing D the revival is greater. Fur-
thermore,by increasingD, bc decreasesi.e. for largerD, Since the relative magnitude of λi(i = 1,2,3,4)
the critical point and hence revival phenomenon occurs depends on the parameters involved, they cannot be
in smaller b. The role of D is more obvious in figs. 2a-2c ordered by magnitude without knowing the values of
whereconcurrenceisplottedintermsofD. Thesefigures the parameters. This prevents one from writing an
show that, when D reaches its critical value defined by analytical expression for concurrence. For particular
parameters, C can be evaluated numerically. The
role of each parameter can be seen by fixing other
D = (η J )2 (b2+J2))/J , (12) parameters and drawing variation of entanglement for
c z z
− − | |
p
5
5
C HaL 0.8
1 4
0.6
0.8 3
C 0.4
T
0.6 0.2 2
0.4 055 1
44
0.2 33D22 11 0 4 3 2 1 0 00 1 2 3 4 5
T D
D
1 2 3 4 5 6
C HbL
FIG. 3: color online) Thermal concurrence vs. T and D.
1
WhereB=4,b=2.5,J =1 and γ =0.3.
0.8
0.6
0.05
0.4 0.1
0.2 =orD0-0.050 Tc1 Tc2 =orD3 0 Tc1 Tc2
f f
1 2 3 4 5 6 D C -0.1 C-0.1
C HcL -0.150 1 2 3 4 5 -0.20 1 2 3 4 5
1 T T
0.8 0.4
0.6
0.3
0.6 =DDc0.2 =D50.4
0.4 for0.1 for0.2
0.2 C 0 Tc2 C 0 Tc2
1 2 3 4 5 6 D 1 2 3 4 5 0 1 2 3 4 5
T T
FIG. 2: The concurrence of the ground state vs. D. a) for FIG. 4: Thermal concurrencevs. T for different valuesof D.
b = 0 (solid line), b = 0.5 (dashed line) and b = 1 (doted Theparametersarethesameasfig. 3. InthiscaseDc ≃4.51.
line),Jz =−1andγ =0.8b)forγ =0.9(solid line),γ =0.5
(dashedline)andγ =0(dotedline),Jz =−1andb=0.75c)
for Jz =−1 (solid line), Jz =0.5 (dashed line) and Jz =0.1
B. The first critical temperature (T ), is the point at
(doted line), b=0.75 and γ =0.7. c1
which λ =λ =λ and hence C =0.
max 4 1
ii) Revival region, for which T < T < T (see below)
c1 c2
and D < D ; in this region λ = λ and hence C
specific values of that parameter. Cross influence of two c max 1 1′
determines the amount of entanglement. According to
parameters can be shown in 3D plots of entanglement.
equation (10), the value of λ is adjustable by changing
1
Thermal entanglement versus system’s parameters is
values of D and ξ (or equivalently D and b) and hence
depicted in figures 3-5. The concurrence as a func-
the parameters D and b play an important role in quan-
tion of T and D is shown in fig. 3. The analysis of
tifying the amount of entanglement, in this region. For
results of this figure are given in the following. Let
T > T , the value of λ and also the rate of enhance-
c1 1
4
Cj′ = 2λj −i=1λi (j = 1,2,3,4), it is evident that the mEnehnatnocfetmheenftunofcttihoenfλu1ncwtiiothnTλ1inwcirtehaTse,scaasuDse Cin1′crreisaesetso.
function Cj P= max{0,Cj′} is the concurrence of the a positive number and thus the entanglement undergoes
system if and only if λj = λmax = max λ1,λ4 (Since a revival. Since the rate of enhancement of the function
{ }
J,γ > 0 we have λ4 > λ3 and λ1 > λ2). We can divide λ1 with T is an increasing function of D, the amount
the regions of fig. 3 in four parts: of revival increases as D increases. When T reaches the
value T = T (second critical temperature), λ tend to
c2 1
i) Region for which T < T (see below) and D < D ; zero again and thus the entanglement vanishes.
c1 c
inthisregionλ =λ andthenC =λ λ λ λ iii) Region for which D >D , for all values of T; in this
max 4 4′ 4− 3− 2− 1 c
determines the amount of entanglement. According to region λ = λ and C is the entanglement indicator.
max 1 1′
equation (10), the values of λ depends on γ and η (or The maximum value of the entanglement occurs in this
4
equivalently γ and B ). Thus in this region we can ma- region. At zero temperature the entanglement has its
nipulate the amountofentanglementby adjusting γ and maximum value and by increasing the temperature the
6
HaL HbL III. THERMAL ENTANGLEMENT
1 TELEPORTATION
0.75
0.2
C C 0.5 For the entanglement teleportation of a whole two-
0.1
0.25
qubit system, a thermal mixed state in Heisenberg spin
066 44b22 0 1.5 1 0.5 0 066 44D22 0 6 4 2 0 cnheal.inNcoawnwbee ccoonnssiiddeerreLdeaesaandgeKniemra’sl dtwepooqlaurbizitintgelcehpaonr--
D B
tation protocol (P ), and use two copies of the above
1
two-qubitthermal state, ρ ρ , as resource[42]. Simi-
T T
⊗
FIG. 5: (color online) (a) Thermal concurrence vs. b and D, lartostandardteleportation,entanglementteleportation
where B=5, (b) Thermal concurrence vs. B and D, where for the mixed channel of an input entangled state is de-
b=2. Here T =0.3,J =1,Jz =0.5 and γ =0.3 stroyedand its replica state appears at the remote place
after applying a local measurement in the form of linear
operators. We consider as input a two-qubit in the spe-
cial pure state ψ = cosθ/210 +eiφsinθ/201 (0
systemloses its entanglementandultimately vanishes at | ini | i | i ≤
θ π,0 φ 2π). The density matrix related to ψ
T =Tc2. In this region no revival phenomenon occurs. ≤ ≤ ≤ | ini
is in the form:
iv) Region for which T T for all values of D; in this
c2
≥
regionallvaluesofCj′ (j =1,2,3,4)havenegativevalues 0 0 0 0
andthenthe entanglementiszeroforallvaluesofD and 0 a c 0
ρ = , (13)
the other parameters. in  0 c b 0 
∗
Notice that, Tc1 and Tc2 are sensitive functions of D. In  0 0 0 0 
fig. 4, we try to demonstrate these facts, by illustrating  
the function of thermal concurrence vs. T for few values where a = sin2θ/2,b = cos2θ/2 and c = 12e−iφsinθ.
corfeDas.ingThfuisncfitgiounreofshDowbsutthfaotr Dfor>DD<, TDc,isTcu1ndisefiandede-. Teihφesrinefθo/re2ccoosnθc/u2rre=ncesinoθf.inTithiael osuttapteutis(reCpilnica=) st2ate|
c c1 |
Incontrast,Tc2 is anincreasingfunction ofD forallval- ρout canbe obtainedby applyingjointmeasurementand
ues of T, i.e. we can create and maintain the nonzero localunitarytransformationoninputstateρin. Thusthe
entanglement at larger temperatures. In summary, fig. out put state is given by [41]
3 shows that: For T < T < T and D < D , there
c1 c2 c
are regions in D T plane where increasing of tempera- ρ = p (σ σ )ρ (σ σ ), (14)
− out µν µ⊗ ν in µ⊗ ν
turefirstincreasestheentanglement(revivalregion)and µ,ν
X
then tends to decrease the entanglement and ultimately
for T > Tc2 entanglement vanishes. The maximum en- where µ,ν = 0,x,y,z (σ0 = I), pµν =
tanglement exists at zero temperature and for large D. tr[Eµρ ]tr[Eνρ ] such that p = 1 and
channel channel µν
In the revival region and region for which D > D , in- ν,µ
c
creasing D causes the entanglement to increase. In this ρchannel represent the state of channelPwhich used for
teleportation. Here E0 = Ψ Ψ , E1 = Φ Φ ,
region, T decreases and T increases as D increases − − − −
c1 c2 E0 = Φ+ Φ+ and E0 =| Ψi+h Ψ|+ where| Ψih =|
and hence the width of the revival region increases. In ±
| ih | | ih | | i
all regions Tc2 is an increasing function of D, thus when (|01i√±2|10i) and |Φ±i= (|00i√±2|11i) are the Bell states.
D is largeenough,the entanglementcanexists forlarger By considering the two-qubit spin system as a quantum
temperatures. Furthermore, the parameter D plays the channel, the state of channel is ρ = ρ given in
channel T
role of parameter b. Fig. 5a, shows the variation of en- the equation (8) and hence one can obtain ρ as
out
tanglementasafunctionofb(inhomogeneityofmagnetic
α 0 0 κ
field) and D. For fixed D, there are three region in this
0 a c 0
figure i) Main region where b < bc; in this region entan- ρout = 0 c′ b′ 0 , (15)
glement is constant ii) Collapse region where b = b ; in ′∗ ′
c
κ 0 0 α
this region entanglement decreases suddenly iii) Revival  
 
region where b > b ; in this region entanglement under-
c where
goes a revival. Amount of D determines b and hence
c
edge of revival region. Furthermore, in the revival re- α=(w1+w2)(µ++µ−),
gionincreasingD increasesthe entanglement. Infig. 5b,
thermal entanglement is plotted vs. magnetic field (B) κ=4Re[z]ν cosφ sinθ,
and D. The role of D is similar to its role in fig. 5a,
enhancement of D improves Bc and increases amount of a′ =(µ++µ−)2cos2 2θ +(w1+w2)2 sin2 θ2, (16)
entanglement in the revival region.
One can use the above entangled two-qubit system for b =(w +w )2 cos2 θ +(µ++µ )2sin2 θ,
′ 1 2 2 − 2
performance the teleportation protocols. The next sec-
tion is spend to this subject. c =2e iφ((Re[z])2+e2iφν2) sinθ.
′ −
7
Followingtheabovewecandetermineconcurrenceofout ρ ρ˜ , i.e. λ s. It is easy to show
out out ′i
put state by calculating positive square roots of R =
out
C2 (cosh2βη e2βJzcosh2βξ)2+4e2βJzcosh2βηcosh2βξ
λ′1,2 = (cid:12)q in −2(coshβη+eβJzcoshβξ)2
(cid:12)
(cid:12)
(cid:12)(cid:12)(cid:12) |Cin((Jηγ)2e2iφsinh2βη+(Jξ)2e2βJzsinh2βξ)| , (17)
± 2(coshβη+eβJzcoshβξ)2 (cid:12)
(cid:12)
λ′3,4 = (cid:12)coshβη cosh(βcξos±hβCηin+(Jηeγβ)J(zJξc)ossihnβhξβ)η2sinhβξcos(cid:12)(cid:12)(cid:12)φ(cid:12)eβJz.
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
Thus Cout = C(ρout) = max{0,2λ′max − λ′i} is HaL HbL
i=1
computable when the parameters of channel arPe known.
The function C is dependent on the entanglement
out
of initial state and the parameters of the channel 0.8 0.2
0.6 0.15
(which determine the entanglement of channel). The Cout0.4 5 Cout0.1 5
0.2 4 0.05 4
pCaoruatmisetnerosnzfoerrowohniclyh Cfocrhapnanretliciuslgarreachteorictehaonf cahcarnintiecla’sl 000 00..2255 23D 000 00..2255 23D
00..55 1 00..55 1
value. The figs. 6 and 8 depict behavior of Cout versus CCiinn 00..7755 CCiinn 00..7755
the parameters of the channel and C . For the case 1 0 1 0
in
J >0,theentanglementofreplicastate(C )increases
z out HcL HdL
linearly as C increases. The rate of this enhancement
in
is determined by D (indeed, by C (D)). But for
channel
the case J < 0, C is zero for small values of D.
z out
As D crosses a threshold value, C increase when 1
C increase with the rate determinedoutby amount of D Cou0t.07.55 2 Cout0.4 2
in 0.25 1.5 0.2 1.5
(equivalently C (D)). The figs. 6b and 6c show 0 0
channel 00 1 b 00 1 b
that the parameter of inhomogeneity (b) can plays role 11 11
22 0.5 22 0.5
of D. DD 33 DD 33
44 5 0 44 5 0
-The Fidelity of entanglement teleportation: Fidelity
between ρ and ρ characterizes the quality of tele-
in out
ported state ρout. When the input state is a pure state, FIG. 6: (color online) Entanglement of output state (Cout)
we can apply the concept of fidelity as a useful indica- vs. the channel’s parameters and Cin. Cout vs. D and Cin
tor of teleportation performance of a quantum channel for a) b = 0.5 and Jz > 0 and b) b = 0.5 and Jz < 0. Also,
[43, 48]. The maximumfidelity ofρ andρ is defined Cout vs. b and D for c) Cin = 1 and Jz > 0 and d)Cin = 1
in out
to be and Jz <0. Where J =1,T =0.1,B =1 and γ =0.3.
1 1
F(ρin,ρout) == {ψTirn[qρo(uρtinψ)i2nρ.out(ρin)2]}2 (18) 2w(hνe2rceosf2cφ=+((Rwe1[z+])2w)2.)T2haenfdunfcqtio=ns21fc−an(wd1f+q awre2)de+-
h | | i
pendent on the channel’s parameters only (we consider
By substituting ρ and ρ from above, we have
in out φ = 0) and hence are relate to the entanglement of the
channel. This formula is the same as the results of ref.
F(ρ ,ρ )=a sin2 θ +b cos2 θ +Re[ce iφ] sinθ(,19) [42][50], but in spite of the werner states, fq can be a
in out ′ 2 ′ 2 ′ − positivenumberforHeisenbergchains. This meansthat,
simplifying the above formula we find that the there exists a channel such that it can teleport more en-
maximum fidelity F(ρ ,ρ ) depends on initial tangled initial state with more fidelity! but it is not a
in out
entanglement(C ): useful claim because, when we choose the parameters of
in
the channel such that fq >0 then fc decreases and ulti-
F(ρ ,ρ )=fc+fq C2 , (20) matelyF(ρ ,ρ )becomessmallerthan 2 whichmeans
in out in in out 3
8
HaL HbL HaL HbL
1 1 1 1
F 00..86 45 F00.0.7.2555 4F5A00.07.2.555 45 Cou0t0.07.2.555 45
0000..2255 23D 00000..2255 23D --011 --00..55 2 3D --011 --00..55 23D
00..55 1 00..55 1 00 1 00 1
CCiinn 00..7755 110 CCiinn 00..7755 110 JJzz 00..55 1 0 JJzz 00..55 1 0
HcL
HdL
FIG.7: (coloronline)F(ρin,ρout)vs. CinandDfora)Jz >0
and b) Jz < 0. Where J = 1,T = 0.1,B = 1,b = 0.5 and
γ =0.3
1 1
F00.0.7.2555 45 Cchanne0l0.0.7.2555 45
tfehraiotrthteoecnlatassnigclaelmceonmtmteulenpicoarttiaotnio.nTohfumsi,xteodostbattaeinistihne- --011--00..55 00 1 23D --011--00..55JJzz00 00..55 123D
same properfidelity, the largerentangledchannelarere- JJzz 00..55 110
10
quired for larger entangled initial state. The fig. 7 em-
phasize the above notes.
TheaveragefidelityF isanotherusefulconceptforchar-
A
acterizing the quality of teleportation. The Average fi- FIG. 8: (color online) a) Average fidelity (a), Output entan-
delity F of teleportation can be obtained by averaging glement(b),Fidelitybetweenρin andρout (c)andtheentan-
A
F(ρ ,ρ ) over all possible initial states: glement (d) of channel versus Jz and D. Where J = 1,T =
in out
0.1,B=1,b=0.5,γ =0.3 and Cin =1
2π π
dφ F(ρ ,ρ )sinθdθ
F = 0 0 out in , (21)
A
4π
R R IV. DISCUSSION
and for our model F can be written as
A
Theentanglementofatwo-qubitXYZHeisenbergsys-
ξ2cosh2βη+e2βJz(2ξ2cosh2βξ+J2sinh2βξ)
F = .(22) tem in the presence of an inhomogeneous magnetic field
A 3ξ2(coshβη+eβJzcoshβξ)2 and spin-orbit interaction is investigated. By turning on
the spin-orbit interaction we can change the behavior of
In the case of an isotropic XXX Heisenberg chain in the
the system without manipulating the other parameters.
absenceofmagneticfieldwithspin-orbitinteraction,this
We haveshownthatthe criticalvalues ofB, b andT are
equation gives the same as results of ref. [45]. The func-
adjustablebyDandwecanimprovethecriticalvaluesof
tion F is dependent on the channel’s parameters. The
A
B, b and T. Increasing D cause to volume enhancement
fig. 8 gives a plot of F , C , F(ρ ,ρ ) and C
A out in out channel
of revival region and also enhancement of entanglement
versusthechannel’sparameters. Thisfigureshowsthat,
in the revival region. Also, entanglement teleportation
in the case of J >0, F , F(ρ ,ρ ) and C decrease
z A in out out
whenD increases. InthiscaseF approaches 2 forlarge via two copies of above two-qubit system is studied. We
A 3 have shown that, by introducing SO interaction, the en-
values of D.
tanglement of replica state and fidelity can be increased
In contrast, in the case of J <0 and for small values
z
for the case of J <0, in spite to the case J >0. When
of D, F and F(ρ ,ρ ) has a constant value (smaller z z
than 32)Aand Coutiins zeoruot. As D becomes larger than a iDs tbheecommaexsimvaelryvallauregef,orthteheficdlealsitsyicaalpcpormoamchuensic32a,tiwonh.ich
threshold value, C undergoes a revival and then
channel
decreases for larger values of D. Since in the revival re-
gion, C has its maximum value, F , F(ρ ,ρ )
channel A in out
and C increase in this region such that for particular
out
interval of D, F becomes larger than 2 and ultimately,
A 3
tends to 2 for larger D. In summary, the fidelity of tele-
3
portation and entanglement of replica state are depen-
dent onthe entanglementof channelwhich is tunable by
the channel’s parameters (such as D,b,B,...). The ef-
fect of D is more desirable in the case of J < 0. In
z
this case for a certain values of D, F becomes greater
A
than 2,thismakethechannelusefulforperformancethe
3
teleportation protocol. For large D, F tends to 2.
A 3
9
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