Turnplate of quantum state

0.48 MB·English

by  Yang Liu

Turnplate of quantum state
Yang Liu and D. L. Zhou∗
Beijing National Laboratory for Condensed Matter Physics,
and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Time-reversalsymmetrybreakingcanenhanceorsuppresstheprobabilityofsuccessforquantum
state transfer (QST), and remarkably it can be used to implement the directional QST. In this
paper we study the QST on a ring with time-reversal asymmetry. We show that the system will
behave as a quantum state turnplate under some proper parameters, which may serve as time
controlled quantum routers in complex quantum networks. We propose to to realize the quantum
stateturnplateinthecoupledresonatoropticalwaveguidebycontrollingthecouplingstrengthand
the phase.
PACSnumbers: 03.67.Ac,03.65.-w
5
1
0
I. INTRODUCTION the physical model and make a general analysis. Then,
2
we analyze the dynamic requirement for the quantum

n Quantumstatetransfer(QST)isoneofthebasictasks stateturnplateinasingleexcitationmodel, andgivethe
a
J in quantum information process. In the past decade energy spectrum and symmetry matching condition for
QST has been studied intensively. Several schemes are quantum state turnplate. In Sec. IV, we study the spec-
8
2 proposed to achieve it by different channels, i.e., spin trum of the system with the cn symmetry. Then, we
chains [1–5], polarized photons in the optical fiber [6, 7], discusstheeffectiveHamiltonianofthesystemusingper-

] coupled-cavity array and so on. Based on cavity quan- turbation method. We come back to the CROW system
h
tum electrodynamics a scheme is to transfer the state in Sec. VI. Finally a summary is given.
p
- of a qubit from a cavity-atom system to another one
t through an optical fiber connecting the two cavities. Us-
n
II. PHYSICAL MODEL
a ing the spin chain as a channel many schemes are re-
u ported, such as, QST along a one dimensional unmod-
q ulated spin chain [1], perfect QST achieved by modu-
[ (a)
lating the coupling strength [3, 5, 8–12], QST without 7 N-1 (b) 7 N-1 JN−1
1 initialization [13, 14], optimizing basis [15, 16] and gen- N J6 N
v eralizing to the high spin QST [17–19]. Other schemes,
9 such as, transferring single-mode photon state through a 6 6 JN
0 coupled-cavity array, are also reported. Recently time- 1 J 1
0 innerpath 5
reversal symmetry breaking is introduced to study the
7 5 5 J
QST [20, 21], where the time-reversal symmetry break- outerpath 1
01.0 ianngdcmanakeenhthanecQeSoTr sduiprpecrteisosntahlebpiarso.baIbniltithyisosfetnhseeQtiSmTe 4 3 2 J4 4 J 3 J22
asymmetry is a new resource for exploring the QST. 3
5
In Refs. [22, 23], it is shown that a synthetic magnetic
1
FIG. 1. (a) The CROW ring. The circles mean the site res-
: fieldcanbeintroducedforphotonsbydifferentialoptical
v onators,andthelinkopticalwaveguideslinktheresonatorsas
paths in system of the coupled resonator optical waveg-
i a ring. Every waveguide between two site resonators has two
X uides(CROW).InthispaperweconsidertheQSTalong
paths,theouterpathandtheinnerpath. Thetwopathshave
r a ring consisting of coupled cavities or coupled resonator different length which induce the complex coupling strength
a optical waveguides with time reversal asymmetry. Be- J. (b) Single excitation graph. The circles correspond to
cause of the time reversal symmetry breaking we hope the resonators in CROW system. The arrow with labels Jl
that the quantum state transfer along the ring one by represents the matrix element Jl l l+1.
|(cid:105)(cid:104) |
one periodically like a turnplate of quantum states. The
quantum turnplate will be useful in building complex IntheCROWthesyntheticmagneticfieldcanbeintro-
quantum networks where it acts as a quantum router. duced by differential optical paths [22, 23]. We consider
In the following of the paper we show that a CROW the CROW system in a ring configuration as shown in
ring will behave as quantum state turnplate under some Fig. 1(a). The Hamiltonian of the ring is
proper parameters.
N
This article is organized as follows. First we give H =(cid:88)(J aˆ aˆ† +J∗aˆ†aˆ ), (1)
R l l l+1 l l l+1
l=1
where J is the coupling strength of between the sites
l
∗ [email protected] l − 1 and l, and N + 1 is interpreted as 1. aˆ (aˆ†) is
l l
2
the annihilation (creation) operator. From Ref. [24] we Through local unitary operator UL, the Hamiltonian
knowthattheconditionfortransferringanysinglemode can be transformed to
photon state from node l to node l(cid:48) is
H(J(cid:48))=ULH(J)UL†,
aˆ (τ)=aˆ , (2)
l(cid:48) l where (cid:80) φ(cid:48) = (cid:80) φ and |J(cid:48)| = |J |. In other words,
l l l l (cid:80) l l
where aˆl (τ) = U†(t)aˆl U(t) with U(t) being the time onlythesumofphases lφl isrelativetotheproperties
evolution(cid:48) operator. It (cid:48)can be easily verified by noting ofQST.Sowecanchooseaproperoperator UL tomake
thattheexpectvalueofanyoperatorinthel(cid:48)-thnodeat allthephasesequal,φ(cid:48) = (cid:80)lφl,withtheQSTproperties
l N
time τ is equal to that of the operator in the l-th node in the time evolution unchanged.
in the initial state. Now we consider the question: In what condition
Using the Heisenberg equation would the system behave like a turnplate of quantum
state? Firstly, we require that the system have the sym-
daˆl(t) =i[H ,aˆ (t)] (3) metry of the cyclic group, cn, for the turnplate having
dt R l n (n ≤ N) scales on it, i.e. N/n is an integer. The
operators of the c group elements can be expressed as
n
and noting that aˆ (t) can be expressed on the operator
l
bases as T , (T )2, ..., (T )n−1, 1
n n n
(cid:88)N where
aˆ (t)= α (t)aˆ ,
l k k
k=1 Tn =eiLn2nπ
the evolution of the operator aˆl(t) can be written as and Ln is a Hermitian operator. From (Tn)n =
1, we know that L has n integer eigenvalues, l ∈
n
idA =−H(J)A, (4) {(cid:98)−n−21(cid:99), (cid:98)−n−21(cid:99) + 1,..., (cid:98)n−21(cid:99)} ïijŇwhere (cid:98)x(cid:99)is the
dt largestintegernotgreaterthanx. Hereweonlyconsider
thecasewherenisodd. FortheHamiltonianwehavethe
where A = [α1(t), α2(t), ...,αN]T with T being the relation[H, L ]=0. Lettheeigenstateofthesystembe
n
transpose operation, and
|ψ (cid:105), that is
l,m
 0 J1 0 ··· JN∗ H|ψl,m(cid:105)=El,m|ψl,m(cid:105),
J∗ 0 J ··· 0
 1 2 
 0 J∗ 0 ··· 0  and
H(J)= 2 , (5)
 ... ... ... ... ...  L |ψ (cid:105)=l|ψ (cid:105).
n l,m l,m
J 0 0 ··· 0
N
Now we prove that the system will be a turnplate
The initial condition is A(0) = [0, 0, ...,1,...,0]T, of quantum states with n scales if the eigenvalues El,m
where 1 is the l-th element. With the above initial con- match the symmetry cn in the following way:
dition, Eq. (4) describes the single excitation evolution
(cid:18) (cid:19)
l
in the ring with coupling strength {−Ji}. El,m = +Zm (cid:15)+(cid:15)0, (6)
So the transfer of any single mode photon state in the n
CROW ring has the same physical picture as the single
where Z is an integer, and (cid:15) correspond to the phase
excitation model, and we do not need to initialize the m 0
that is not an observable in physics.
state of other sites except the input one.
Let the initial state of the system is |ψ (cid:105). It can be
0
easily proved that at time τ = 2π,
(cid:15)
III. SINGLE EXCITATION RING |ψ(τ)(cid:105)=eiπ(cid:15)0/(cid:15)Tn|ψ0(cid:105), (7)
Firstly, we consider the single excitation model. To meaning that quantum state of the system turn one
sketch our central idea, we consider the system depicted scale every time interval τ. The energy and symme-
byagraphinFig.1(b)consistingofN sitesasaring,with try matching condition can be seen as the generalization
the Hamiltonian H(J) having the form given in Eq. (5), of the energy and parity matching condition mentioned
where J is the the set {J }. Let us denote Φ as the set in Ref. [25], where the parity matching condition corre-
l
{φ },whereφ isthecomplexphaseofJ . FromRef.[21] sponds to the case n=2 and (cid:15)=2E .
l l l 0
we know that the phase Φ can give rise to the time- Here we look at the particular case N = n = 3.
reversal asymmetry if the graph is non-bipartite graph The relation [H, T ] = 0 requires that J = J = J .
3 1 2 3
(N is odd). In this article we concentrate on the cases l ∈ {0, ±1} and using the property of the c group we
3
where N is odd. known H and L have the same eigenvectors |φ (cid:105) =
n l
3
√13[1, ω3l, ω32l]T with ω3 = ei23π. When J1 = J2 = where g(x) is an arbitrary function of x. So the deter-
J3 = eiπ6, that i√s the total pha√se is π/2, the eigenval- minate of AN with odd N is
ues are El=0 =√ 3, El=1 = − 3, El=−1 = 0. So we detA =−λdetB −J2detB
get (cid:15)0 = 3(cid:15) = 3 and the time interval is τ = 32√π3. We N N N−1 N 1 N−2
numericallystimulatethetimeevolutionofthiscaseand (cid:89) (cid:89)
+ J eiφ+ J e−iφ−J2 detB
show the probability of the wave function in the Fig 2. l l N N−2
l=1 l=1
(cid:89)
=λf(λ2)+λg(λ2)+2 J cosφ
1 l
l
(cid:89)
=λF(λ2)+ J 2cosφ
l
0.8
l
where φ is the total phase. When φ = π + kπ, k =
0.6 2
0, ±1, ±2, ···, the spectrum has the structure {0, ±E }
) l
t
( that is the spectrum is symmetric around 0. Let us con-
P
0.4 siderthesystemthathastheCn symmetryandcontains
N = n×p sites. The eigenvalues of the operator L is
n
0, ±1, ..., ±n−1, which we label as l, and every eigen-
2
0.2 value has p-fold degeneracy. We can easily write out the
eigenvector of L ,
n
0
0 1τ 2 2τ 3 3τ 4 5 |l, i(cid:105)= √1 (cid:0)|i(cid:105)+ωl|i+p(cid:105)+ω2l|i+2p(cid:105)
t n n n
(cid:17)
+···+ω(n−1)l|i+(n−1)p(cid:105) .
n
FIG. 2. (Color online) Numerical stimulation of the time
evolution of the system consisting of 3 sites with the param- (cid:80)
eters J1 =J2 =J3 =eiπ6 and the initial state 100 . P(t) is Let Pl = i|l,i(cid:105)(cid:104)l,i|. From the relation [H, L] = 0, we
| (cid:105) know that
the probability of the wave function. The solid line, dashed
lineanddashdottedlinedescribetheprobabilityatsite1,site
2 and site 3 respectively. At time τ the excitation transfers PlHPl =0 for l(cid:54)=l(cid:48).
(cid:48)
fromsite1tosite2andittransferstosite3afteranexttime
intervalτ. Thesystemactslikeaturnplateoftheexcitation. This means that the Hamiltonian is block diagonalized
under the bases {|l, i(cid:105)}. Because the system has the c
n
symmetry, we have the relation J = J . So there are
Given the single excitation condition, it was proved i i+p
only p parameters, J , J , ..., J . We can always let
that there is the pretty good state transfer between any 1 2 p
J = 1 and other parameters be the ratio to J . Then
twositesofauniformringwithtotalphaseasπ/2+kπ,if 1 1
the property of the system don’t change up to the time
thenumberofthesiteoftheringisprime,inRef.[26]. It
scales. So there are p−1 parameters that we need to
indicatesthattheenergyspectrumandsymmetrymatch-
consider.
ingcondition,Eq.(6)isapproximatelysatisfiedwhenthe
Using the bra ket form of the Hamiltonian
length of the ring is prime.
H =J |N(cid:105)(cid:104)1|+J∗|1(cid:105)(cid:104)N|
N N
N−1
(cid:88)
IV. STRUCTURE OF THE SPECTRUM + (J |k(cid:105)(cid:104)k+1|+J∗|k+1(cid:105)(cid:104)k|), (8)
k k
k=1
In this section we analyze the energy spectrum of
and acting the projector P on both sides of Eq. (8) we
the system described by the Hamiltonian in Eq. (5). l
can directly give
We start with the definitions of some notations. We
denote the characteristic polynomial as A , that is
N P HP =ωlJ |p(cid:105)(cid:104)1|+ωl∗J∗|1(cid:105)(cid:104)p|
A = det(H −λ). And denote B as B = l l n p n p
N N N N
det(H (J =0)−λ),whereH (J =0)representsthe p−1
N N N N (cid:88)
Hamiltonianwherethecouplingbetweenfirstsiteandthe + (Jk|k(cid:105)(cid:104)k+1|+Jk∗|k+1(cid:105)(cid:104)k|). (9)
last site is zero, i.e., the chain is an open one. From the k=1
Ref. [27] we know that,
Comparing Eq. (8) and Eq. (9) we get that in the every
(cid:40) block the Hamiltonian is equivalent to the Hamiltonian
λg(λ2) if N is odd,
B = of the ring with length p and the moduli of the cou-
N g(λ2) if Nis even, pling strength are not changed just with the total phase
4
changing from φ/n to φl = nφ + 2nlπ, see Fig. 3. So the (a) 40 (b)
characteristic function can be written as
20
p
(cid:89)
λF(λ2)+2 J cosφ =0,
k l
4 2 2 4
k=1 − −
20
−
wherel=−n−1, ..., n−1. Everyfunctionmeansacurve
2 2 40
that cross with the axis of the variable p times corre- −
sponding to the p roots, see Fig. 4(a). All the curves
have the same shape. When the total phase φ= π +kπ, FIG. 4. Spectrum structure of the Hamiltonian H9 with
thecurvecorrespondingtol=−n+1 (orl= n−1)2crosses length N =π9 and C3 symmetry. The coupling strength is
4 4 J1 =J3 =e18, J2 =4J1. So the total phase is φ=π/2. (a)
the original point and we call it curve-0. Other curves
Pictures of the characteristic polynomials of three equivalent
(cid:81)
caalonngbethgeotvefrrotimcalthaexicsu.rSveo-0ifb(cid:81)ypkt=r1anJsklaatree2littkleJekncoousgφhl Hlinaemiislttohneiacnusrvgeo-0t fcroormrestphoenHdianmgitltooln=ian−H1,9a.nTdhtehdeaosthhdeortttwedo
thespectrumoftheHamiltonianhastheshapeindicated linescanbegotfromcurve-0bytranslatedalongthevertical
in Fig. 4(b), that is, the spectrum consists of separated axis. (b)SpectrumoftheHamiltonianH9. Everyenergyelver
groups. correspondtoonecrosspointofthecurveandthehorizontal
axis in (a).
J
1 8 J
2
7 where φ = (cid:80) φ . So we consider the Hamiltonian H(cid:48).
J3 9 k k
Let J(cid:48) (cid:29) J(cid:48) and write the Hamiltonian H(cid:48) into two
6 J3 terms2, H(cid:48) =1,H3 (cid:48) +V. Given that the H(cid:48) is represented
0
in the site basis |i(cid:105), H(cid:48) consists of the terms containing
0
J2 1 J2(cid:48) , and V consists the terms containing J1(cid:48),3. V is the
perturbation compared with H(cid:48).
5 J1 The eigenvalues of the Ha0miltonian H(cid:48) are α ∈
0
J1 2 {0, ±J2(cid:48)} and every energy level has three-fold degener-
4 J acy. So the energy levels are separated into three groups
3 2
J3 (manifold) corresponding to three αs. Using i label the
2π 4π different bases we denote the three manifold as |i, α(cid:105). In
3 J3 3 J3e 3 3 J3e 3 the manifold with α=0 the three bases are
|1(cid:105), |4(cid:105), |7(cid:105).
J 1 J 1 J 1
2 2 2
And the manifolds with α = ±|J | are spanned by the
2
2 J1 2 J1 2 J1 bases
FIG. 3. The ring with 9 nodes and c3 symmetry. Under the |X2+3(cid:105), |X5+6(cid:105), |X8+9(cid:105)
bases l,i theHamiltonianisblockdiagonalizedconsisting3
blocks|wh(cid:105)ichisimpliedbyc3. IneveryblocktheHamiltonian and
represents an ring with length 3 and the coupling strength is
the same as the original ring in the site bases with the total |X−(cid:105), |X−(cid:105), |X−(cid:105)
23 56 89
phase being φ/3, φ/3+2π/3 and φ/3+4π/3 where φ is the
total phase of the original ring. respectively, where
|0(cid:105) +|1(cid:105) |0(cid:105) −|1(cid:105)
|X+(cid:105)= i√ j, and |X−(cid:105)= i√ j.
ij ij
2 2
V. EFFECTIVE HAMILTONIAN
We take V as perturbation then compute the effective
Hamiltonian in the manifold α = 0. And the effective
Now we introduce the approximate method based on
Hamiltonian is
the spectrum structure. To discuss concretely, we con-
siderthesystemwithninesites(N =9)andthec sym-
3
metry. Its configure is shown in Fig. 3. The eigensystem  0 geiφ g
oHf(cid:48)twheithHamiltonian H is equivalent to the Hamiltonian Heαff=0 =−ge−iφ 0 g,
g g 0
J(cid:48) =|J |, for k (cid:54)=1
k k where g = J1J3. Hα=0 is identical to the representa-
J1(cid:48) =|J1|eiφ, tion of the HJa2miltoneiffan of the ring consisting of three
5
nodes with uniform coupling strength g and total phase which is not in the manifold M . So when V is
α=0 k,m
φ. From the analysis in the Sec. III we know that when much less than E(0) (the gap on the zero energy level),
φ = π2 +kπ the system is a turnplate of quantum state themanifoldMα=k0 isclose,i.e.,iftheinitialstateis|m(cid:105)
with three scales and the time interval of transfer state then the system is govern by the effective Hamiltonian
from one node to next is 2√π . We numerically simu- Hα=0 whichrepresentsaHamiltonianofthen-sitecycle.
3g 3 eff
late the time evolution of the system with parameters So the system with N = n×p and c symmetry can be
n
J1 =J3 =1, J2 =100 and φ=−π2 from the initial state reducedtothesystemwithN =nandwithcnsymmetry.
|ψ(0)(cid:105)=|1(cid:105) in Fig. 5. At time τ =120.92 the excitation
transfer from site 1 to site 4 and after the same time in-
terval it transfer to site 7 then back to site 1 circularly.
The sites except the sites 1, 4 and 7 can’t be excited. J 1 Jp J 1
0
1 J1 J p J1 ··· ···
0.8
p 1 p 1
− −
0.6
) FIG. 6. Spectrum of the H0 which is the main part of
(t theHamiltonian. EveryzeroenergyleveldenotestheHilbert
P
space of the sites labeled by number (1+l p), and every
0.4 ×
groupedp 1levelsdenotetheHilbertspaceofthesitesform
−
(2+l p)top+(l p)respectively,wherel=0,1,...,n 1.
× × −
0.2
0
0 100 200 300 400 500 VI. QUANTUM TURNPLATE ON THE CROW
t RING
FIG.5. (Coloronline)Numericallysimulatethetimeevolu- Nowwecomebacktothephysicalsystem, theCROW
tion of the system with 9 sites. J1 = J3 = 1, J2 = 100 and ring. For the ring containing three resonators, they can
φ=−π2. Theinitialstateis|ψ(0)(cid:105)=|100000000(cid:105)with|1(cid:105)as be connected by three identical connecting waveguides,
inputstate. Solidlinepresentsthefidelityofthestatetrans-
which contribute the same coupling strength |J | and
l
ferforthestateofthefirstsiteatdifferenttime. Thedashed
phase φ . In order to make a quantum turnplate we
line and the dashdotted line corresponding to the fidelity for l
just need to modify the optical path to make total phase
the seventh site and fourth site respectively.
φ = π/2+kπ. Initially we input the photonic state to
thenode1. Thenwewillseethatthephotonicstatewill
The correspondence between the propriety of the sys-
transfer from node 1 to node 2, node 3 and back to node
tem with site number N = n × p and N = n, where
1 cyclicly with perfect fidelity every time interval τ. For
n is decided by the symmetry of the large system, can
the ring contains N = 3×p resonators, which has the
be generalized to general case. Let J (cid:28) J then
1(p) l(cid:54)=1(p) c symmetry, we need to modify the coupling strength
the Hamiltonian can be written as H = H +V, where 3
0
between resonators and connecting waveguides to satisfy
V is perturbation term consisting of the terms contain-
theconditionJ (cid:28)J andchangetheopticalpath
ing J , and H is other terms. The spectrum of the 1(p) l(cid:54)=1(p)
1(p) 0 tomakethetotalphasebeφ=π/2+kπ. Thenphotonic
Hamiltonian H has the form depicted in Fig. 6. The
0
states can be transfer among the site 1, p+1, and 2p+1
zeroenergylevelisn-folddegeneracywithdegenerateket
cyclicly with high fidelity.
|m(0)(cid:105), where m = 1+l×p and l = 1,2,...,n−1. All
We simulate the QST along the CROW ring consist-
the |m(cid:105)s span the manifold M . The energies greater
α=0
andlessthanzero distributetwosides ofthezeroenergy ing 9 resonators with c3 symmetry. The parameters are
levelsymmetricallywithanenergygapandeveryenergy |J1| = |J3| = 1, |J2| = 100 and the total√phase φ = π/2.
level is n-fold degeneracy. From the perturbation theory Initially the state |ψ(cid:105)=(|0(cid:105)+|1(cid:105)+|2(cid:105))/ 3 is input into
in the degenerate case we know that up to the first or- thefirstresonator. Thenweobservetransferof|ψ(cid:105)along
dertheeigenketsofHamiltonianH correspondingtothe the ring. In Fig. 7 we plot the time evolution of the fi-
manifold M are delity
α=0
|φ(cid:105)=(cid:88)c |m(cid:105)− (cid:88) |k(0)(cid:105)Vk,m, F(t)=(cid:104)ψ|ρi(t)|ψ(cid:105)
m (0)
E
m k∈/M k
of sites 1, 4 and 7 and it behaves as a quantum state
where V = (cid:104)k|V|m(cid:105) and |k(0)(cid:105) is the eigenket of H turnplate. ρ means the reduced density matrix of site i.
k,m 0 i
6
1 for the annihilation operator of photon in the CROW
ring. So the basis for the quantum state in different site
shouldbeidentified. Forexampleintheringwith9sites
0.8 the bases for site 1 and site 4 should be {|0(cid:105),|1(cid:105),|2(cid:105),...}
and {|0(cid:105),eiπ/3|1(cid:105),ei2π/3|2(cid:105),...} respectively. Light scat-
teredfromtheresonatorscanbeimagedusingainfrared
) 0.6 camera. Directlythequantumturnplatewillbeobserved
t
F( from the image of the camera.
0.4
0.2
0 100 200 300 400 500
In summary, we study the QST on the ring of coupled
t
cavities with time reversal asymmetry. The transfer of
any single mode photon state in the CROW ring has the
FIG. 7. (Color online) Numerically simulate the time
samephysicalpictureasthesingleexcitationmodel,and
evolution of the fidelity of the CROW system with 9 res-
onators. J1 = J3 = 1, J2 = 100 and φ = π2. The initial we do not need to initialize the state of other sites ex-
state is √1 (0 + 21 + 32 )(1000000 + 1100000 ) with cept the input one. To act as a quantum state turnplate
28 | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105)
√1 (0 +21 +32 )astheinputstate. Solidlinerepresents the eigenvalues of the equivalent single excitation model
14 | (cid:105) | (cid:105) | (cid:105) should satisfy the matching condition Eq. (6). And we
the fidelity of the state transfer for the state of the first site
at different time. The dashed line and the dashdotted line show that when number of site N = 3 and the total
correspond to the fidelity for the fourth site and seventh site phase φ = π/2+kπ the matching condition is satisfied.
respectively. Further, we study the structure of the spectrum of the
single excitation ring in general condition and prove the
QST equivalent between the ring consisting n sites and
VII. DISCUSSION AND SUMMARY the one consisting n×p sites with c symmetry. Uti-
n
lizing the time reversal asymmetry the CROW consist-
Using the similar method we used to get the Eq. (7) ing 3×p resonators can sever as a quantum turnplate
we have the equation without of initialization, which can also be observed in
experiments. Quantum state turnplates would be useful
aˆl(τ)=eiπ(cid:15)0/(cid:15)aˆl−p, (10) to build a complex quantum network.
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