Perfect state transfer in XX chains induced by boundary magnetic fields

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by  Thorben Linneweber

Perfect state transfer in XX chains induced by boundary magnetic fields
Thorben Linneweber,1 Joachim Stolze,1,∗ and Go¨tz S. Uhrig1
1Technische Universita¨t Dortmund, Institut fu¨r Physik, D-44221 Dortmund, Germany
(Dated: January 26, 2012)
A recent numerical study of short chains found near-perfect quantum state transfer between the
boundarysitesof aspin-1/2 XXchain ifasufficiently strongmagnetic field actson thesesites. We
show that the phenomenon is based on a pair of states strongly localized at the boundaries of the
system and providea simple quantitativeanalytical explanation.
PACSnumbers: 03.67.Hk,03.65.Yz,75.10.Pq,75.40.Gb
2
1 I. INTRODUCTION ceptional energy eigenstates of the chain. These states
0 are exponentially localized at the boundaries with local-
2 ization length (lnh)−1. The PST is completely de-
The transfer of quantum states between different ele- ∼
n termined by these states and the dynamically relevant
ments of a quantum computer is an important and chal-
a energydifferencesbecomeexponentiallysmallasthesys-
lenging task in quantum information processing. Chains
J
tem size grows. Correspondingly the time scales become
ofinteractingspin-1/2particleswereproposedasameans
5 exponentially large, making this kind of PST unfortu-
to achieve this task. Many different types of quantum
2
nately impractical for long chains. Furthermore, since
spinchainshavebeen discussed,asevident,for example,
the PST depends on an exponentially weak link between
] fromthe reviews1and2. Inthe mostfrequentlystudied
h transmitter and receiver(the centralregionof the chain,
scenario the first site of a finite spin chain is preparedin
p where the amplitudes ofthe relevantstates areexponen-
a definite state which is then supposed to be transferred
- tially small) one might speculate that any slight imper-
t to the last site by the natural Hamiltonian dynamics.
n fection is bound to stronglyaffectthe PST performance.
a Since the elementary excitations of spin chains are of-
The robustness of state transfer against disorder in the
u ten dispersive, the initially sharply localized quantum
couplings was recently studied for several kinds of both
q state broadens as it propagates and thus quantum infor-
[ mation degrades3 so that the state received is not iden- engineered and boundary-controlledspin chains17,18.
2 tical to the state transmitted. In view of limits to fault In Sec. II of this paper we presentanalyticalsolutions
for the energy eigenstates and eigenvalues of the homo-
v tolerance in quantum information processing it is, how-
geneous semi-infinite chain with a magnetic field at the
6 ever, desirable to achieve perfect state transfer (PST).
8 Engineered chains were shown to be attractive means in firstsite. Theresultsthusobtainedsuggestasimpleper-
6 reachingthis goal2,4–9. Inthese chains,allspin-spincou- turbative calculation and an approximate expression for
2 the exact result for the finite chain with boundary fields
plingsand/orlocalfieldsaresetatcertainvalueswhich
. at both ends, which we discuss in Sec. III and which
1 guarantee PST.
1 explains the numerical observations in Ref. 16.
However, from a practical point of view, it would be
1
more desirable to achieve PST by simpler means, for ex-
1
v: ample by manipulating only O(1) systemparameters in- II. THE SEMI-INFINITE CHAIN: EXACT
steadofthe (N) coupling constants whichmustbe ad-
i O RESULTS
X justed for an N-site engineered PST chain. This line of
thoughthasbeenpursuedinanumberofstudiesinwhich
r The spin-1/2XX chainwith boundary magnetic fields
a the endsofahomogeneousspinchain,i.e.,onewithcon-
is defined by the Hamiltonian
stant couplings, were connected to a single transmitting
and receiving qubit, respectively, by adjustable coupling
N−1
constants10–15 which influence the transfer of states via H = 2J SxSx +SySy +h(Sz +Sz ), (1)
different mechanisms, depending on other details of the i i+1 i i+1 1 N
i=1
systems employed. X (cid:0) (cid:1)
In a similar vein, Casaccino et al.16 suggested to use where Sx, Sy, and Sz are the usual spin-1/2 operators
i i i
sufficiently strong magnetic fields acting exclusively on with eigenvalues 1/2. The most popular scenario in
±
the boundary spins of an otherwise homogeneous spin quantum information transport studies uses an initial
chain. In their numerical results, they indeed observed product state, ψ(t=0) = ϕ N , with ϕ =
| i | i i=2|↓ii | i
near-perfect state transfer, however, with the transfer α +β ,i.e.,asuperpositionofthestatewithnospin
time growing strongly with both the chain length and up|↑aind a|↓sitate with one single spNin up located at site 1.
the magnetic field strength. SincetheHamiltonianconservesthenumberofspins‘up’
In this paper, we explain the effect observed in Ref. and since the zero spin up component does not evolve in
16 by analytical considerations. We show that bound- time it suffices to follow the evolution of the single spin
ary magnetic fields of sufficient strength h produce ex- up component. The corresponding N-dimensional sub-
2
space is spanned by the states i =S+ ... and the Obviously δ has a singularity for h>J at some q. How-
| i i |↓↓ ↓i
Hamiltonian matrix H is given by ever, that singularity is only a jump from π/2 to π/2
ij
−
(orviceversa)correspondingtoanirrelevantsignchange
Hi,i+1 =Hi+1,i =J;H11 =HNN =h. (2) in the eigenvector components.
IfN isfinite(butlarge)theamplitudesofallextended
Forh=0the eigenvectors~uν ofthe matrix(2)aregiven
eigenstates scale as N−1/2, while the amplitude (8) of
by their components
the localized state at the boundary site is constant and
1 approaches unity as h grows. This is the key to the
2 2 νπl
uν = sin (3) (quasi-) PST reported in Ref. 16. As N or h grow the
l N +1 N +1
(cid:18) (cid:19) localized states at the boundaries will increasingly dom-
inate the transfer of localized quantum information be-
with ν =1,...,N, and the energy eigenvalues are
tween the ends of the chain, and we expect the speed
νπ
E =2Jcos . (4) of that transfer to be proportional to the overlap be-
ν
N +1 tween the states localized at the left and right ends of
the chain, respectively, which by (5) and (6) is propor-
By the Hellmann-Feynman theorem19–22 all eigenvalues
grow if a magnetic field h > 0 is switched on. To see tional to e−κN = J N.
h
more precisely what happens for h = 0 we now consider
6 (cid:0) (cid:1)
the caseN ,i.e., weeffectivelystudy asemi-infinite
→∞ III. THE FINITE CHAIN: APPROXIMATE
chainwithboundarymagneticfieldh. Sincewearelook-
RESULTS AND PERFECT STATE TRANSFER
ing for states localized near the boundary l = 1 we use
the Ansatz
We now consider the full model (1), with a strong
ul =e−κl (5) boundary magnetic field h J at both ends of the N-
≫
site chain. We then have two extremely localized energy
which turns out to be an eigenvector for h>J, with
eigenstates with energies far above all other states. The
h system can thus be approximately described by two iso-
κ=ln (6) lated spins in a strong field and an (N 2)-site chain
J −
in between. The corresponding Hamiltonian matrix is
and obtained from (2) by setting
J2
E =2Jcoshκ=h+ . (7) H12 =H21 =HN,N−1 =HN−1,N =0. (12)
h
Taking parity into account, the two localized states are
Wethusfindacriticalfieldvalue,abovewhichoneenergy
given by the amplitudes
eigenstate splits off from the quasi-continuum (4) and
moves upward with growing h. The localization length
1
of that state is proportionalto (lnh/J)−1 and its ampli- u± = u± = ; u± =0 otherwise (13)
1 ± N √2 l
tude at the boundary taking into account normalization
is given by both states have energy E = h. The N 2 extended
±
−
states are given by
2
J
(u )2 =1 . (8)
1 − h 1
(cid:18) (cid:19) uν =uν =0; uν = 2 2 sinνπ(l−1) (14)
The remaining eigenstates of the system are extended 1 N l N 1 N 1
(cid:18) − (cid:19) −
and similar to (3); the appropriate Ansatz is
(ν =1,...,N 2), with energy eigenvalues
−
u =sin(ql+δ), (9)
l νπ
E =2Jcos . (15)
ν
where q is a wave vector between 0 and π. For finite N N 1
−
the value of q is fixed by the boundary condition at the
Returning to the full problem defined by the Hamil-
farendofthesystem,leadingtoq =νπ/(N+1)forh=0
tonian matrix (2) we expect modified energy eigenvalues
as in Eq. (3), while for N q becomes continuous
and can take any value betw→een∞0 and π. The state (9) E± for the formerly degenerate isolated boundary states
. The initial and final states of the desired perfect
yields the eigenvalue |±i
state transfer can be written in terms of the states (13):
E =2Jcosq (10)
1 1
1 = (+ + ) ; N = (+ ) (16)
and the phase shift δ obeys | i √2 | i |−i | i √2 | i−|−i
hsinq andconsequentlythetransfertime τ isequaltothe time
tanδ = . (11)
J hcosq during which a phase difference π develops between the
−
3
energy eigenstates + and , that is basis 1 , N are (compare (14))
| i |−i {| i | i}
π
τ = . (17) 1
2 2 νπ
E+ E− 1V ν =Juν =J sin (19)
− h | | i 2 N 1 N 1
(cid:18) − (cid:19) −
The energy difference E E can be determined ap-
+ −
−
proximately by treating the matrix elements J between and
sites 1 and 2 and sites N 1 and N, respectively, as
−
perturbations. The corresponding perturbation opera- N V ν =Juν =( 1)ν+1 1V ν . (20)
tor V then is a N N matrix with four nonzero entries h | | i N−1 − h | | i
×
J. The matrix elements of V between the unperturbed
The diagonalelements (n=n′) of the 2 2 matrix (18)
eigenstates (13) vanish and thus first-order degenerate ×
thusturnouttobeequal,leadingtoamereenergyshift.
perturbation theory does not lift the degeneracy. In sec-
The non-diagonalelements are both equal to
ond order, the corrections to the energy are given23 by
the eigenvalues of the matrix
N−2 1V ν ν V N 2J2 N−2( 1)ν+1sin2 νπ
nV ν ν V n′ hh| 2|Jihco|s |νπi = N 1 −h 2Jcos νNπ−1,
ν6=Xn,n′ h E| n(0|)i−h E|ν(0|) i, (18) νX=1 − N−1 − Xν=1 − N−1(21)
leading to an energy splitting equal to twice the non-
where n and n′ aretheunperturbeddegenerateenergy diagonal element (21):
| i | i
(0)
eigenstates with eigenvalue E = h, and ν runs over
n
| i
all other unperturbed eigenstates (14), with eigenvalues 4J N−2( 1)ν+1sin2 πν
Eν(0) (15). Inourcasethedegeneratesubspaceisspanned E+−E− = N 1 −h 2cos πNν−1. (22)
by + and , or equivalently, by 1 and N , compare − νX=1 J − N−1
| i |−i | i | i
(16). The matrix elements of the perturbation in the
An asymptotic expansion of (22) in the small parameter J/h reads
J N−2 J 2 (N2 3N 2) J 4 J 6
E E =2J 1+(N 3) + − − + . (23)
+ −
− (cid:18)h(cid:19) " − (cid:18)h(cid:19) 2 (cid:18)h(cid:19) O (cid:18)h(cid:19) !#
This confirms the behavior which was to be expected starting from κ = ln h (compare Eq.(6)). To lowest
0 J
based on the results of Sec. II and it explains the expo- order in (J/h)N−2 this yields
nentialgrowthofthetransfertime(17)withbothN and
h/J. J N−2 J 2 J 2N−4
E E =2J 1 + .
TheexactexpressionoftheenergydifferenceE+ E− +− − h − h O h
can be found and approximated by the following−sim- (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17) (cid:16)(cid:16) (cid:17) (2(cid:17)7)
ple consideration. The boundary-localized symmetric For comparison to the numerical results of Ref. 16 we
and antisymmetric eigenstates are determined using an identify h with ∆E from that reference and set J =2.
ansatz of the form
h Ref. 16 exact diag. perturbative order (J/h)N−2
u± =e−κl e−κ(N+1−l). (24) 10 99 107 90 107
l ± 20 8010 801 770 801
30 2665 2674 2627 2674
The correspondingenergyeigenvaluesareagaingivenby
40 6260 6315 6252 6315
50 12294 12311 12233 12311
E =2Jcoshκ , (25)
± ±
TABLE I: Transfer times for a chain of length N = 5 with
(compare Eq.(7)) but now the values κ± fulfill the tran- differentvaluesoftheboundaryfieldhasobtainedbyvarious
scendental equation methods. Seetext for details.
h h In Table I we compare the transfer time for N =5 as
eκ± = e−Nκ± eκ± 1 (26)
J ± J − obtained by different methods and for different values of
(cid:18) (cid:19)
h. The column labelled “Ref. 16” contains the numbers
which results from the condition for the eigenvalue at reported in that reference (with an obvious typing error
sites 1 or N. This equation can be solved iteratively, for h = 20). The next column shows the transfer time
4
(17) from the difference of the two dominant eigenvalues where we haveseparatedthe dominant(localized)eigen-
E and E as obtained from an exact numerical diag- states from the remainig (extended) eigenstates µ ,
+ −
|±i | i
onalization of the Hamiltonian (2), whereas the fourth eachofwhichyieldsacontribution (1/N),withaquasi-
O
columnisobtainedfromtheperturbationcalculationde- random phase. The transfer time τ is the instant when
scribed above. the two dominant terms in (29) interfere constructively,
The asymptotic expansion (23) agrees to the expres- maximizing f and thus leading to (compare (8))
1N
sion(22)toamuchhigherprecisionthanthatofthenum-
bers in the table. However, it can be seen that the per- 2
J
turbative result approaches the exact one only for fairly 1e−iHτ N 1 (30)
|h | | i|≈ − h
largevalues of h. Onthe other hand, the lowestorderin (cid:18) (cid:19)
J N−2, given in Eq. (27), agrees with the exact values
h or
to all digits given (compare the last column in Table I).
(cid:0) (cid:1)
We have performed similar comparisons for chain
J 2
lengths N = 7 and N = 12, obtaining similar results,
f (τ)=1 2 . (31)
1N
thatis,forsufficientlystrongboundaryfields thepertur- − h
(cid:18) (cid:19)
bation calculation yields the transfer time to a precision
of one percent or better while the lowest order fits the The dominant time dependence of f (t) is a slow os-
1N
exact result within 10−7 for N =7 and within 10−14 for cillationbetween zero andthe maximum value (31) with
N =12, as was to be expected from Eq. (27). However, period 2τ, with superimposed rapid small-amplitude os-
at N = 12 the difference between the two leading en- cillations from the subdominant terms in (29). This is
ergy eigenvalues approaches the precision limits of stan- preciselywhatnumericalcalculationsforsmallN show.
dard numerical procedures and the corresponding trans- The valuef (τ) doesnotdependonthe chainlength
1N
fer times become prohibitively long by any standards. N and it grows (increasingly slowly) as h grows. Note,
Finally, we briefly discuss the degree of fidelity which however,that the transfer time τ grows exponentially as
can be reached by the scheme discussed here. A reason- h grows. For a given N the choice of h thus is the result
able measure of the transfer fidelity between the states ofatrade-offbetweentransfertime(17)andfidelity(31),
1 and N is dependingontherelativeimportanceassignedtoeachof
| i | i
these two figures of merit. Hence, our analytical results
f (t)= 1e−iHt N 2. (28)
1N fortransfertimeandfidelityprovidethebasisforfinding
|h | | i|
an optimum trade-off in practice.
Employingtheenergyeigenstates ν andtheeigenvalues
| i
E we can write
ν
1e−iHt N = 1ν ν N e−iEνt (29)
h | | i h | ih | i Acknowledgments
ν
X
= 1+ +N e−iE+t
h | ih | i JS wishes to thank Analia Zwick for helpful discus-
+ 1 N e−iE−t
h |−ih−| i sions. GSU is supported by the DFG through grant UH
+ 1µ µN e−iEµt, 90/5-1.
h | ih | i
µ
X
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