Quantum communication beyond the localization length in disordered spin chains

0.2 MB·English

by  Jonathan Allcock

Quantum communication beyond the localization length in disordered spin chains
Jonathan Allcock∗ and Noah Linden†
Department of Mathematics, University of Bristol,
University Walk, Bristol BS8 1TW, United Kingdom
(Dated: January 27, 2008)
Westudytheeffectsoflocalization onquantumstatetransferinspinchains. Weshowhowtouse
quantum error correction and multiple parallel spin chains to send a qubit with high fidelity over
arbitrary distances; in particular distances muchgreater than the localization length of thechain.
The reliable communication of quantum states from thedistancepropagatedandthedegreeofdisorderinthe
8 one physical location to another is most likely necessary chain. Wethenshowhowtousemultiplespinchainsand
0 if large-scale quantum computing is ever to be realised. quantum error correction to achieve high fidelity quan-
0
Several years ago Bose [1] proposed using spin chains tuminformationtransferoverarbitrarydistances;inpar-
2
as the medium for such quantum state transfer. Since ticular over distances much greater than the localization

n
then, there has been much interest in this area, and nu- length. By considering a concatenation scheme we show
a
merous protocols have been put forward which develop that, provided the disorder is not too great, the number
J
this idea, improving the efficiency and fidelity of trans- of spin chains required scales only polylogarithmically
1
3 fer using methods suchas encoding the informationover with the distance over which we wish to communicate.
multiple qubits in the chain [2, 3, 4, 5], using engineered Inwhatfollowsweshallusethefollowingnotationand

] couplings [6, 7], multi-rail encodings [8, 9, 10] and local conventions: A spin in the 1 state will be called an
h | i
memory [11]. excitation. For a system of N spins, we shall use a bold
p
- Unfortunately, anyrealspinchainwillinevitably have font 0 = 00...0 to denote the zero excitation state,
t | i | i
n anelementofdisorderinherentinthesystem. Aspointed and j = 00...010...0 to denote the single excitation
| i | i
a state with the jth spin in the state 1 and all the other
out in [12], this can cause a phenomenon now known
u spins in state 0 . Finally, σ~ =(σx,|σyi,σz), where σx,y,z
q as Anderson localization [13] to take place. This is the | i i i i i i
arethePaulimatricesactingontheith spininthechain.
[ process in which the energy eigenstates of a disordered
Let us consider the following communication scenario.
lattice become localized in space, rather than extending
1 Alice and Bobare at opposite ends of a chain of N spin-
throughout the system as they would in the absence of
v
1/2particlesdescribedbysomenearest-neighbourHamil-
7 disorder. This is turn inhibits state transfer beyond a
tonian H. Alice’s task is to send, with as high a fidelity
6 distance known as the localization length of the chain.
8 While this may be useful for localizing qubits for quan- as possible, an unknown qubit state to Bob. We shall
4 assume that the following conditions hold: (C1): The
tum computing [14], it provides a major challenge for
1. the use of spin chains in quantum communication. Al- system is isolated from the environment, and thus there
0 though localization has been extensively studied in solid arenoexternalsourcesofnoise. (C2): H commuteswith
8 state physics, its implications for quantum information thetotalZ-spinoperator Nσz andhenceconservesthe
0 k k
are not well understood [12],[5],[15, 16, 17]. number of excitations on the chain. (C3): The system
: P
v starts in the initial state 0 . For example, H could be a
Xi lemInsddeeude, tiot ilsocnaolitzactlieoanr afitt fiwristthignlatnhcee shtoawndtahredperrorbor- simpleHeisenbergcouplin|gi(intheabsenceofanexternal
field)
r paradigm considered in quantum information theory.
a
Firstly, rather than being due to some coupling of the
H = (J/2) σ~ σ~ (1)
spin chain with the environment, localization is an in- − i· j
<i,j>
trinsic source of error. Even in the absence of disor- X
der, the excitations carrying the quantum information
where denotes that the sum is over all adjacent
<i,j>
become spread out along the chain. In the presence of
spins i and j, and J is the coupling constant between
disorder, only an exponentially small part of the signal P
spins. Notethatwehavenotyetintroducedanydisorder
reaches beyond the localization length. Also, although
into the system.
we are only trying to communicate a single qubit, the
The communication proceeds according to [2] and [3].
localization errors take place in the higher-dimensional
We assume that Alice and Bob each have access to a
Hilbert space of the entire spin chain.
numberN , N respectively,ofspinsattheirendsofthe
A B
In this Letter we look more closely at the effect local- chain. Tobegin,Aliceencodestheinputstatea 0 +b 1
| i | i
ization has on spin chain state transfer and find that, as a state of her N spins in such a way that the follow-
A
for a class of standard spin chain protocols, localization ing two conditions hold. (C4): 0 is encoded as 0 .
| i | iA
caneffectively be viewedasasourceofamplitude damp- (C5): 1 is encoded as 1 . Here A denotes Al-
| i | ENCiA
ingerrors,wherethedampingparameterisdependenton ice’s addressable spins, A denotes all other spins, and
2
1 lies in the linear span of j , i.e. it is a su-
| ENCiA | iA
perposition of single excitation states, where the excita-
tions lie in Alice’s domain. The system then undergoes
unitary time evolution for some time t. Note that as a
resultofAlice’sencoding,andthefactthatthe Hamilto-
nian preservesthe total number of excitations, the chain cj 2
| |
dynamicsremainrestrictedtotheN+1dimensionalsub-
space spanned by the zero and single excitation states.
Finally, Bob applies a decoding unitary to his address-
able spins. This concentratesthe stateonto asinglespin
which he then takes as the output of the transfer. The
whole process is equivalent to sending Alice’s original
state down an amplitude damping channel with time-
FIG. 1: Effect of localization on state propagation for a dis-
dependent channel parameter γ(t) = 1 (t), where
− CB ordered Heisenberg chain with N = 501, J = 1/√2. The
0 C (t) 1. The corresponding average fidelity is
B wavepackets have been plotted at time increments δt = 25.
≤ ≤
given by 1/2+ CB(t)/3+CB(t)/6. Diagonaldisorderwasdrawnfromanormaldistributionwith
Thereisaconvenientwaytovisualisethetransferpro- mean zero and standard deviation δ. From top to bottom,
p
cess [2]. At time t the state of the system can be ex- δ=0, 0.1, 0.2 and 0.3.
pressedas a 0 +b N c (t) 0 , where N c (t)2 =
| i j=1 j | i j=1| j |
2
1. Byplotting the qPuantities |cj(t)| againPstsite number pressed. Withadisorderedchain,the channelparameter
j we can produce a graph of the state. Then, the quan- (t)dependsonboththetimeandtheparticularreal-
B,δ
tity B(t)isgivenbytheareaunderthegraphsupported Cization of the disorder (that is, the specific values of the
C
by Bob’s accessible sites. To achieve high fidelity trans- ǫ in (2)). However, for a given , the specific values
j δ
fer,the strategyistochoose|1ENCiA tobe awavepacket of ǫj can only be determined probPabilistically. In other
withaparticularshapethatleadstominimaldispersion. words,γ γ (t)=1 (t) is a stochastic function of
δ B,δ
What happens now if there is disorder present in the time, par≡ameterized b−yCδ.
chain? Any real chain will, due to engineering limita- We have investigated this claim in more depth by nu-
tions and thermal fluctuations, have spin-spin couplings merically evaluating γ (t) for various values of t and δ,
δ
that are not described by an ideal Hamiltonian such as with chosento be the uniformdistribution. Since the
δ
(1). Ingeneraltheremaybecomplicatedtime-dependent outcoPme is stochastic,we averageovermany trials in or-
perturbations to the Hamiltonian, possibly coupling the der to build up a mean surface (see Fig 2). This was
zero and single excitation subspaces with subspaces of found to have the empirical form
larger numbers of excitations. However, as in [12], here
we consider a simpler model of randomness where the γ (t)=1 e−αt(δ2+βδ) (3)
δ
−
total Hamiltonian H is given by
ǫ
for some constants α and β. The same empirical form
Hǫ =H + ǫj|jihj| (2) (n3o)rmwaalsdfiosutrnidbuttoiohnosl,dalftohroPugδhchdoiffseenrentot vbaeluCesauocfhαy aanndd
j
X
β wereobserved. Thus,ifweknowthe valueofδ,wecan
where H is the Hamiltonian (1) and the ǫ are i.i.d real deducehowfarBobcanbefromAlicebefore,onaverage,
j
random variables drawn from some underlying distribu- thefidelitydropsbelowacertainthreshold. Asexpected,
tion withboundeddensity,characterisedbyadisorder forfixedδ thefidelitydecaysexponentiallyquicklyasthe
δ
P
parameterδ (forinstance,theuniformdistributioninthe state propagates along the chain.
interval [ δ,δ]). Theidentificationofadisorderedspinchainasanam-
−
What are the implications of this for state transfer? plitude damping channel (albeit one with a stochastic
In spite of the disorder, the all zero state 0 trivially damping parameter) immediately opens up the possi-
| i
remains an eigenstate of H and, furthermore, the chain bility of using quantum error correction to improve the
ǫ
evolution remains restricted to the zero and single exci- channel fidelity. Alice can encode her message state into
tation subspaces. This implies that the disordered chain a state of multiple qubits and send each of these qubits
still behaves like an amplitude damping channel, where down a separate, parallel, spin chain. Bob will then re-
the damping parameter depends on the area under the ceive the multiple qubits and apply standard error cor-
graph in Bob’s domain. Fig 1 shows how increasing the rection techniques. Indeed, quantum codes specifically
degree of disorder causes the graph of the state to suffer tailoredtotreatingamplitudedampingerrorsarealready
from dispersion and reflection as it propagates. Con- known [18, 19]. However,if Alice and Bob are separated
sequently, the area under the graph in Bob’s domain, byadistancelargerthanthelocalizationlength,theam-
andhencethetransferfidelity,becomesincreasinglysup- plitude damping will become too severe and this error
3
γ (t) ger,andletǫ be a positive constantinthe interval(0,1].
δ
Then it is possible to send a qubit, a distance mL, with
fidelity greater than 1 ǫ/2, using a number of chains n
−
polylogarithmic in m:
m 3 15p −3
n=5k > ln ln . (4)
ǫ − 2
(cid:16) (cid:17) (cid:18) (cid:19)
To show this, we need the following facts regarding
quantum channels:
Lemma 1. Channel Twirling [22]. Any channel Λ
t δ
can be turned into a depolarizing channel with the same
entanglement fidelity F and average channel fidelity f as
FIG.2: Meansurfaceofγ(t,δ). Red: numericaldata. Green: Λ.
1 e−αt(δ2+βδ), for α = 2.56, β = 0.029. Diagonal disorder
− A depolarizing channel with parameter p is a quantum
drawn from a uniform distribution in therange [ δ,δ] .
− channel which, with probability (1 p) leaves an input
−
stateρuntouched,andwithprobabilitypreplacesρwith
the maximally mixed state.
correction protocol will fail [23].
However we have the option of performing error cor- Lemma 2. Consider the following procedure: (a) En-
rectionatregularintervals (shorterthan the localization coding a single qubit in the space of 5k qubits according
length)alongthechain,interveningbeforetheamplitude to the 5 qubit code concatenated k times. (b) Sending
dampingbecomes toogreat. Thusaslongas δ is nottoo each qubit down an identical depolarizing channel with
large, error correction can be used to correct localiza- parameter p. (c) Error correcting and decoding back to
tion errors, and achieve high fidelity state transfer over the space of a single qubit. This procedure is equivalent
distances much larger than the localization length. Fur- tosendingthequbit down adepolarizing channelwith pa-
thermore, since the wavepackets propagate with a well- rameter p < 2 15p 2k.
defined groupvelocity,the transmissiontakes a time lin- k 15 2
ear in the distance over which we want to propagate. Proof. Consider (cid:0)the s(cid:1)impler procedure: (a) Encoding a
The key result of this Letter is that this scheme is single qubit in the space of 5 qubits according to the
scalable. That is, the number of parallel chains needed 5 qubit code. (b) Sending each qubit down an iden-
to faithfully communicate a qubit growsfavourablywith tical depolarizing channel with parameter p. (c) Error
the distance we wish to send it. We show this by con- correcting and decoding back to the space of a single
sidering the following protocol. Alice encodes her initial qubit. Explicit calculation shows that this procedure is
qubit in the space of n = 5k qubits according to the 5 equivalent to sending the single qubit down a depolariz-
qubit code [20, 21] concatenated k times. Each qubit is ing channel with parameter 15p2 25p3 + 15p4 3p5,
2 − 2 2 − 2
then sent down a separate spin chain - modified via a which, in our range of interest 0 p 1, is monoton-
≤ ≤
channel twirling process [22] - encoded over Alice’s N ically non-decreasing in p and (for p not zero) strictly
A
sites asa minimally-dispersivewavepacketinaccordance less than 15p2. The effect ofconcatenatingour quantum
2
with [2] and [3]. (The twirling is applied to ensure that code then follows by induction.
theconcatenationpreservesthestructureofthechannel.)
Lemma 3. Composing m identical depolarizing chan-
At periodic intervals of distance L we will error correct,
nels, each with channel parameter p, gives a depolarizing
until the wavepacketsreach Bob’s end of the chain. Bob
channel with parameter 1 (1 p)m.
then decodes each of the (distorted) wavepackets back − −
down to the space of single qubits, before finally decod- Suppose we haveatour disposalmany quantumchan-
ing these 5k qubits down to the space of a single qubit. nels, each with physical length L. Let us call these basic
Provided that the number of parallel chains n scales channels. By lemma 1 we can assume, without loss of
polylogarithmically in the distance we wish to commu- generality, that these are depolarizing channels with pa-
nicate over, we show that it is possible to send a qubit rameter p. Now consider combining 5k of these channels
an arbitrary distance, with arbitrarily high fidelity us- in parallel to form a block of length L and depth (i.e.
ing this protocol. More specifically, let L be a non-zero number of channels in parallel) 5k. By lemma 2 we can
length chosenso that after twirling, eachlengthL of the useerrorcorrectiontosendaqubitdownthis block,and
chain has the depolarizing parameter p. (For the time the net effect is equivalent to sending the qubit down a
being we will consider, for convenience, that each chan- depolarizing channel with parameter p . Let us call this
k
nel has the same value of p; we will discuss at the end of resultingchannelak-block,inviewofthefactthatk lev-
the Letter how to relax this). Let m be a positive inte- els of concatenation are involved. If we now compose m
4
oftheseblockstogethertoformachannelofdepth5k and be used to deal with rather more general errors arising
total length mL, lemma 3 tells us that this is equivalent in quantum communication in spin chains and not just
to a depolarizing channel with parameter those arising from localization.
2k m Inthetextwehaveconsideredthateachchannelhasan
2 15
identicalerrorparameter. Thisis,ofcourse,anunreason-
p <1 1 p . (5)
total
−" − 15(cid:18) 2 (cid:19) # ableassumptionsince the errorsarestochastic. However
itisnottoodifficulttoseethattheprotocolstillleadsto
Now, (4) means that
polylogarithmic scaling if, for example, the channels are
ln( 2 (1+ m)) log25 guaranteed to have error parameter p below some given
5k > 15 ǫ threshold. Itwouldbe aninterestingquestionforthe fu-
ln15p ! turetoanalysehowwelltheprotocolsucceedsifoneonly
− 2
knows, for example, the distribution of possible channel
ln(15(1 e−ǫ/m)
2k > 2 − . (6) parameters..
⇒ ln15p !
2 It is worth making explicit the nature of the inter-
ventions needed along the chain in order to perform our
2k protocol. The syndrome measurements may be done co-
2 15
p < 1 e−ǫ/m herently, rather than by performing von Neumann mea-
⇒ 15 2 −
(cid:18) (cid:19) surements. Inotherwords,theindividualchainscanhave
p < 1 e−ǫ
total (passive)unitaryinteractionsbetweenthemthatperform
⇒ −
< ǫ. (7) the errorcorrection. This requirescleanqubits – sources
of low entropy – to be coupled into the chains at regular
We have used the inequalities 1 e−x > x for x > 0
− 1+x intervals.
and 1 e−x <x for x>0 in (6) and (7) respectively.
− Finallyweobservethatourresultsmightbeinteresting
So provided that the parameter p of the fundamen-
inthecontextofsolidstatephysics. Wehaveshownthat
tal depolarizing channels is less than 2/15, we are free
parallel disordered one-dimensional spin chains can sup-
to choose any m, arbitrarily large, and any ǫ, however
port high fidelity “conduction” of quantum information
close to zero, and the overall channel will have parame-
over arbitrary distances; the number of required chains
ter p < ǫ as long as the channel depth satisfies (4).
total scaling only polylogarithmically with the distance. This
Theaveragefidelityofsendingaqubitdownthischannel
is perhaps at odds with the intuition one might have
is then f >1 ǫ/2.
− from the well-known fact that in one dimension, disor-
We can now directly apply this result to the case
der inevitably prevents propagation. Of course there is
where our basic channels are the sections of the dis-
no true contradiction here. For example, our system is
ordered spin chains. As mentioned before, each chain
notstrictlyonedimensional;theerrorcorrectionleadsto
acts like an amplitude damping channel with identical
subtle coupling of the one-dimensional chains. However
parameter γ. If we now apply channel twirling to our
we believe that our techniques from quantum informa-
chains they become depolarizing channels with parame-
tionmayoffernewinsightsintolocalizationinsolidstate
terp= 1 γ+2 1 √1 γ . Werequirethatp<2/15
3 − − systems.
so, given a degree of disorder δ, we choose a value of L
such that(cid:2)this is(cid:0)true, at lea(cid:1)s(cid:3)t with high probability. In WewouldliketothankTobyCubittandAndreasWin-
other words, the degree of disorder fixes the maximum ter for many helpful discussions. J.A. also gratefully ac-
possiblelengthofourbasicchannels,orequivalently,the knowledges the support of the Dorothy Hodgkin Foun-
length before which we must error correct. We can now dation.
form k-blocks from our twirled chains, and compose m
ofthese blockstogethertoformachanneloftotallength
mL. It then follows that it is possible, using a number
of disordered chains polylogarithmic in m/ǫ, to send a
∗ Electronic address: [email protected]
qubit a distance of mL with overall fidelity greater than † Electronic address: [email protected]
1 ǫ/2. [1] S. Bose, Phys. Rev.Lett. 91, 207901 (2003).
−
Wehaveshownthatforaclassofspinchainprotocols, [2] T. Osborne and N. Linden, Phys. Rev. A 69, 052315
namely those that satisfy conditions (C1)-(C5), that lo- (2004).
calizationcan be viewed as a source of amplitude damp- [3] H. Haselgrove, Phys.Rev. A 72, 062326 (2005).
[4] D. Burgarth, V.Giovannetti, and S.Bose, Phys.Rev.A
ing errors. Althoughwe havefocused hereonthe caseof
75, 062327 (2007).
diagonal disorder, the same holds true for more general
[5] D. Burgarth, Eur. Phys. J. Special Topics 151, 147
models of disorder, provided that the disorder does not
(2007).
couple together subspaces of different numbers of exci- [6] M. Christandl, N. Datta, A. Ekert, and A. Landahl,
tations. Indeed it will be realized that our protocol can Phys. Rev.Lett. 92, 187902 (2004).
5
[7] M. Christandl, N. Datta, A. Ekert, and A. Landahl, Phys. Rev.A 72, 012323 (2005).
Phys.Rev.A 71, 032312 (2005). [17] C. Burrell and T. Osborne, Phys. Rev. Lett. 99, 167201
[8] D. Burgarth and S. Bose, Phys. Rev. A 71, 052315 (2007).
(2005). [18] D. Leung, M. Nielsen, I. Chuang, and Y. Yamamoto,
[9] K.Shizume,K.Jacobs, D.Burgarth, andS.Bose, Phys. Phys. Rev.A 56, 2567 (1997).
Rev.A 75, 062328 (2007). [19] A. Fletcher, P. Shor, and M. Win, quant-ph/0710.1052
[10] D. Burgarth, V. Giovannetti, and S. Bose, J. Phys. A: (2007).
Math. Gen. 38, 6793 (2005). [20] C.Bennett,D.DiVincenzo,J.Smolin,andW.Wootters,
[11] V. Giovannetti and D. Burgarth, Phys. Rev. Lett. 96, Phys. Rev.A 54, 3824 (1996).
030501 (2006). [21] R. Laflamme, C. Miquel, J. Paz, and W. Zurek, Phys.
[12] J. Keating, N. Linden, J. Matthews, and A. Winter, Rev. Lett.77, 198 (1996).
Phys.Rev.A 76, 012315 (2007). [22] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.
[13] P.Anderson, Phys.Rev. 109, 1492 (1958). Rev. A 60, 1888 (1999).
[14] M. Dykman, L. Santos, M. Shapiro, and F. Izrailev, [23] In one dimension, for any non-zero degree of disorder,
Quant.Inf. and Comp. 5, 335 (2005). the localization length is finitewith probability one.
[15] D.Burgarth and S.Bose, New J. Phys.7, 135 (2005).
[16] G. D. Chiara, D. Rossini, S. Montangero, and R. Fazio,