Proposed all-versus-nothing violation of local realism in the Kitaev spin-lattice model

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by  Ming-Guang Hu

Proposed all-versus-nothing violation of local realism in the Kitaev spin-lattice model
Ming-Guang Hu, Dong-Ling Deng, and Jing-Ling Chen∗
Theoretical Physics Division, Chern Institute of Mathematics,
Nankai University, Tianjin 300071, People’s Republic of China
(Dated: January 22, 2009)
We investigate the nonlocal property of the fractional statistics in Kitaevs toric code model. To
this end, we construct the Greenberger-Horne-Zeilinger paradox which builds a direct conflict be-
tweenthestatisticsandlocalrealism. Itturnsoutthatthefractionalstatisticsinthemodelispurely
a quantum effect and independent of any classical theory.We also discuss a feasible experimental
schemeusing anyonic interferometry to test thiscontradiction.
9 PACSnumbers: 03.65.Ud,03.67.Mn,75.10.Jm,42.50.Dv
0
0
2 Quantum theory can predict results which are never in Kitaev’s model is purely a quantum effect and inde-
achievablefromthe localrealism(LR)[1]. Bydefinition, pendent of any classicaltheory. In experiment, the GHZ
n
LR consists of two constraints of realism and locality: paradox was tested only by using multi-photon systems
a
J Any observable has a predetermined value, regardless of [11,12]. Themodeldiscussedherealsoprovidesapoten-
whether it is measured or not, and the choice of which tial platform to test the GHZ paradox in the future and
2
2 observabletomeasureononepartyofamultipartitesys- this is discussed briefly at the end.
temdoesnotaffecttheresultsoftheotherparties. These The Kitaev’s toric code spin-lattice model [4] is intro-

] constraintsleadnotonlytothewell-knownBellinequali- duced as follows. Considering a k k square lattice on
h ties[2]whichputboundsonthecorrelationsandarevio- a torus T2 (see Fig. 1), one spin or×qubit is attached to
p
- lated statistically by certain quantum states, but also to each edge of the lattice. Thus there are 2k2 qubits. For
nt the so-called Greenberger-Horne-Zeilinger (GHZ) para- eachvertex and eachface , consideroperatorsof the
V F
a dox [3] which derives directly the inconsistent values of following forms:
u the correlations from LR and quantum mechanics. Such
[q paradox is tested by the nonstatistical measurements, AV = Yσjx, BF = Yσjz,
yielding succinctly an all-versus-nothing proof between
j∈V j∈F

3 LR and quantum mechanics.
v In this work we investigate LR in the context of Ki- where the σX denotes the Pauli matrix with X =x,y,z
7 j
taev’s toric code spin-lattice model, which is an exactly and it acts on the j-th qubit of a vertex or face
5 V
solvable model and is crucial for fault-tolerant topologi- . These four-bodyinteracting operatorscommute with
1
F
1 cal quantum computation (TQC) [4, 5]. This model has eachother because a vertex anda boundary consist
V F
. themerits: Itsdegenerategroundstatesyieldatopologi- of either 0 or 2 common qubits. From their definitions,
0
1 callyprotectedsubspacethatprovidesrobustnessagainst weknowthatoperatorsAV andBF haveeigenvalues±1.
noise and quasilocal perturbations, arousing much inter- Summing them together, it constructs the model Hamil-
8
0 est in condensed matter and quantum optical physics tonian as
: to realize and control it [6, 7, 8]; Excitations of the
v
i ground states known as anyons possess a class of frac- H0 =− X AV − X BF,
X tional statistics [9] intervening between the bosonic and V∈T2 F∈T2
r fermionic statistics, that is, the quantumstate ofanyons
a can acquire an unusual phase factor when one anyon is of which the ground states g satisfy AV g = g ,
| i | i | i
exchanged with another one, in contrast to usual values BF g = g forall , . Duetothetopologicalproperty
| i | i V F
+1 for bosons and 1 for fermions. of torus, the ground states are four-fold degenerate and
−
Since anyons are at the heart of TQC [10], it is natu- construct a four-dimensionalHilbert subspace, one basis
ral and important to ask whether the anyonic statistics, of which can be explicitly written as
though defined in quantum mechanics can be described
byLR.Withthisaim,weconstructtheGHZparadoxby g0 = Y (1+AV)0 ⊗2k2, (1)
| i J | i
using the string operators that are used in the model to V∈T2
move anyons on the lattice. According to this paradox,
the results derived from anyonic statistics will contra- with a normalization constant , while the remaining
J
dict irreconcilably that derived from LR. In this way we threecanbegivenafterweintroducetheconceptofstring
conclude straightforwardly that the fractional statistics operators [4, 13].
Here we describe anyons as the quasiparticle excita-
tions of the spin-lattice system with H0. There are two
types of anyons: z-particles living on the vertices and x-
∗Electronicaddress: [email protected] particles living on the faces of the lattice. These anyons
2
globalphasefactor 1ispickedupinfrontoftheground
−
state, i.e., D1 g = g . Likewise, as for the operations
D2 = SLz8z72SLx|xSiLz3z,−a|ndi D3 = SLz8z7651SLxxSLz3z, they have
the same interpretations as D1 but with different loops
Lx,z. ThelinksconstructedbyLx,z forD2,3 areshownin
Fig. 2(b) and 2(c), respectively and due to the anyonic
statistics, we have D2,3 g = g . As for the operation
D4 =SLxx, it has a stra|igihtfor−w|arid correspondence with
AV operator and its loop Lx constructs a simple unknot
FIG. 1: (Color online) An illustration of the Kitaev spin- shown in Fig. 2(d). Therefore when acting it on the
lattice model: Each black dot on the edge of the lattice rep- ground state, no change would appear, i.e., D4 g = g .
| i | i
resents a spin-1/2 particle or qubit; The interactions of the Writing the above statements of composite string op-
Hamiltonian H0 are along edges that bound a face F, and erations into more convenient forms, we have
edgesthatmeetatavertexV. ThestringPx,z indicatepaths
ofproductsofσx,z operatorsthatarelogicaloperatorsonthe D1 g = σ6zσ5zσ4xσ3xσ2yσ1y g = g , (3a)
| i | i −| i
qubits. D2 g = σ8zσ7zσ4xσ3yσ2yσ1x g = g , (3b)
| i | i −| i
D3 g = σ8zσ7zσ6zσ5zσ4xσ3yσ2xσ1y g = g , (3c)
| i | i −| i
are created in pairs (of the same type) by string opera- D4 g = σ4xσ3xσ2xσ1x g = g , (3d)
tors: ψz(P ) =Sz g and ψx(P ) =Sx g and they | i | i | i
live at| the eznid of sPtzri|nigs, wh|ere x i Px| i in which the algebraic relation σjzσjx = iσjy has been
used. In the viewpoint of measurements, suppose there
SPzz = Y σrz, SPxx = Y σrx, (2) aspreine(iSgheetFobigs.er2v)eirns tahnedmeoacdhel.ofOtnhetmhehi-atshascpcines,sthtoe coonre-
r∈Pz r∈Px
respondingobservermeasurestheobservableσX without
i
arestring operatorsassociatedwithstring Pz onthe lat- disturbing other spins andthe measurementresultis de-
tice and string Px on the dual lattice, respectively (Fig. noted by mX. Since these results must satisfy the same
i
1). Notethattwoanyonsofthe sametypewouldannihi- functional relations satisfied by the corresponding oper-
lateeachotherwhentheymeetandthisissocalledfusion
ator, then from Eqs. (3), we can predict that, if all the
rule. ThenwecanseethatAV andBF arejusttwoclosed operators in Eqs. (3) are measured, their results must
stringoperators. Especially,therearefournonequivalent
satisfy
classes of closed strings that are not contractible, e.g.,
x1, z1, x2, z2 in Fig. 1. The corresponding string mz6mz5mx4mx3my2my1 = 1, (4a)
mo{tapCuteitroeantwCoriertslha{tCASioVCxnx,s1BC,aSFsCz}{azσ1n,1xdS,tCσxhx1zu2,,sσSH2xCz,0z2,σ}w2z}hhaiacvnhedcthoanellssoeafqmutheenecmtolmycgmoimvue-- mz8mz7mmz6z8mmz5z7mmx4x4mmy3y3mmx2y2mmx1y1 == −−−11,, ((44bc))
out the remaining three bases of the ground state sub- mx4mx3mx2mx1 = +1. (4d)
space through {SCxx1|g0i, SCxx2|g0i, SCxx1SCxx2|g0i} [4, 13]. After obtaining this, we next reveal how it produces the
In addition, if one utilizes string operators to move an
contradiction according to LR.
x- (or z-) particle around a z- (or x-)particle one loop,
Noting that Eqs. (3) contain only local operators, the
a global phase factor 1 would be picked up in front of
− operators in each equation thereby commute and can all
the initial wavefunction. This is the unusual statistical
simultaneouslyhavetheireigenvalues. Thus,fromLRwe
property of abelian anyons.
canassociateanelementofrealitytoeachoftheeigenval-
Now, we present our idea for the construction of GHZ
ues in Eqs. (4). For instance, the observers on particles
paradox by using the string operators. We define four
(2,3,4,5,6)measure, without disturbing each other, the
compositestringoperationsdenotedbyD1,2,3,4 byvirtue observables (σz,σz,σx,σx,σy), respectively and if the
of string operators (Fig. 2). For D1, it can be writ- multiplierofth6eir5resu4lts3is12(or 1),thenfromEq.(4a)
ten as D1 = SLz6z51SLxxSLz2z. When it acts on a ground theycanpredictwithcertaintyth−attheresultofmeasur-
state g , anyons will be created, moved and annihi- ing σy will be 1 (or 1). That is, they can predict with
| i 1
lated as follows: First, a pair of z-particles are cre- certainty the v−alue of quantity σy by measuring other
ated by the string operator Sz = σz with Lj denot- 1
L2 2 z particles without disturbing particle 1, and therefore an
z
ing the edge where the j-th qubit crosses the loop Lz elementofrealitycanbeassociatedtothephysicalquan-
as shown in Fig. 2(a); Then a pair of x-particles are tityσy. Analogously,wecanassociateelementsofreality
1
created, moved and annihilated along the loop Lx by to all the physical quantities in Eqs. (3). Then we can
tohpeersattroinrgSoLzp65e1ra=torσS6zLσxx5zσ=1z σm1xoσv4xeσs3xσon2xe; Aotf ltahset,zt-hpearsttircilnegs sinuiptipaollsyethhidadtetnhiisnretshueltorwigaisnsaolmsteahtoewopfrtehdeestyersmteimne.dSauncdh
z
to meet and hence annihilate another one. As a result, predictions with certainty would lead us to assignvalues
we can see from Fig. 2(a) that the loops L and L +1or 1toalltheobservablesinEqs.(3). However,such
x z
−
construct a link. According to the anyonic statistics, a assignment cannot be consistent with rules of quantum
3
which still needs an investigation in the future.
Attheend,letusdiscussbrieflyafeasibleexperimental
implementation of the above consideration. The Kitaev
spin-lattice model could be realized through dynamic
laser manipulation of trapped atoms [6] or molecules [7]
inanopticallattice. Inaddition,anapproachofanyonic
interferometry in atomic systems was as well suggested
recentlybyJiangetal. [8]tomeasuretopologicaldegen-
eracy and anyonic statistics, enabling the measurement
ofthestatisticalphaseassociatedwitharbitrarybraiding
paths. By using this approach, it suffices to implement
theoperationsD1,2,3,4 inEqs. (3)andtodetectthesign.
We briefly introduce this in the following.
Consider a spin lattice of trapped atoms or molecules
inside anopticalcavity (as shownin Fig. 2a ofRef. [8]),
whichprovidesamodelHamiltonianH0andonwhichthe
spins are called memory qubits. Except for the memory
qubits, an additional ancilla spin is needed to probe the
FIG. 2: (Color online) Using string operations to illustrate sign change before ground states and hence is called the
Eqs. (3). (a) The operator D1 acts on a ground state |gi; probe qubit. To achieve controlled-string operations, an
it means that first a pair of z-particles and x-particles are optical cavity associated with the quantum nondemoli-
created at the qubit labeled by 2, then one of the x-particles
tion interaction between the common cavity mode and
is moved along the loop Lx and annihilate with the other selected spins is used to implement, e.g., a z-type string
one, and finally the string operator SLz6z51 = σ6zσ5zσ1z moves operation
one of the z-particles to meet and hence annihilate another
onealongtheloop Lz. (b),(c)TheoperatorsD2 andD3 act Λ[SCz]= 1 A 1 SCz+ 0 A 0 I, (5)
on the ground state respectively and they have the similar | i h |⊗ | i h |⊗
meanings as D1. (d) The operator D4 acts on the ground where the probe qubit is spanned by 0 A, 1 A . It
stateandcontainsonlyoneloopLx,whichshowsthemoving means: If the ancilla spin is in state 0{A|,ino|opier}ation
of x-particles. | i
is applied to the memory qubits; If the ancilla spin is in
state |1iA, the operation SCz is applied to the topological
memory. Forourspin-latticesystemwithH0,weprepare
mechanicsbecauseifwemultiplyEqs.(4a)-(4c)together, its initial state Ψinitial to be a ground state g and
idticwtsillElqea.d(4tdo),.mTx4hmerx3emfox2rem,x1w=e c−on1c,lwudheicthhdatirtehctelyfocuornptrrae-- Dgoju|Ψndinsittiaaltie=g ±is||Ψwihnaittiawliie. nHeeedretothoebsseigrvne.inTfhreonfot|llooifwtinhge
dictions of quantum mechanics given by Eqs. (3) cannot interference|exiperimentcanbe usedto measurethe sign.
bereproducedbyLR.Thiscompletestheconstructionof First,wepreparetheprobequbitinasuperpositionstate
theGHZparadoxinthecontextoftheKitaevspin-lattice (0 + 1 )/√2. We then use controlled-string opera-
A A
model. t|ioinsto|acihieveinterferenceofthefollowingtwopossible
Further, the above GHZ paradox applies to more gen- evolutions: If the probe qubit is in state 0 A, no opera-
| i
eral situations. We can enlarge the loops Lx,z in Fig. 2 tionisappliedtothememoryqubits;Iftheprobequbitis
to generalize these D1,2,3,4 operations. For each set of instate 1 A,theoperationDj isappliedtothetopologi-
D1,2,3,4 when acting on a ground state, it can admit the calmem|oriy,whichpicksuptheextraphasefactoreiθj we
GHZ paradox only if they satisfy all of the following re- want to measure. After the controlled-string operations,
quirements: (i) the loopLx for allof them shouldbe the the probequbitwillbe instate(0 A+eiθj 1 A)/√2. Fi-
| i | i
same,no matterhowlargeareathey enclose;(ii)there is nally, we project the probe qubit to the basis of ξ±
| i ≡
a loop Lz for each of D1,2,3 that should construct a link (0 A eiφ 1 A)/√2 with φ [0,2π), and measure the
when combined with Lx; (iii) when we merge the Lz of o|peirat±or σφ| i ξ+ ξ+ ξ−∈ξ− . The measurement of
D1 with the Lz of D2 together with overlapping edges σφ versus φ≡s|houihld h|a−ve| friinhges| with perfect contrast
vanishing, the resultant loop should be the same as the ahndiamaximumshiftedbyφ=θ for σ =cos(φ θ ).
j φ j
Lz of D3. In this case, the string operators of anyons Inother words,for sigmaoperationsDhj (ij =1,2,3−),the
giveusasimpleyeteffectiveapproachtolookforvarious φshifts ofthe maximal σφ willdiffer fromthoseforD4
sets of D1,2,3,4 operators to construct the GHZ paradox. by π. h i
To sum up, it turns out that the GHZ paradox is very In summary, we have shown the GHZ paradox in the
commonintheKitaev’storiccodemodel. Theall-versus- contextofKitaev’storiccodespin-latticemodelbyusing
nothing violation of LR above well shows the anyonic the anyonic string operations. It shows that the anyonic
statisticsinthemodelasapurequantumeffect. Inaway, statistics in the model cannot be described by LR but
italsoindicatesthattheanyonicstatisticsmaybeatthe be a purely quantum effect. In return, the Kitaev model
conflictive regime between LR and quantum mechanics, provides a potential platform for testing the GHZ para-
4
dox or LR in the future. A feasible experimental consid- Besides, the groundstates of the Kitaev model belong
eration by using the anyonic interferometry is discussed to graphstates, which is crucial in quantum information
to test such a contradiction at the end. It is worth not- application. Bellinequalities havebeenshownto discuss
ing that the measurement employed in the above exper- LR for graph states [17]. Our construction here actually
imental scheme is non-destructive and can be repeated also contributes to the subject.
without disturbing the ground state. It is a predomi-
nant advantage of using Kitaev’s model compared with M. G. H. is indebted to J. Du for enlightenment at
the experimental tests by using multi-photon systems. USTC. This work was supported in part by NSF of
Also recent experiments demonstrated string operations China (Grants No. 10575053 and No. 10605013), Pro-
on small networks of interacting NMR qubits [14] and gram for New Century Excellent Talents in University,
non-interacting optical qubits [15, 16], on the basis of theProject-sponsoredbySRFforROCS,SEM,andLui-
whichitispossibletorealizeourconstructioninadvance Hui Center for Applied Mathematics through the joint
on a small-scale qubit system. project of Nankai and Tianjin Universities.
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