Quantum chaos algorithms and dissipative decoherence with quantum trajectories

0.25 MB·English

by  Jae Weon Lee

Quantum chaos algorithms and dissipative decoherence with quantum trajectories
Jae Weon Lee, and Dima L. Shepelyansky
Laboratoire de Physique Th´eorique, UMR 5152 du CNRS,
Universit´e Paul Sabatier, 31062 Toulouse Cedex 4, France
(Dated: January 21, 2005)
Usingthemethodsofquantumtrajectoriesweinvestigatetheeffectsofdissipativedecoherencein
aquantumcomputeralgorithm simulatingdynamicsinvariousregimesofquantumchaosincluding
dynamicallocalization,quantumergodicregimeandquasi-integrablemotion. Asanexampleweuse
the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum
gates. It is shown that the fidelity of quantum computation decays exponentially with time and
thatthedecayrateisproportional tothenumberofqubits,numberofquantumgatesandpergate
dissipation rate induced by external decoherence. In the limit of strong dissipation the quantum
5 algorithm generates a quantum attractor which may have complex or simple structure. We also
0 compare theeffects of dissipative decoherence with theeffects of static imperfections.
0
2 PACSnumbers: 03.67.Lx,05.45.Mt,03.65.Yz

n
a
Themainfundamentalobstaclesinrealizationofquan- lattice in a magnetic field and the kicked Harper model
J
tum computer [1] are external decoherence and inter- [23]. Thequantumalgorithmforthequantumbakermap
1
nal imperfections. The decoherence is produced by cou- hasbeenimplementedexperimentallywithaNMRbased
2
plings between the quantum computer and the external quantum computer [27].

1 world (see e.g. review [2]). The internal imperfections However till now the quantum chaos algorithms have
v appear due to static one-qubit energy shifts and resid- been used only for investigationof unitary errorseffects.
0 ual couplings between qubits which exist inside the iso- This is always true for internal static imperfections but
2 lated quantum computer. These imperfections may lead the external decoherence generally leads also to dissipa-
1
to emergence of quantum chaos and melting of quan- tiveerrors. Thefirststepintheanalysisofdissipativede-
1
tum computer eigenstates [3, 4]. The effects of unitary coherenceinquantumalgorithmshasbeendonein[28]on
0
quantum errors produced by decoherence and imperfec- arelativelysimpleexampleofentanglementteleportation
5
0 tions on the accuracy of quantum algorithms have been along a quantum register (chain of qubits). After that
/ studied by different groups using numerical modeling thisapproachhasbeenappliedtostudythefidelitydecay
h
of quantum computers performing quantum algorithms inthequantumbakermapalgorithm[29]. In[28,29]the
p
with about 10 20 qubits. The noisy errors in quantum decoherenceisinvestigatedintheMarkovianassumption
-
t gates produced−by external decoherence are analyzed in using the master equationfor the density matrix written
n
a [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] while the errors in the Lindblad form [30]. Already with nq = 10 20
u inducedbyinternalstaticimperfectionsareconsideredin qubits in the Hilbert space of size N = 2n the num−eri-
q
q [17, 18, 19, 20, 21, 22, 23]. The analyticaltreatment[21] cal solution of the exact master equation becomes enor-
:
v basedontherandommatrixtheoryallowstocomparethe mouslycomplicatedduetoalargenumberofvariablesin
i accuracy bounds for these two types of errors for quan- the density matrix which is equal to N2. Therefore the
X
tum algorithms simulating complex quantum dynamics. onlypossibilityfornumericalstudiesatlargen istouse
q
r the method of quantum trajectories for which the num-
a In fact, a convenient frame for investigation of quan-
berofvariablesisreducedtoN withadditionalaveraging
tum errors effects in quantum computations is given by
overmany trajectories. This quantumMonte Carlotype
models of quantum chaos [24]. Such models describe a
method appeared as a result of investigations of open
quantum dynamics which is chaotic in the classicallimit
dissipative quantum systems mainly within the field of
andwhichhasanumberofnontrivialpropertiesincluding
quantum optics but also in the quantum measurement
dynamicallocalization of chaos,quantum ergodicity and
theory (see the original works [31, 32, 33, 34]). More re-
mixing in phase space (see e.g. [24]). It has been shown
centdevelopmentsinthisfieldcanbefindin[35,36,37]).
that for many of such models the quantum computers
In this paper we investigate the effects of dissipative
withn qubits cansimulatethequantumevolutionofan
q decoherence on the accuracy of the quantum sawtooth
exponentially large state (e.g. with N =2nq levels) in a
map algorithm. The system Hamiltonian of the exact
polynomialnumberofelementaryquantumgatesn (e.g.
g map reads [17, 19]
with n = O(n2) or n = O(n3)). The quantum algo-
g q g q
rithmsarenowavailableforthequantumbakermap[25], H (nˆ,θ)=Tnˆ2/2+kV(θ) δ(t m) . (1)
the kicked rotator [26], the quantum sawtooth [17, 19] s X −
m
and tent [21] maps, the kicked wavelet rotator [18], the
quantumdouble-wellmap [10]. Their further generaliza- Here the first term describes free particle rotation on
tion and development gave quantum algorithms for the a ring while the second term gives kicks periodic in
Anderson metal-insulator transition [20], electrons on a time and nˆ = i∂/∂θ. The kick potential is V(θ) =
−
2
(θ π)2/2 for 0 θ < 2π. It is periodically re-
− − ≤
peatedforallotherθ sothatthe wavefunction ψ(θ)sat- -2
isfies the periodic boundary condition ψ(θ)=ψ(θ+2π).
The classical limit corresponds to T 0, k with
ln W
→ → ∞
K = kT = const. In these notations the Planck con- n
stantis assumedto be ~=1 while T playsthe roleof an
effective dimensionless Planck constant.
-4
The classical dynamics is described by a symplectic
area-preservingmap
n=n+k(θ π), θ =θ+Tn. (2)
−
Using the rescaledmomentum variable p=Tn it is easy -6
to see that the dynamics depends only on the chaos pa-
rameter K = kT. The motion is stable for 4 < K < 0 -40 -20 0 20 40
−
and completely chaotic for K < 4 and K > 0 (see [17] n
−
and Refs. therein). The map (2) can be studied on the
cylinder (p ( ,+ )), which can also be closed to
∈ −∞ ∞ FIG. 1: Probability distribution Wn over momentum eigen-
form a torus of length 2πL, where L is an integer. statesninthequantumsawtoothmap(3)attimet=30. The
The quantum propagationon one map iteration is de- quantum evolution is simulated by the quantum algorithm
scribedbyaunitaryoperatorUˆ actingonthewavefunc- with nq = 6 qubits in presence of dissipative decoherence.
tion ψ: The dissipation rate per gate is Γ = 0.001 and the map pa-
rametersarek=√3,K =√2withthetotalnumberofstates
ψ =Uˆψ =Uˆ Uˆ ψ =e−iTnˆ2/2e−ikV(θ)ψ . (3) N =2nq =64. Thefullcurverepresentstheexactsolutionof
T k
theLindblad equation (4). Symbolsshow theresult of quan-
tumtrajectoriescomputationwiththenumberoftrajectories
ThequantumevolutionisconsideredonN quantummo-
M = 20(+), M = 50(o), M = 200(x), M = 1000( ). The
mentum levels. For N = 2nq this evolution can be im- △
initial state is n=0. Thelogarithm is natural.
plemented on a quantum computer with n qubits. The
q
quantum algorithm described in [17] performs one itera-
tionofthequantummap(3)inn =3n2+n elementary
g q q This evolution of ρ can be efficiently simulated by av-
quantum gates. It essentially uses the quantum Fourier
eraging over the M quantum trajectories which evolve
transform which allows to go from momentum to phase
accordingtothefollowingstochasticdifferentialequation
representation in n (n + 1)/2 gates. The rotation of
q q for states ψα (α=1, ,M):
quantum phases in each representation is performed in | i ···
approximately n2 gates. Here we consider the case of
q 1
oneclassicalcell(toruswithL=1whenT =2π/N)[17] |dψαi=−iHs|ψαidt+ 2X(hL†mLmiψ (5)
and the case of dynamical localization with N levels on m
a torus and K 1, k = K/T 1 [19]. Here and below
the time t is me∼asured in numb∼er of map iterations. L† L )ψα dt+  Lm 1 ψα dN ,
To study the effects of dissipative decoherence on the − m m | i Xm q L†mLm ψ − | i m
accuracy of the quantum sawtooth algorithm we follow h i
the approach with the amplitude damping channelused where represents an expectation value on ψα and
ψ
in [29]. The evolution of the density operator ρ(t) of hi | i
dN are stochastic differential variables defined in the
m
open system under weak Markovian noises is given by
same way as in [29] (see Eq.(10) there). The above
the master equation with Lindblad operators L (m =
m equation can be solved numerically by the quantum
1, ,nq): Monte Carlo (MC) methods by letting the state ψα
···
| i
jump to one of L ψα /L ψα states with prob-
ρ˙ =−~i[Heffρ−ρHe†ff]+Xm LmρL†m , (4) aiHbielfitfydt/d~p)m|ψα≡i/p|L1mm−||ψPαiim|2|ddptmm|[29w]iit|hor pervooblavbeilittoy (11 −−
dp . Then,thedensitymatrixcanbeapproximately
where the system Hamiltonian Hs is related to the ef- Pexpmressmed as
fective Hamiltonian H H i~/2 L† L and
eff ≡ s − Pm m m
m marks the qubit number. In this paper we assume
M
1
that the system is coupled to the environment through ρ(t) ψ(t) ψ(t) = ψα(t) ψα(t) , (6)
anamplitude dampingchannelwithL =aˆ √Γ,where ≈h| ih |iM M X| ih |
m m α=1
aˆ is the destruction operator for m th qubit and the
m
−
dimensionless rate Γ gives the decay rate for each qubit where representsanensemble averageoverM quan-
M
per one quantum gate. The rate Γ is the same for all tum trhaijectories ψα(t) . Hence, an expectation value of
| i
qubits. an operator O is given by O =Tr(Oρ) O .
M
h i ≈h i
3
0
10
-2
10
-4
f10 γ
0
10
-6
10 -1
10
Γ
eff
-2
-8 10
10 -1 0 1
10 10 10
20 40 60 80 100
t
FIG.3: Fidelityf asafunctionofiterationtimet. Theupper
two curves are for Γ = 0.0005 and the lower two curves are
for Γ = 0.001. Here M = 50,nq = 8, k = 2nqK/2π and
K = 0.5(fullcurves)K =0.5 (dottedcurves),respectively.
−
Theinitialstateis n=0 . Theinsetshowsthefidelitydecay
| i
rate γ as a function of Γeff nqngΓ with nq =4,6,8. Here
≡
K = 0.5 (+) and K = 0.5( ), respectively. The straight
− △
line is thebest fit γ =0.08 Γ .
eff
tories we compare its results with the exact solution of
the Lindblad equation for the density matrix ρ (4). The
comparison is done for the case of dynamical localiza-
tion of quantum chaos and is shown in Fig. 1. It shows
FIG. 2: (Color online) First top row: classical phase space
that the dynamical localizationis preservedat relatively
distribution obtained from the classical sawtooth map with
weak dissipation rate Γ. It also shows that the quan-
a Gaussian averaging over a quantum cell (N = 256 quan-
tum cells inside whole classical area; see text). Second row: tum trajectories method correctly reproduces the exact
thecorrespondingHusimifunctionforthequantumsawtooth solutionofthe Lindbladequationandthatitissufficient
mapatnq =8andΓ=0. Thirdrow: theHusimifunctionob- to use M = 50 trajectories to reproduce correctly the
tainedwithM =50quantumtrajectoriesinpresenceofdissi- phenomenonofdynamicallocalizationinpresenceofdis-
pative decoherence with rate Γ=0.0005 and nq =8. Fourth sipative decoherence of qubits.
bottomrow: sameasforthethirdrowbutwithnq =10. Here To analyze the effects of dissipation rate Γ in a more
K = 0.5, T = 2π/N corresponding to L = 1 and N = 2nq
− quantitative way we start from the quasi-integrable case
quantum states in the whole classical area. Columns show
K = 0.5withoneclassicalcellL=1(T =2π/N). The
distributions averaged in the time intervals: 0 t 9 (left), −
≤ ≤ classical phase space distribution, averaged over a time
40 t 49(middle),90 t 99(right). Theinitialstateis
≤ ≤ ≤ ≤ intervalanda Gaussiandistributionovera quantumcell
n 0.1N. Color represents the density from blue/black (0)
to≈red/gray (maximal value). withaneffectivePlanckconstant,isshowninFig.2inthe
firsttoprow(thereareN =2nq quantumcellsinsidethe
whole classical phase space). Such Gaussian averaging
For the quantum sawtooth algorithm the dissipative ofthe classicaldistributiongivesthe resultwhichis very
noise is introduced in the quantum trajectory context close to the Husimi function in the corresponding quan-
(Eq. (6)) after each elementary quantum gate and cal- tum case at Γ=0 (Fig. 2, second row). We remind that
culated by the MC methods. The same physical process the Husimi function is obtainedbya Gaussianaveraging
can be described by density matrix theory. The evolu- oftheWignerfunctionoveraquantum~cell(see[38]for
tion of density matrix after single iteration of quantum details). In our case the Husimi function h(θ,n) is com-
sawtooth map is described by putedthroughthewavefunctionofeachquantumtrajec-
toryandafterthat itis averagedoverallM trajectories.
ρ′ =U U ρU†U† . (7) The effect of dissipative decoherence with Γ = 0.0005 is
k T T k
shown in the third row of Fig. 2. At Γ = 0 the phase
Toincludethedissipativenoiseeffectsthedensitymatrix space distribution remains approximately stationary in
furtherevolvesaccordingtoEq. (4)withH =0between time while for Γ > 0 it starts to spread so that at large
s
consecutive quantum gates composing U and U . times the typical structure of the classical phase space
k T
Totesttheaccuracyofthemethodofquantumtrajec- becomes completely washed out. This destructive pro-
4
102 101
ξ ξ/ξ
0
1
10
0
10
-4 -3 -2 -1
0 1 2 10 10 10 10
10 10 10 Γ
t
FIG. 5: (Color online) Dependence of the ratio of ξ to its
FIG. 4: (Color online) Dependence of the IPR ξ on time t valueξ0 intheidealalgorithm onthedissipativedecoherence
at Γ = 0.001 and M = 50 shown by full curves for nq = 4 rate Γ for nq = 4 (green/gray o), nq = 6 (blue/black x),
(green/gray curve),nq =6 (blue/black curve), nq =8 (black nq =8(black+)frombottomtotop. Thevaluesξ andξ0 are
curve), bottom to top respectively. The dashed curves show averaged in the time interval 30 t 40. Other parameters
≤ ≤
the same cases at Γ=0 (bottom dashed curve is for nq =4, are as in Fig. 4.
top dashed curve is for nq = 6 and nq = 8 where ξ values
arepracticallyidentical). Heretheinitialstateis n=0 and
k=√3, K =√2. | i bakermap. This shows that the dependence (9) is really
universal. Its physical origin is rather simple. After one
gate the probability of a qubit to stay in upper state
cessbecomesmorerapidwiththeincreaseofthenumber dropsbyafactorexp( Γ)foreachqubit(weremindthat
of qubits even if Γ remains fixed (Fig. 2, fourth bottom Γisdefinedasaperg−atedecayrate). Thewavefunction
row). One of the reasons is that Γ is defined as a rate ofthetotalsystemisgivenbyaproductofwavefunctions
per gate and the number of gates ng = 3n2q +nq grows of individual qubits that leads to the fidelity drop by a
with nq. However,this is notthe only reasonas it shows factor exp( CnqΓ) after one gate and exp( CnqngΓ)
the analysis of the fidelity decay. after n ga−tes leading to the relation (9). In−principle,
g
Thefidelityf ofquantumalgorithminpresenceofdis- one may expect that the decay of f(t) is sensitive to a
sipative decoherence is defined as number of qubit up states in a given wavefunction since
there is no decay for qubit down states. However, in
1
f(t) ψ (t)ρ(t)ψ (t) ψ (t)ψα(t) 2 , (8) a context of a concrete algorithm this number varies in
≡h 0 | | 0 i≈ M X|h 0 | Γ i|
time and only its averagevalue contributes to the global
α
fidelity decay.
where ψ (t) is the wave function given by the exact
0 The result (9) gives the time scale t of reliable quan-
| i f
algorithm and ρ(t) is the density matrix of the quantum tum computation in presence of dissipative decoherence.
computerinpresenceofdecoherence,botharetakenafter
On this scale the fidelity should be close to unity (e.g.
t map iterations. Here, ρ(t) is expressed approximately f =0.9) that gives
throughthesumoverquantumtrajectories(seealso[29]).
The dependence of fidelity f(t) on time t is shown in t 1/(n n Γ); N =1/(n Γ). (10)
f q g g q
≈
Fig. 3. At relatively short time t < 50 the decay is ap-
Here N = n t is the total number of quantum gates
proximately exponential f(t) exp( γt). The decay g g
≈ − which can be performed with high fidelity (f > 0.9) at
rate γ is described by the relation
given n and Γ. The comparison with the results ob-
q
γ =CΓ =Cn n Γ, (9) tained for static imperfections [21] of strength ǫ shows
eff q g
that for them N drops more rapidly with n : N
g q g
∼
whereC =0.08isanumericalconstant(seeFig.3inset). 1/(ǫ2n n ). Therefore the static imperfections destroy
q g
The important result of Fig. 3 is that the decay of f(t) the accuracy of quantum computation in a more rapid
is notverysensitivetothe mapparameters. Indeed,it is way compared to dissipative decoherence.
notaffected by a changeofK which significantlymodify Itisalsointerestingtoanalyzetheeffectsofdissipative
the classical dynamics which is quasi-integrable at K = decoherence on the dynamical localization. For that, in
0.5 and fully chaotic at K = 0.5. Another important addition to the probability distribution as in Fig. 1, it
−
result is that up to a numericalconstant the relation(9) is convenientto use the inverseparticipationratio(IPR)
follows the dependence found in [29] for the quantum defined as ξ = 1/ ψ 4 1/ ψ 2 2 where
Pn| n| ≃ Pn|h| n| iM|
5
introduces some noise which destroys localization.
However, there is also another effect which becomes
visible at relatively large Γ. It is shown in Fig. 5 which
gives the ratio of ξ to its value ξ in the ideal algorithm.
0
Thus, at small Γ the ration ξ/ξ growswith the increase
0
ofΓwhileitstarstodropatlargeΓ. Thisisamanifesta-
tion of the fact that in absence of algorithm the dissipa-
tiondrivesthequantumregistertothestate n=0 with
| i
all qubits in down state. Even in presence of the quan-
tum algorithm this dissipative effect becomes dominant
at large Γ leading to a decrease of the ratio ξ/ξ .
0
The dissipative effect of decoherence at large values of
Γ is also clearly seen in the case of quantum chaos er-
godicinoneclassicalcell(L=1). AtlargeΓ the Husimi
distribution relaxes to the stationary state induced by
dissipation n=0 (third row in Fig. 6 at Γ=0.1). In a
| i
sense this correspondsto a simple attractor in the phase
space. The stationary state becomes more complicated
with a decrease of Γ (second row in Fig. 6 at Γ = 0.05).
And at even smaller Γ=0.01 the stationary state shows
a complex structure in the phase space (top first row in
Fig.6). Itis importanttostressthatthisstructureisin-
dependent of the initial state (bottom row in Fig. 6). In
this sense we may say that in such a case the dissipative
decoherence leads to appearance of a quantum strange
attractor in the quantum algorithm. Of course, this sta-
tionaryquantumattractorstateisverydifferentfromthe
Husimi distribution generated by the ideal quantum al-
gorithm. However, it may be of certain interest to use
the dissipative decoherence in quantum algorithms for
FIG.6: (Coloronline)EachpanelshowstheHusimidistribu- investigation of quantum strange attractors which have
tion for the quantum sawtooth map algorithm with K = 1, been discussed in the context of quantum chaos and dis-
T = 2π/2nq and nq =8. The top three rows show the cases sipation (see e.g. Refs. [39, 40, 41]). At the same time
with the rate Γ = 0.01, Γ = 0.05, and Γ = 0.1, respectively
we should note that the dissipation induced by decoher-
from top to third row. The initial state is n = 60 and
| i ence acts during each gate that makes its effect rather
M = 50. The distribution is averaged in the time interval
nontrivial due to change of representations in the map
0 t 9 (left column), 40 t 49 (middle column),
≤ ≤ ≤ ≤ (3).
90 t 99 (right column). The fourth bottom row shows
≤ ≤
the distribution for another initial state n = 0 averaged In conclusion, our studies determine the fidelity de-
| i
in the time interval 90 t 99 for Γ = 0.01 (left panel), cay law in presence of dissipative decoherence which is
≤ ≤
Γ=0.05(middlepanel),andΓ=0.1(rightpanel)(compare
in agreement with the results obtained in [29] for a very
with the right column of top three rows). Color represents
different quantum algorithm. This confirms the univer-
thedensityfromblue/black(0)tored/gray(maximalvalue).
sal nature of the established fidelity decay law. These
studies also show that at moderate strength the dissipa-
tive decoherence destroys dynamical localization while a
notestheaverageoverM quantumtrajectories.
M
h|···|i strong dissipation leads to localization and appearance
Thisquantityisoftenusedintheproblemswithlocalized
complex or simple attractor. The effects of dissipative
wave functions. In fact ξ gives an effective number of
decoherence are compared with the effects of static im-
statesoverwhichthetotalprobabilityisdistributed. The
perfections and it is shown that in absence of quantum
dependence of ξ on time t is shown in Fig. 4. It shows
error correction the later give more restrictions on the
thatinpresenceofdissipativedecoherencethedynamical
accuracy of quantum computations with a large number
localization is destroyed. Indeed, at large t the value
of qubits.
of ξ grows with n while for the ideal algorithm it is
q
independent of n . The physical meaning of this effect This work was supported in part by the EU IST-FET
q
is rather clear. As in Fig. 2 the dissipative decoherence project EDIQIP.
[1] M.A. Nielsen and I.L. Chuang Quantum Computa- Cambridge (2000).
tion and Quantum Information,Cambridge Univ.Press,
6
[2] W.H.Zurek, Rev.Mod. Phys. 75, 715 (2003). [24] B. V. Chirikov, in Chaos and Quantum Physics, Les
[3] B. Georgeot, and D. L. Shepelyansky, Phys. Rev. E 62, HouchesLectureSeries52Eds.M.-J.Giannoni,A.Voros,
3504 (2000); ibid. 62, 6366 (2000). and J. Zinn-Justin (North-Holland, Amsterdam, 1991),
[4] D.L. Shepelyansky,Physica Scripta T90, 112 (2001). p.443; F. M. Izrailev, Phys.Rep. 196, 299 (1990).
[5] J.I.CiracandP.Zoller,Phys.Rev.Lett.74,4091(1995). [25] R. Schack,Phys. Rev.A 57, 1634 (1998).
[6] C. Miquel, J. P. Paz and R. Perazzo, Phys. Rev. A 54, [26] B. Georgeot, and D.L. Shepelyansky, Phys. Rev. Lett.
2605 (1996). 86, 2890 (2001).
[7] C. Miquel, J.P. Paz and W.H. Zurek, Phys. Rev. Lett. [27] Y.S. Weinstein, S. Lloyd, J. Emerson, and D.G. Cory,
78, 3971 (1997). Phys. Rev.Lett. 89, 157902 (2002).
[8] P.H.Song and D.L.Shepelyansky, Phys. Rev. Lett. 86, [28] G.G. Carlo, G. Benenti and G. Casati, Phys. Rev. Lett.
2162 (2001). 91, 257903 (2003).
[9] B.GeorgeotandD.L.Shepelyansky,Phys.Rev.Lett.86, [29] G.G. Carlo, G. Benenti, G. Casati and C. Mejia-
5393 (2001); ibid. 88, 219802 (2002). Monasterio, Phys. Rev.A 69, 062317 (2004).
[10] A.D. Chepelianskii, and D.L. Shepelyansky, Phys. Rev. [30] G. Lindblad, Commun. Math. Phys. 48, 119 (1976);
A 66, 054301 (2002). Gorini,A.Kossakowski,andE.C.G.Sudarshan,J.Math.
[11] D.Braun, Phys. Rev.A 65, 042317 (2002). Phys. 17, 821 (1976).
[12] M. Terraneo, B. Georgeot and D.L. Shepelyansky, Eur. [31] H.J.Carmichel,AnOpenSystemsApproach toQuantum
Phys.J. D 22, 127 (2003). Optics (Springer,Berlin, 1993).
[13] P.H.Song and I. Kim, Eur. Phys.J. D 23, 299 (2003). [32] R.Dum,P.Zoller,andH.Ritsch,Phys.Rev.A45,4879
[14] B.L´evi,B.Georgeot andD.L.Shepelyansky,Phys.Rev. (1992).
E 67, 046220 (2003). [33] J.Dalibard,Y.Castin,andK.Mølmer,Phys.Rev.Lett.
[15] S. Bettelli and D.L. Shepelyansky, Phys. Rev. A 67, 68, 580 (1992).
054303 (2003). [34] N. Gisin, Phys.Rev.Lett. 52, 1657 (1984).
[16] S.Bettelli, Phys.Rev.A 69, 042310 (2004). [35] R. Schack, T.A. Brun, and I.C. Percival, J. Phys. A 28,
[17] G. Benenti, G. Casati, S. Montangero and D. L. Shep- 5401 (1995).
elyansky,Phys.Rev.Lett. 87, 227901 (2001). [36] A. Barenco, T.A. Brun, R. Schack, and T.P. Spiller,
[18] M.TerraneoandD.L.Shepelyansky,Phys.Rev.Lett.90, Phys. Rev.A 56, 1177 (1997).
257902 (2003). [37] T.A. Brun, Am. J. Phys. 70, 719 (2002);
[19] G. Benenti, G. Casati, S. Montangero and D. L. Shep- quant-ph/0301046.
elyansky,Phys.Rev.A 67, 52312 (2003). [38] S.-J. Chang and K.-J. Shi, Phys.Rev.A 34, 7 (1986).
[20] A.A. Pomeransky, D.L. Shepelyansky,Phys. Rev. A 69, [39] T. Dittrich and R. Graham, Annals of Physics 200, 363
014302 (2004). (1990).
[21] K. Frahm, R.Fleckinger and D.L.Shepelyansky, Eur. [40] G. Casati, G. Maspero and D.L. Shepelyansky, Physica
Phys.J. D 29, 139 (2004). D 131, 311 (1999).
[22] A.A. Pomeransky, O.V.Zhirov and D.L. Shepelaynsky, [41] G. Carlo, G. Benenti, G. Casati and D.L. Shepelyansky,
Eur. Phys.J. D 31, 131 (2004). cond-mat/0407702.
[23] B.L´eviandB.Georgeot,Phys.Rev.E70,056218(2004).