Unification of Dynamical Decoupling and the Quantum Zeno Effect

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by  P. Facchi

Unification of Dynamical Decoupling and the Quantum Zeno Effect
P. Facchi,1 D.A. Lidar,2 and S. Pascazio1
1Dipartimento di Fisica, Universit`a di Bari I-70126 Bari, Italy,
and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy
2Chemical Physics Theory Group, Chemistry Department, University of Toronto,
80 St. George Street, Toronto, Ontario M5S 3H6, Canada
(Dated: February 1, 2008)
WeunifythequantumZenoeffect(QZE)andthe“bang-bang”(BB)decouplingmethodforsup-
pressingdecoherenceinopenquantumsystems: inbothcasesstrongcouplingtoanexternalsystem
orapparatusinducesadynamicalsuperselectionrulethatpartitionstheopensystem’sHilbertspace
into quantum Zeno subspaces. Our unification makes use of von Neumann’s ergodic theorem and
avoids making any of the symmetry assumptions usually made in discussions of BB. Thus we are
4 able to generalize BB to arbitrary fast and strong pulse sequences, requiring no symmetry, and to
0 show the existence of two alternatives to pulsed BB: continuous decoupling, and pulsed measure-
0 ments. Our unified treatment enables us to derive limits on the efficacy of the BB method: we
2
explicitly show that the inverse QZE implies that BB can in some cases accelerate, rather than

n inhibit,decoherence.
a
J PACSnumbers: 03.67.Pp,03.65.Xp,03.65.Yz,03.67.Lx

1
2
I. INTRODUCTION from the same physical considerations, and hence can

2 be unified under the same conceptual and formal frame-
v work. Furthermore, they appear as particular cases of a
Recentyearshavewitnessedasurgeofinterestinways
2 more general dynamics in which the system of interest
toprotectquantumcoherence,drivenmostlybydevelop-
3 is “strongly”coupledto anexternalsystemthat(loosely
1 ments in the theory of quantum information processing
speaking) plays the role of a measuring apparatus.
3 [1]. A number of promising strategies for combatting
We use these insights to (i) generalize the BB method
0 decoherence have been conceived and in some cases ex-
3 perimentally tested, including quantum error correcting to pulse sequences with no symmetry; (ii) to point out
0 thattheBBpulsescanhavetheoppositefromthedesired
codes and topological codes (for a review see [2]), deco-
/ effect(asituationwellknownfromtheQZEliteratureas
h herence free subspaces and (noiseless) subsystems (for a
the “inverse”or“anti”Zeno effect) [14, 15]; (iii) to show
p reviewsee[3]),and“bang-bang”(BB)decoupling[4,5,6]
- (for an overview see [7]). Two recent papers have shown that alternatives to the unitary pulse controlscheme are
t availabletosuppressthesystem-environmentinteraction,
n thatthesevariousmethodscanbeunifiedunderageneral
a algebraic framework [8]. Here, using a very different ap- namely: a)continuous unitaryinteraction,andb)pulsed
u measurements.
proach,we continue this development for BB decoupling
q
and the quantum Zeno effect (QZE).
:
v
The idea behind BB is that the application of suffi-
i II. SIMPLEST BB CYCLE
X ciently strong and fast pulses, with appropriate symme-
r try (notions we make precise later), when applied to a
a system,candecoupleitfromitsdecoheringenvironment. Consider the “BB-evolution” induced by the two-
element control set (not necessarily a group) {I,U },
The notion of a strong and fast interaction with a quan- 1
where I is the identity operator, in which the controlled
tum system is also the key idea behind the QZE [9] (for
system Q alternately undergoes N “kicks” U (instanta-
reviews see [10, 11]). The standard view of the QZE 1
neous unitary transformations) and free evolutions in a
effect is that by performing frequent projectivemeasure-
time interval t
mentsonecanfreezetheevolutionofaquantumstate(“a
watched pot cannot boil”). However, recently it has be- U (t)=[U U(t/N)]N. (1)
N 1
comeclearthatthisviewoftheQZEistoonarrow,intwo
main respects: (i) The projective measurements can be WetakeU =exp(−iHt),withH the(time-independent)
replacedbyanotherquantumsysteminteractingstrongly Hamiltonian ofQ, its environmentand their interaction,
with the principal system [11, 12]; (ii) The states of the andwillsometimesabbreviateU(t/N)byU. Wepresent
principal system need not be frozen: instead the gen- anewderivationofthis “BB-evolution”thatallowsfora
eral situation is one of dynamically generated quantum transparent connection to the formulation of the QZE.
Zeno subspaces, in which non-trivial coherent evolution In the large N limit, the dominant contribution to
can take place [13]. It is therefore not only physically U (t) is UN. We therefore consider the sequence of uni-
N 1
reasonable,but also logicallyappealing to view the QZE tary operators
as a dynamical effect: in this broader context, both BB
decoupling and the QZE can be understood as arising V (t)=U†NU (t). (2)
N 1 N
2
Observe that V (0)=I for any N and decomposition (9). Here is the proof. For any vector ψ
N
in the Hilbert space H, we get, using Eq. (9)
N−1
d dU
i V (t) = U†N (U U)k U i (U U)N−k−1
dt N 1 1 1 dt 1 1 N−1 1 N−1
k=0 (cid:18) (cid:19) U†kHUkP ψ = U˜kφ, (13)
X N 1 1 µ N
N−1
= U†N 1 (U U)kU HU†(U U)†k(U U)N Xk=0 Xk=0
1 N 1 1 1 1 1
kX=0 where U˜ = (U1eiλµ)† is a unitary operator whose eigen-
= H (t)V (t), (V (0)=I) (3) projection P has eigenvalue 1 and φ = HP ψ ∈ H.
N N N µ µ
Recallnowanergodictheoremdue tovonNeumann[16,
with p. 57] that states that if U˜ is a unitary operator on the
1 N−1 Hilbert space H and Pµ its eigenprojection with eigen-
H (t)= U†N(U U)kU HU†(U U)†kUN. (4) value 1 (U˜P =P ), then for any φ∈H
N N 1 1 1 1 1 1 µ µ
k=0
X
N−1
The limiting evolution operator lim 1 U˜kφ=P φ. (14)
µ
N→∞N
U(t)≡ lim VN(t) (5) kX=0
N→∞
Asaconsequence,bytakingthelimitof(13),weget(12).
satisfies the equation
Notice that the intertwining property (12) holds also
d foranunboundedH whosedomainD containstherange
idtU(t)=HZU(t), (U(0)=1) (6) of Pµ, namely PµH ⊂ D(H). For a generic unbounded
Hamiltonian, we can still formally consider (10) as the
with the “Zeno” Hamiltonian limiting evolution, but the meaning of P HP and its
µ µ
domain of selfadjointness should be properly analyzed.
H ≡ lim H (t). (7)
Z N In conclusion
N→∞
Therefore U(t) = exp(−iHZt). In order to study the U(t)=exp(−iHZt)=exp[−i PµHPµt] (15)
behavior of the limiting operator we first observe that
µ
for N →∞ we can neglect the free evolution U(t/N) in X
Eq. (4) and so and, due to Eqs. (2) and (5),
1 N−1 1 N−1 U (t) ∼ UNU =UNexp(−iH t)
H ∼ U†NUk+1HU†k+1UN = U†kHUk. N 1 1 Z
N N 1 1 1 1 N 1 1
= exp[−i (Nλ P +P HP t)]. (16)
k=0 k=0 µ µ µ µ
X X
(8) µ
X
Next we will show that for any bounded H and any U
1
with a pure point spectrum, namely This proves that the “BB-evolution” (1) yields a Zeno
effect and a partitioning of the Hilbert space into “Zeno
U = e−iλµP (9) subspaces”, in the sense of [13].
1 µ
µ We emphasize that no cyclic group properties are re-
X
quired for pulse sequences. This extends previous stud-
[λ 6=λ (mod 2π) for µ6=ν, P P =δ P ], one gets
µ ν µ ν µν µ ies, in which “symmetrization” was thought to play an
important role in order to obtain decoupling and sup-
N−1
1
H = lim U†kHUk = P HP ≡Π (H), pression of decoherence [24]. Indeed the dynamics (1) is
Z N→∞N 1 1 µ µ U1 different from the dynamics [U†U(t/2N)U U(t/2N)]N,
k=0 µ 1 1
X X
(10) originally proposed in [4], because it is only constructed
wherethe mapΠ isthe projectionontothe centralizer with a single “bang” U , without the second “bang” U†
U1 1 1
(or commutant) of U , which would close the group. We will further elaborate
1
on this issue in Sec. IV.
Z(U )={X|[X,U ]=0}. (11)
1 1 Bytaking H tobe asystem-bathinteractionHamilto-
nian,weseethattheeffectoftheU “kicks”istoproject
First we show that the (strong) limit H in Eq. (10) 1
Z the decohering evolution into disjoint subspaces defined
is a bounded operator which satisfies the intertwining
by the spectral resolution of U . A proper choice of U
property 1 1
can either eliminate this evolution or make it proceed
H P =P HP =P H (12) in some desired fashion. To give the simplest possible
Z µ µ µ µ Z
example, suppose
for any eigenprojection Pµ of U1, with eigenvalue e−iλµ.
Equation (10) follows whenever U admits the spectral H =σ ⊗B, U =σ . (17)
1 x 1 z
3
H generates“bit-flips”andthe projectionoperatorsare n ≥ 2 (n = 2 is typical of quantum dots [17]). The free
decay rate is
1
P = (I±σ ) (18)
± 2 z γ =2πκ(ω0), (23)
with eigenvalues λ =±1. Thus ω being the energy difference between the two qubit
± 0
states (Fermi golden rule). The modified decay rate can
H = P HP = P σ P ⊗B =0, (19) be shown to read [4, 18]
Z µ µ µ x µ
µX=± µX=± ∞
γ(τ)= lim t dω κ(ω)
so the decohering evolution is completely cancelled. t→∞ Z−∞
The physical mechanism giving rise to the Zeno sub- ω−ω ω−ω
spacesintheN →∞limitcanbeunderstoodbyconsid- ×sinc2 0t tan2 0τ , (24)
2 4
eringthecaseofafinitedimensionalHilbertspace. Then (cid:18) (cid:19) (cid:18) (cid:19)
the limit (10) reads whereτ =t/N istheperiodbetweenkicksandsinc(x)≡
x−1sinx. By expanding for large values of N one gets
1 N−1 1 N−1 [19]
U†kHUk = P HP eik(λµ−λν) (20)
N 1 1 µ νN
k=0 µ,ν k=0 8 2π
X X X γ(τ)∼ κ , τ →0 . (25)
π τ
andoneseesthatthelastsumis1forµ=ν andvanishes (cid:18) (cid:19)
asO(1/N)otherwise. [rememberthatλµ 6=λν (mod2π) Notice that, according to (25), for small values of τ the
for µ 6= ν in Eq. (9)]. The appearance of the Zeno sub- modified decay rate γ(τ) is proportional to the “tail” of
spaces is thus a direct consequence of the fast oscillating the spectral density κ(ω). By defining a characteristic
phases between different eigenspaces of the kick. This is transition time τ∗, solution of the equation
equivalentto aprocedureofphaserandomization,andis
analogousto the case of strong continuous coupling [13]. 2π π π2
κ ≃ γ = κ(ω ), (26)
τ∗ 8 4 0
(cid:18) (cid:19)
one obtains
III. IMPLICATIONS OF INVERSE ZENO
EFFECT
γ(τ)<γ for τ <τ∗,
γ(τ)>γ for τ >τ∗. (27)
The above conclusions are correct in the (mathemat-
ical) limit of large N. However it is known that, if N
Decoherence is suppressed in the former case, but it is
is not too large, the form factors of the interaction play enhanced in the latter situation (which is analogous to
a primary role and can provoke an inverse Zeno effect
what one calls IZE in the case of projective measure-
(IZE), by which the decohering evolution is accelerated,
ments). This shows that an “inverse Zeno regime” is a
rather than suppressed [14, 15]. Reconsider the example
serious drawback also in the case of dynamical decou-
(17),withB couplingQtoagenericbathwithathermal pling. Since the limit τ < τ∗ can be very difficult to
spectral density attain, for a bona fide dissipative system, the efficacy of
BB as a method for decoherence suppression must be
κ(ω)= dtexp(iωt)hB(t)Bi, (21) carefully analyzed. For instance, in the Ohmic case (22)
Z at low temperature T ≪ ω0 ≪ ωc, one easily gets from
(25)
where B(t) = eiHBtBe−iHBt is the interaction-picture
evolved bath operator, H the free bath Hamiltonian 1
B π2ω 2n−1
and h...i the average over the bath state. For in- τ∗ ≃2πω−1 0 ≪2πω−1, (28)
c 4 ω c
stance, one can consider the linear coupling B = (cid:18) c(cid:19)
dω f(ω) a(ω)+a†(ω) , where [a(ω),a†(ω′)] = δ(ω −
aconditionthatmaybedifficulttoachieveinpractice. In
ω′) are boson operators and f(ω) a form factor, while
fact,weseeherethattherelevanttimescaleisnotsimply
RHB = dω(cid:0)ωa†(ω)a(ω).(cid:1)The form factor of the interac- the inverse bandwidth ω−1, but can be much shorter if
tion (together with the bath state) determines the spec- c
ω ≪ω ,asistypicallythe case. Ithasalreadybeenob-
R 0 c
tral density (21). For instance, for an Ohmic bath,
served that the Ohmic bath is a particularly demanding
setting for BB, and that spin-boson baths with decay-
ω ω
κ(ω)∝ 1+(ω/ωc)2 n coth(cid:16)2T(cid:17), (22) itnhgermspaelctsrpaelcdtreanlsidtyenIsi(tωy)κ[n(ωot)],tosubcehcaosnf1u/sfed, awreithmtohree
(cid:16) (cid:17) amenable to successful BB decoupling [18]. We will re-
where ω is the frequency cutoff, T the temperature of consider this issue from the point of view of the IZE in
c
the bath (Boltzmann’s constant k =1) and n an integer [19].
4
IV. BB CYCLE OF SEVERAL PULSES In conclusion,
U (t) ∼ (U ···U )NU(t)=(U ···U )Nexp(−iH t)
We now generalize the previous result to the situation N g 1 g 1 Z
whereeachcycleconsistsofg kicks. Thiswillallowusto
= exp −i (Nλ P +P H¯P t) . (38)
show how the procedure of “decoupling by symmetriza- µ µ µ µ
!
tion” [6], i.e., the standard view of the BB effect, arises Xµ
asaspecialcaseofsuchcyclesandisrelatedtotheQZE.
It is clear that also in this case we get a QZE, with rel-
WeconsiderN cyclesofg instantaneouskicksU ,...,U
1 g evant Zeno subspaces [13]. The only difference from the
in a time interval t single-kickcaseisthatthe HamiltonianH¯ [Eq.(37)]and
t t t N the product of the cycle Ug···U2U1 [Eq. (36)] take the
U (t)= U U ···U U U U . place of H and U , respectively.
N g 2 1 1
gN gN gN
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) It is important to observe again that no symmetry or
(29)
groupstructureisrequiredfromthe“kick”sequence(29):
Weusethesamenotationasabove,sometimesabbreviat-
theaboveformulasareofgeneralvalidity,astheyrelyon
ing U(t/gN)byU,unless confusionmayarise. Similarly
the von Neumann ergodic theorem. They reduce to the
to the single-kick case, in the N → ∞ limit, the domi-
usual expression in the case of a finite closed group of
nant contribution is (U ···U U )N and it is convenient
g 2 1 unitaries G with elements V , r = 1,...,g and V = I.
r 1
to consider the sequence of unitary operators
Indeed, decoupling by symmetrization[6] is recoveredas
a particular case by considering the unitary operators
V (t)=(U ···U )†NU (t). (30)
N g 1 N
U =V V†, (r =1,...,g−1), U =V†. (39)
The differential equation is again r r+1 r g g
A single cycle yields
d
i V (t)=H (t)V (t), (V (0)=I) (31)
N N N N
dt t t
U (t)=V†U V ···V†U V , (40)
cycle g gN g 1 gN 1
where (cid:18) (cid:19) (cid:18) (cid:19)
N−1 while
1
H (t) = (U ···U )†N(U U···U U)k
N N g 1 g 1 U ···U =V†V V† ···V†V =I. (41)
k=0 g 1 g g g−1 2 2
X
×H¯N(UgU···U1U)†k(Ug···U1)N, (32) We therefore reobtain, as a special case of the QZE, the
well-known BB result [6]:
with
N→∞
1 U (t)=V (t) ∼ exp(−iH t), (42)
H¯ = U HU†+(U UU )H(U UU )†+··· N N eff
N g g g g g−1 g g−1
where H =H and
+(U UU(cid:2) ···U UU )H(U UU ···U UU )†(3.3) eff Z
g g−1 2 1 g g−1 2 1
g
1
We can now follow through the same calculations a(cid:3)s in HZ =ΠI(H¯)=H¯ = g Vr†HVr =ΠG(H). (43)
the single-kick case, substituting U HU† everywhere by r=1
1 1 X
H¯ , and U by U ···U . It is then straightforward to
N 1 g 1
verify that in the N →∞ limit we get
V. ORIGIN OF EQUIVALENCE BETWEEN
CONTINUOUS AND PULSED FORMULATIONS
U(t)≡ lim V (t), (34)
N
N→∞
The equivalence between the ways in which the QZE
which againsatisfies Eq.(6), with the Zeno Hamiltonian
canbegeneratedviaobservationandviaHamiltonianin-
teractionhavebeendiscussedin[13]. Wenowexplainthe
H =Π (H¯)= P H¯P , (35) equivalence between the continuous and pulsed Hamil-
Z Ug···U1 µ µ tonian interaction pictures, in generating the Zeno sub-
µ
X spaces. In fact, the two procedures differ only in the
where order in which two limits are computed. We recall that
the continuous case deals with the strong coupling limit
U ···U U = P e−iλµ, (36) [13]
g 2 1 µ
µ
X H =H +KH , K →∞ (44)
1 tot 1
H¯ = [H +···+(U ···U )†H(U ···U )
g−2 1 g−2 1
g
andtheZenosubspacesaretheeigenspacesofH . Onthe
1
+(U ···U )†H(U ···U )]. (37) otherhand,thekickeddynamicsentailsthelimitN →∞
g−1 1 g−1 1
5
in (1) and the Zeno subspaces are the eigenspaces of U . VI. CONCLUSIONS
1
This evolution is generated by the Hamiltonian
In this work we have shown the formal equivalence of
Htot =H +τ1H1 δ(t−nτ2), τ2 →0 (45) the quantum Zeno effect (QZE), which has been known
n since vonNeumann laid downthe mathematical founda-
X
tions ofquantummechanics [20,p.366]andhasbeen the
whereτ istheperiodbetweentwokicksandtheunitary
2 subject of intense investigations since the seminal paper
evolution during a kick is U = exp(−iτ H ). The limit
1 1 1
[9], to the recently introduced [4, 5, 6] “bang-bang” de-
N →∞in(1)correspondstoτ →0. Thetwodynamics
2 couplingmethod(BB)forreducingdecoherenceinquan-
(44)and(45)arebothlimiting casesofthe followingone
tum information processing [25]. The QZE is tradition-
allyderivedbyconsideringaseriesofrapid,pulsedobser-
t−n(τ +τ /K)
2 1
Htot = H +KH1 g , (46) vations [9]. This became almost a dogma and motivated
τ /K
n (cid:18) 1 (cid:19) interesting seminal experiments [15, 21]. Later formu-
X
lations emphasized that the QZE can also be generated
where the function g has the properties
by continuous Hamiltonian interaction [12, 13, 22]. The
BBmethod, onthe otherhand, employsa seriesofrapid
g(x−n) = 1 (47)
pulsed interactions. Here we have shown that both the
Xn QZE (in its continuous-interaction formulation) and the
lim Kg(Kx) = δ(x). (48)
BBmethodcanbeunderstoodaslimitsofasingleHamil-
K→∞
tonian,Eq.(46),givingrisetoeitherpulsedorcontinuous
Forexamplewecanconsiderg(x)=χ (x),where dynamics, with a resulting partitioning of the controlled
[−1/2,1/2]
χ is the characteristic function of the set I. In Eq. (46) system’sHilbertspaceintoquantumZenosubspaces,de-
I
theperiodbetweentwokicksisτ /K+τ ,whilethekick fined by Eqs. (9)-(10). This unified view not only offers
1 2
lasts for a time τ /K. By taking the limit τ →0 in Eq. the advantage of conceptual simplicity, but also has sig-
1 2
(46), i.e., a sequence of pulses of finite duration τ /K nificant practical consequences: it shows that the scope
1
without any idle time among them, and using property of all the methods analyzed here (QZE, BB and contin-
(47), one recovers the continuous case (44). Then, by uous interaction) are wider than previously suspected,
taking the strong coupling limit K → ∞ one gets the leading to greater flexibility in their implementation. In
Zeno subspaces. On the other hand, by taking the K → particular, since all formulations of the QZE are physi-
∞ limit, i.e., the limit of shorter pulses (but with the cally equivalent, and BB is equivalent to the kicked uni-
same global—integral—effect), and using property (48) taryformulationofthe QZE,it is clear thatBB canalso
and the identity δ(t/τ )=τ δ(t), one obtains the kicked be formulated in terms of a continuous interaction and
1 1
case (45). Then, by taking the vanishing idle time limit pulsedmeasurements. Thecontinuousinteractionversion
τ →0 one gets again the Zeno subspaces. In short, the ofBBavoidsthefrequentlycriticisedoff-resonanttransi-
2
mathematicalequivalencebetweenthetwoapproachesis tionsassociatedwiththelargebandwidthpulsesrequired
expressed by the relation inthepulsedBBimplementation[23]. Wehavenotstud-
ied the practical advantages or drawbacks of the pulsed
lim lim Htot = lim lim Htot, (49) measurement formulation of BB.
K→∞ τ2→0 τ2→0 K→∞
We emphasize that ourconclusionsaboutgreaterflex-
(foralmostallτ )withtheleft(right)sideexpressingthe ibilityinthe practicalimplementationoftheBBmethod
1
continuous (pulsed) case. Note that this formal equiva- aresupportedby the factthat experiments with largeN
lence mustphysically be checkedona caseby casebasis, have been performed, proving both the quantum Zeno
and it is legitimate only if the inverse Zeno regime is [15, 21] and the inverse quantum Zeno effect [15], and
avoided and the role of the form factors clearly spelled showing that the strong coupling regime is attainable in
out. That is, physically the relevant timescales play a real physical systems.
crucialrole,andinpracticetherecertainlycanbeadiffer- Another consequence of our work is that the Zeno-
ence between kicked dynamics and continuous coupling, subspacedynamics,initspulsedformulation,canbegen-
in spite of their equivalence in the above mathematical eratedbyasequenceofarbitrary (fastandstrong)pulses,
limit. without any (symmetry) assumptions about the relation
Another key issue of physical relevance, in particular betweenpulses. Thisgeneralizesallpreviouslypublished
if one is interested in possible applications, is played by formulations of the BB method, which assumed such re-
the physical meaning of “strong” when one talks of the lations.
strongcoupling regime. We showedthatstrong coupling Finally, owing perhaps to its longer history, the QZE
is equivalent to large N (number of interruptions) and, has been more thoroughly studied than the BB method,
since experiments with large N have been performed, and it has been recognized that in physically relevant
provingboththequantumZenoandtheinversequantum limits an inverse QZE can arise. We have shown that
Zeno effect [15], the strong coupling regime is attainable the same conclusion applies to the BB method, with the
in real physical systems. importantimplicationthatinsomecasesBBcanactually
6
enhance,ratherthanreducedecoherence. This issuewill brought us together in Trieste. D.A.L. is supported
be the subject of further investigations [19]. in part by the DARPA-QuIST program (managed by
AFOSR under Grant No. F49620-01-1-0468).
Acknowledgments
We thank the organizers of the Conference on “Ir-
reversible Quantum Dynamics,” (August 2002) who
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