PHYSICAL REVIEW A 83, 022311 (2011)

Physically realizable entanglement by local continuous measurements

Eduardo Mascarenhas,1,2 Daniel Cavalcanti,2 Vlatko Vedral,2,3,4 and Marcelo Franca Santos1,2

1Departamento de Fısica, Universidade Federal de Minas Gerais, Belo Horizonte, Caixa Postal 702, 30123-970, MG, Brazil2Centre for Quantum Technologies, National University of Singapore, Singapore

3Department of Physics, National University of Singapore, Singapore4Department of Physics, University of Oxford, Clarendon Laboratory, Oxford, OX1 3PU, United Kingdom

(Received 15 June 2010; published 14 February 2011)

Quantum systems prepared in pure states evolve into mixtures under environmental action. Continuouslyrealizable ensembles (or physically realizable) are the pure state decompositions of those mixtures that can begenerated in time through continuous measurements of the environment. Here, we define continuously realizableentanglement as the average entanglement over realizable ensembles. We search for the measurement strategyto maximize and minimize this quantity through observations on the independent environments that cause twoqubits to disentangle in time. We then compare it with the entanglement bounds (entanglement of formation andentanglement of assistance) for the unmonitored system. For some relevant noise sources the maximum realizableentanglement coincides with the upper bound, establishing the scheme as an alternative to protect entanglement.However, for local strategies, the lower bound of the unmonitored system is not reached.

DOI: 10.1103/PhysRevA.83.022311 PACS number(s): 03.67.Mn, 03.65.Ta, 03.67.Pp, 42.50.Lc

I. INTRODUCTION

Decoherence is the process in which the exchange ofinformation between a quantum system and an externalenvironment continuously downgrades quantum properties ofthe former [1]. This dynamics turns initially pure entangledstates into mixed less entangled ones, destroying the capacityof the system for quantum applications. In this picture, thedescription of the properties of the system in time resultsfrom ignoring the information that leaks into the environmentand only considering the general statistical characteristicsof the reservoir. If, however, one is able to recover someof this information by, for example, continuously observingchanges into the environment, the time evolution of the systemundergoes a different dynamics that is conditioned on theresults of these observations. Such conditional evolutions areusually referred to as quantum trajectories [1] because if thesystem is initially prepared in a pure state, the sequence ofmeasurements performed on the environment determines arespective sequence of other pure states for the system. Eachtrajectory corresponds to one realization of the experiment andthe open system evolution is recovered when averaging overall possible trajectories. Such trajectories have already beenobserved both in massive particles and light [2].

There are infinitely many different ways to observe aphysical environment. The measurement schemes are com-monly referred to as unravelings. Each one of them definesa different set of experimentally realizable trajectories. How-ever, even though at any given time the incoherent sum overall possible trajectories reproduces the quantum state of theunmonitored system, that does not mean that all its pure statedecompositions are available, because not all ensembles canbe achieved through this continuous monitoring process [3].The fact that one can reconstruct the state allows one to recoverthe average value of any observable in time by averagingover the trajectories. Obviously, the average value of anylinear quantity (any observable) is the same for all ensembles.However, when it comes to obtaining the entanglement of thesystem (a nonlinear quantity that may present different average

values for different ensembles), the restrictions imposed by theunraveled time evolution may be too strong [4].

In this paper, we investigate equivalent measurementstrategies over the independent reservoirs that cause twoentangled qubits to lose coherence and entanglement in time.The restriction on the ensembles that can be continuouslygenerated in time becomes clear when we compare theminimum and maximum average entanglement [5], resultingfrom realizable unravelings, to the extremes over all possibledecompositions of a given density matrix, respectively, theentanglement of formation EF [6] and the entanglement ofassistance EA [7]. It would not seem to be far fetched toexpect global unravelings to present larger values of theaverage entanglement and local unravelings to do the opposite.However, we show that local operations (in the exampleswe address) never suffice to obtain the entanglement offormation (which is possible with global unravelings) and thatin general not even global unravelings yield the entanglementof assistance, which shows how restricted may be the class ofphysically realizable ensembles. One may additionally restrictthe unravelings to local measurements, which means that themeasurements performed over independent reservoirs should,themselves, be independent as well. This local condition isessential whenever the decohering qubits are distributed overdifferent and distinct channels, as in quantum teleportation,communication, or likewise.

We analyze both dephasing and spontaneous emission asthe decoherence sources. For dephasing, we show that it isnot only possible to continuously reach the entanglement ofassistance of the system but the protocol to do so does not evenrequire any classical communication between the observers,as long as local phase feedback is allowed. In fact, in thisparticular case, we show that monitoring the local reservoirs isenough to fully protect the entanglement of the system. On theother hand, we also show that the entanglement of formationcan never be achieved through local observations, even withthe addition of classical noise to the measurement process [8]and the increase of the entropy of the collected information.

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MASCARENHAS, CAVALCANTI, VEDRAL, AND SANTOS PHYSICAL REVIEW A 83, 022311 (2011)

This becomes even more explicit for dissipative reservoirswhere EA can only be achieved for some particular initialstates. This means that not only some decompositions arenot produceable through continuous monitoring of the envi-ronment, but, more importantly, neither is the entanglementof formation of ρ(t) (under the restriction of local and/ornondiffusive unravelings) nor the entanglement of assistance(in general). Once again, the fact that the entanglement offormation (minimum over possible decompositions) is nevercontinuously reached but EA (maximum) can be approachedalso means that the independent monitoring of local reservoirsis indeed a good strategy to preserve entanglement for quantumcommunication and teleportation protocols, especially inquantum feedback schemes [9].

II. MIXED-STATE DECOMPOSITIONSAND UNRAVELINGS

The physical system we describe is that of two noninter-acting qubits initially prepared in a pure state σ = |σ 〉〈σ |and coupled to independent reservoirs in the Born-Markovapproximations [1]. Their unmonitored time evolution isdetermined by a master equation of the type ρ = LAρ + LBρ,where

Lρ =∑

i

γi

2(2�iρ�

†i − {�†

i �i,ρ}), (1)

and the set {�i} represents the subsystem-reservoir couplings.ρ(t) can be formally obtained by integrating this equation intime. It also corresponds to a completely positive map in theKraus form [10] acting on the initial state,

ρ = KσK† =∑

i

KiσK†i , (2)

with K representing a vector of Kraus operators K =[K0,K1, . . . ,Kn] and K† = [K†

0,K†1, . . . ,K

†n]T. The ith Kraus

operator can be identified as a measurement operator suchthat probability is conserved, meaning

∑i K

†i Ki = 1, and

the measurement process is characterized by a non-negativedistribution 〈K†

i Ki〉 � 0. The K unraveling gives output statesrepresenting a decomposition of the system state ρ = ��†,with K|σ 〉 = � and �i = Ki |σ 〉 = |ψi〉 (the tilde indicates aunnormalized state).

This is one of infinitely many possible decompositions ofρ and any other decomposition ρ = ��† can be obtainedby a unitary rearrangement of �, �T = U�T, such thatρ = ��† = �UTU∗�† = ��†, where U is a unitary matrixrespecting (U†U)T = UTU∗ = 1. The authors of Ref. [11]show a one-to-one relation between unravelings and pure stateensembles by demonstrating that the same freedom to makeunitary transformations applies to the unraveling choice, or inother words, that the same unitary takes the unraveling K thatgenerates � to another unraveling G that generates �, withGT = UKT:

ρ = KσK† = KUTσU∗K† = GσG†. (3)

Their result shows that in principle all possible ensemblesthat decompose the decohered state at a given time t can beobtained through measurements on a purification of the state.These include, in particular, the decompositions that handle the

maximum and minimum average entanglement, respectively,EA and EF .

However, the measurement apparatuses that lead to somedecompositions are not compatible with a continuous measure-ment process of the environments, or, as defined in Ref. [3], thecorresponding ensembles are not physically realizable throughthe continuous monitoring of the independent reservoirs.

If one is to monitor the reservoirs continuously, then the evo-lution of the system respects a new equation where at each timestep the system is projected onto a new state. Assuming thatthe system is initially prepared in a pure state, the evolution isthen given by a sequence of pure states {|σ0〉,|σdt 〉, . . . ,|σn dt 〉}which correspond to a quantum trajectory of the system [1].This process is also described by a completely positive (CP)map in the form of (2) but now there is an extra restrictionimposed over the different possible unravelings K. Physically,this is related to the fact that some K’s represent operationsthat combine measurements performed on the reservoir atdifferent times (a measurement that is global in time),while the Markovian approximation that leads to the masterequation imposes reservoirs with no memory that, underconstant monitoring, must produce independent measurementoutcomes. Furthermore, by imposing local observation of thereservoirs that act on each qubit, one also creates an extrarestriction on the class of unitary transformations U thatcombine the different unraveling operations to operations ofthe type UA ⊗ UB.

Note that, in the limit of continuous monitoring there is yetanother class of possible decompositions that is associated tothe addition of classical noise to the measurement records(as in homodyne detection, for example, Ref. [12]). Inthis case, the decomposition reads GT

±∗ = 1√2[� ± UKT

∗ ],where ∗ indicates the absence of K0 and G0, and G0 =1 − 1

2

∑i=1[G†

i+Gi+ + G†i−Gi−]. The vector � (whose ele-

ments are complex constants) represents the external entitythat adds the extra noise. When |�| tends to infinity, wereach the quantum-state diffusion limit [8]. The measurementoperators can then be described by DJ = K0 + G∗J†

√dt ,

with G∗ = K∗UT and J dt = 〈GT∗ + G†

∗〉T dt + dW, such thatdWi are infinitesimal Wiener increments (gaussian distributedvariables of variance dt) and dWT dW = 1 dt . The short timeexpansion of the map that reproduces the density matrix isrecovered with

ρ(dt) =∫

dµ(J)DJσD†J, (4)

with dµ(J) being the normalized gaussian measure over themeasurement outcomes.

When restricted to U = UA ⊗ UB, this decompositionalways decreases the average entanglement (as compared tothe corresponding one without classical noise) as should beexpected given that the added noise blurs the outcomes of themeasurements on the reservoirs. This fact is used in Ref. [4]to minimize the average concurrence.

We show below that the local and continuous restrictionsmade on the monitoring protocols of the reservoirs imposelimitations on the minimum and maximum average entangle-ment E attainable from realizable ensembles. Here, E(�) =∑

i piE(|φi〉), where the entanglement of realizable state |φi〉,which happens with probability pi = 〈φi |φi〉, is obtained by

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PHYSICALLY REALIZABLE ENTANGLEMENT BY LOCAL . . . PHYSICAL REVIEW A 83, 022311 (2011)

calculating the Von Neumann entropy S(ρ) = −tr{ρ log2 ρ}on the reduced density matrix over either one of the qubits,E(|φi〉) = S(trB{|φi〉〈φi |}).

III. CONTINUOUSLY REALIZABLE ENTANGLEMENT

Let us first consider the example of both qubits initiallyprepared in a maximally entangled state (for example, oneof |�±〉 = 1√

2(|00〉 ± |11〉) and subjected to independent

dephasing reservoirs represented by Kraus operators givenby A0 = (1 − γ dt

2 )1 and A1 = Z√

γ dt in the short timeexpansion (up to first order in dt). Here Z is the usual Paulimatrix, γ is the system-environment coupling, and the sameholds for observer B. This particular unraveling (UA(B) = 1)already gives the EA of the system in each time interval dt [13].Dephasing implies that the system stochastically switchesbetween |�+〉 and |�−〉, and by monitoring the reservoir oneknows if the phase flip (the jump) has occurred or not. Thisresult can still be classically communicated between the partsthat then completely preserves the entanglement of the system.If, however, the observer is capable of phase flipping its localqubit, then not even the communication is required and theprotocol becomes entirely local, similar to the one presented inRef. [14] for spontaneous emission. This also means that in thiscase EA[ρ(t)] corresponds to a locally realizable ensemble.Also note that because local phase flip is an unitary operation,this result is general and holds for any initial state.

Changing the unraveling also changes the decompositionsand E, however, the EF of the unmonitored system isunreachable as shown in Fig. 1 for the optimized local

0 0.5 10

0.2

0.4

0.6

0.8

1

t0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

t

F boundE boundMAXUMINUE

F

LocalDiffusionE

F

EA & MAXU

MINU

FIG. 1. (Color online) (Left-hand side) Dephasing reservoir for aBell state: EA and the maximized locally unraveled entanglement(MAXU) coincide. At the other extreme, the minimized locallyunraveled entanglement (MINU) and the one obtained by localquantum-state diffusion also coincides, but there is always a gapbetween both these quantities and the entanglement of formation EF

of the unmonitored ρ(t). (Right-hand side) Dissipative reservoir fora state presenting sudden death of entanglement |σ 〉 = 1√

8(|00〉 +√

7|11〉): Once again, local state diffusion and minimized locallyunraveled entanglement (MINU) give the local minimum averageentanglement that never goes to zero. In this case, the gap betweenunraveled entanglement and EF is more drastic because by locallymonitoring the reservoirs, entanglement sudden death is prevented.At the other extreme, EA is only defined within the eigenstate (Ebound) and the fidelity (F bound) bounds and in this case, even thebest global unraveling strategies (MAXU) is mostly smaller than bothbounds, even though it helps redefining the lower bound for largertimes (γ t ∼ 1.2).

unraveling that minimizes E. The minimum local unraveling isgiven by the equal and local superposition of the measurementoutcomes and coincides with the result obtained for localquantum-state diffusion. Further minimization might still bepossible but those would require global unravelings withclassical noise [15], which means entangling the reservoirsbefore performing the measurements. Note, however, that suchunravelings are experimentally challenging, particularly inprotocols where the qubits are far apart, because they requirethe joint manipulation of independent reservoirs.

Turning to local dissipative environments, theU = 1 unrav-eling features Kraus operators of the type A0 = 1 − γ

2 L†Ldt

and A1 = L√

γ dt , with L = |0〉〈1|. For initial states inthe subspace {|01〉,|10〉} like, for example, |�+〉 = (|01〉 +|10〉)/√2, the entanglement of assistance of ρ(t) can onceagain be exactly obtained and coincides with the fidelity bound[7]. This derives from the particular time evolution of the sys-tem that leads to a mixture of a maximally entangled state anda separable one, ρ(t) = p(t)|�+〉〈�+| + [1 − p(t)]|00〉〈00|,and EA[ρ(t)] is given by p(t) = e−γ t . This maximum isexactly the one obtained by the UA(B) = 1 unraveling [13,14],which means that once again, local strategies can allow forthe maximum extraction of entanglement from the system.In this case, classical communication is required given thata single decay completely destroys the entanglement of thesystem. Also note that, in this case, it is particularly importantto use EF as a lower bound for the average entanglementbecause the Wootters concurrence, for example, still gives thesame average values for all unravelings, as shown in Refs. [4]and [15].

From the point of view of the detection of unrealizableensembles, the most interesting example, however, appearswhen we analyze initial states that present finite time dis-entanglement, such as |σ 〉 = 1√

8(|00〉 + √

7|11〉) [16]. In thiscase, neither extremes of the entanglement of the system areachievable through local measurements on the independentreservoirs, as shown in Fig. 1(b). For the entanglement ofassistance, this derives immediately from the fact that localstrategies cannot increase the average entanglement while EA

increases for times smaller than 1/γ . Note that, soon afterEF [ρ(t)] vanishes (in finite time [16]), the entanglement ofassistance reaches the maximum value. At this moment thegap between the entanglements defined for ρ(t) (EF and EA)and those obtained by continuous reservoir monitoring (MINUand MAXU) reach maximum values. Note that in this case,as discussed in the final section of the paper, we used the bestglobal unraveling strategy to define MAXU, which means thatfor most times even global unravelings do not produce EA.Also note that because sudden death [17–19] is prevented,by locally monitoring the environments, the observer alwayspreserves some entanglement in the system.

IV. INFORMATION OF LOCAL UNRAVELINGS

In this section, we analyze the persisting gap between E

and EF from an informational point of view. The entropyof the measurement outcomes of a general unravelling isgiven by S(G) = −∑

i pi log2 (pi), with pi = 〈G†i Gi〉 being

the probability of the ith outcome. The information acquired

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MASCARENHAS, CAVALCANTI, VEDRAL, AND SANTOS PHYSICAL REVIEW A 83, 022311 (2011)

from the reservoir is maximum if the entropy goes to zero,and minimum if the entropy is maximum. We will proceedto show how this information can be related to the upper andlower bounds of the continuously realizable entanglement E

of the system.In order to exemplify our analysis, we will concen-

trate first in the U = 1 unraveling, because this unravelingalready locally produces the entanglement of assistance bothfor dephasing and spontaneous emission in the {|01〉,|10〉}subspace. In the short time expansion, the U = 1 unraveling isgiven by an operator K1 proportional to

√dt (jump operator)

and its complementary K0 = 1 − 12K

†1K1. Its entropy, given

by S(K) = −p1 log2 (p1) − (1 − p1) log2(1 − p1), with p1 ∝dt , tends to zero for small time steps dt → 0, maximizing,thus, the information available in the reservoirs readout. Inboth situations mentioned above, the most probable event (nojump) is correlated to preserving entanglement in the system.In fact, for dephasing, both click and no click produce equallyentangled states, hence EA always coincides with the initialentanglement of the system, as already discussed in previousparagraphs, whereas for spontaneous decay for initial states inthe subspace {|01〉,|10〉}, no click still preserves entanglement(if both local reservoirs have the same decay rate) becausethe entire subspace is unaffected by the no-jump evolution.However, differently from the dephasing case, this time a clickin the reservoir kills entanglement completely by taking thesystem to its ground separable state |00〉, which explains why,in this case, EA also decays despite still being reproducible byE (for U = 1).

On the other hand, any other unraveling that combinesthe jump and no-jump operators [20], necessarily increasesthe entropy (less information is obtained) and mixes theabove mentioned correlations. In the worst-case scenario ofa 50-50 superposition (UW = [

√α −√

1−α√1−α

√α

] with α = 1/2)all the information (regarding clicks and no clicks) is lostand the average entanglement is minimized [correspond-ing, for example, to the MINU curve in Fig. 1(a)]. Tomake it more explicit we can evaluate the entropy of localunravelings S(G) = −∑

i gi log2(gi), with g1 = αp1 + (1 −α)(1 − p1) − √

α(1 − α)p1 and g2 = (1 − α)p1 + α(1 −p1) + √

α(1 − α)p1 (1 p1). The entropy is maximumS(G) → 1 for α = 1/2, which means that equally superposingthe measurement outcomes maximally mixes the readoutinformation.

For dephasing, it means that an event at time t (either clickor no click) will be correlated to both |�±〉 states, thereforeunavoidably recombining the pre-event state and decreasing itsentanglement. For example, if the initial state is |�+〉 and thebest U = 1 is used, in most realizations, no click is observedand the state is preserved, and even when sometimes one clickis detected, the sate is still projected onto the other maximallyentangled state |�−〉. However, if the worst UW unraveling isused, and a (K0 + K1)/

√2 click is detected, then the new (non-

normalized) less entangled state given by (1 − γ dt/2)|�+〉 +√γ dt |�−〉 is produced.Much in the same way, for the {|01〉,|10〉} decay, the U = 1

unraveling protects entanglement if no jump is detected, whilein any other combination of the reservoirs readout, and inparticular UW , a component of |00〉 is unavoidably added to

the state in which case even the no jump will help degradingthe entanglement.

In both examples, U = 1 and UW establish, respectively,the local upper and lower unraveling bounds and any otherlocal unraveling (more general U’s) will produce averageentanglement in between them.

A similar analysis can be carried out in the quantum-state diffusion limit. In this case, the addition of classicalnoise, ultimately represented by dWi , blurs the outcomes anddestroys the acquisition of information from the reservoir,which leads the diffusive limit also to the minimum localaverage entanglement, as we have shown in Fig. 1 of thepaper. In this case, the mixing of the readout information yieldsan entropy of S(J ) = − ∫

p(J ) ln[p(J )]dJ , with p(J )dJ =dµ(J )〈D†

J DJ 〉. In a short time expansion it is fair to assume〈D†

J DJ 〉 ≈ 1, because large values of |J | will not contributeto the integral because of the gaussian weight and thereforethe entropy reduces to S(J ) ≈ 1

2 ln( 2πdt

) + 12 → ∞. Similar

to the G unraveling, the quantum-state diffusion limit alsomixes the information provided by the measurements per-formed on the reservoirs, although the infinite value of S(J )may mislead one into thinking that the latter causes greaterinformation losses. However, note that in both cases there is atmost one bit of information available in the reservoir readoutand the origin of the divergence of the diffusive scenario iscommonly encountered in continuous variable distributions (inour case the distribution tends to a delta function if dt → 0).This is clear from the fact that the diffusive limit also givesthe minimum local average entanglement, as we have shownin Fig. 1 of the paper. Once again, all the information acquiredfrom the reservoirs is mixed and the realizable entanglementreaches its minimum but not the EF [ρ(t)].

Finally note that the the gap between the lowest realizableE and the entanglement of formation of the system necessarilymeans that ignoring the information that goes to the reservoir(master equation evolution of ρ) is different from continuouslyacquiring the least amount of information from it.

V. A FEW WORDS ABOUT GLOBAL UNRAVELINGS

Before concluding, let us comment on the possible changesrelated to lifting the local observation restriction. That meansallowing for operations that coherently superpose the resultsobserved in each independent reservoir. In Fig. 1, we show thatthis strategy may also increase the average entanglement of thesystem. This happens to be the case for initial states presentingfinite time disentanglement for spontaneous emission. In thiscase, both the no-jump trajectory and the superposed jumpsare correlated to increasing entanglement for the first part ofthe evolution (times of the order of 1/γ ). While the no-jumptrajectory performs the optimum singlet conversion protocol[14,21], correlated jumps of the form G± = 1√

2[A1 ⊗ 1 ± 1 ⊗

B1] produce Bell states for both qubits, increasing then theaverage entanglement of the system. This strategy is still notenough to reach the EA of the unmonitored evolution, but forcertain times it redefines the lower bound for this quantityand it is the maximum continuous realizable entanglement.We can make this statement clear by obtaining the differentialequation for concurrence (for simplicity) for an arbitrary initial

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PHYSICALLY REALIZABLE ENTANGLEMENT BY LOCAL . . . PHYSICAL REVIEW A 83, 022311 (2011)

pure state σ undergoing a general unraveling (respectingthe no-jump asymmetry) as show in Ref. [4]. If we assumenondiffusive unravelings we obtain the equation of motion forthe average concurrence c,

dc = −γ c(σ ) dt + 4√

β(1 − β)γ |σ11(t)2| dt. (5)

The state component σ11(t) = 〈11|σ |11〉 = σ11(0)e−γ t is inde-pendent of the state and unraveling, therefore to maximize theaverage concurrence as it evolves one should choose β = 1/2,with 1/2 � β � 0, such that the unraveling is completely localfor β = 0 and maximally global for β = 1/2.

In Ref. [15], the authors use a global state-dependentstrategy plus classical noise (diffusive unraveling) to reach theother extreme, continuously reproducing the concurrence (andalso EF ) of the unmonitored system. By slightly extendingtheir results, we can show that by choosing the symmetricphase of their minimization scheme θmax = θmin + π (in theirpaper θmin = θopt), one reaches the same average concurrencefor the global jump unraveling with β = 1/2 as defined in (5).The maximum and minimum average concurrence are thengiven by

cmaxmin

(t) = e−γ t [c(σ ) ± 2|σ11|2(1 − e−γ t )], (6)

such that the maximum is attainable both from a jump (state-independent) unraveling and a diffusion (state-dependent)unraveling. However, the minimum is only reached by dif-fusion (in a state-dependent manner), because we are applyingjump unravelings that respect the no-jump asymmetry andinformation is only mixed by the classical noise in the diffusiveunraveling.

VI. CONCLUSIONS

We have studied the behavior of entanglement in quantumopen systems when the environments coupled to the systemare continuously monitored. We have analyzed two differentsources of decoherence, namely, dissipation and dephasing,when applied to initially pure and entangled states. In thesespecific cases, we have shown a gap between the locallyrealizable entanglement and the extremes entanglement ofassistance and formation of the unmonitored system. Wehave also related the minimum attainable entanglement to theincrease of the entropy of the information provided by thereadout of the environment. On its turn, the above-mentionedgap means that completely scrambling the informationacquired through the reservoirs is fundamentally differentfrom not collecting it. This fact is better exemplified whenthe system is initially prepared in a pure entangled stateand presents finite time disentanglement under dissipation,in which case it is impossible to locally unravel an ensemblewith zero average entanglement at finite time. There is alwaysat least one entangled trajectory, which means that the averageentanglement can only go to zero asymptotically in time. Thisfact in turn provides a method to protect entanglement andavoid entanglement sudden death.

ACKNOWLEDGMENTS

The authors would like to thank A.R.R. Carvalho foruseful discussions. This work was financially supported by theNational Research Foundation and the Ministry of Educationof Singapore. E.M. would also like to thank CNPq for financialsupport.

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