PHYSICAL REVIEW A 89, 042112 (2014)

Robertson-Schrodinger-type formulation of Ozawa’s noise-disturbance uncertainty principle

Catarina Bastos*

Instituto de Plasmas e Fusao Nuclear, Instituto Superior Tecnico Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

Alex E. Bernardini†

Departamento de Fısica e Astronomia, Faculdade de Ciencias da Universidade do Porto,Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Orfeu Bertolami‡

Departamento de Fısica e Astronomia, Centro de Fısica do Porto, Faculdade de Ciencias,Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Nuno Costa Dias§ and Joao Nuno Prata‖Universidade Lusofona de Humanidades e Tecnologia, Avenida Campo Alegre, 376, 1749-024 Lisboa, Portugal

and Grupo de Fısica Matematica, UL, Avenida Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal(Received 17 October 2013; published 22 April 2014)

In this work we derive a matrix formulation of a noise-disturbance uncertainty relation, which is akin tothe Robertson-Schrodinger uncertainty principle. Our inequality is stronger than Ozawa’s uncertainty principleand takes noise-disturbance correlations into account. Moreover, we show that for certain types of measurementinteractions it is covariant with respect to linear symplectic transformations of the noise and disturbance operators.Finally, we also study the tightness of our matrix inequality.

DOI: 10.1103/PhysRevA.89.042112 PACS number(s): 03.65.Ta, 03.75.Dg

I. INTRODUCTION

One of the hallmarks of quantum mechanics is commonlyknown as the uncertainty principle. This is a generic designa-tion for a variety of constraints or bounds on the simultaneouslocalization or measurement accuracy of a pair of observables.

Some of these bounds stem from kinematical restrictionsimposed upon the states of a quantum system. They are usuallybuilt into the mathematical framework, where the state ofa system is represented by a unit ray in some appropriateHilbert space. The actual measurement process or the accuracyof measuring devices has no bearing on these preparationuncertainty principles [1]. The most standard example isRobertson’s uncertainty principle. Let C be some observableand write �C = C − 〈ψ |C|ψ〉 for a given state ψ . If σ (C,ψ)denotes the mean standard deviation σ 2(C,ψ) = 〈ψ |�C2|ψ〉,then Robertson’s uncertainty relation is formulated as

σ (A,ψ)σ (B,ψ) � |〈ψ |[A,B]|ψ〉|2

(1)

for some pair of of observables A,B. We denote by [A,B] =AB − BA the commutator and by {A,B} = AB+BA

2 the anti-commutator. The relation (1) is a simple consequence of theCauchy-Schwartz inequality.

*catarina.bastos@ist.utl.pt†On leave from Departamento de Fısica, Universidade Federal

de Sao Carlos, PO Box 676, 13565-905, Sao Carlos, SP, Brasil;alexeb@ufscar.br‡orfeu.bertolami@fc.up.pt§ncdias@meo.pt‖joao.prata@mail.telepac.pt

In particular, if A = X is the particle’s position and B = P

its momentum, then [X,P ] = i� and we obtain the familiarinequality

σ (X,ψ)σ (P,ψ) � �

2. (2)

There is another preparation uncertainty principle involvingthe mean standard deviations σ (X,ψ),σ (P,ψ). It is known asthe Roberson-Schrodinger uncertainty principle (RSUP) andcan be stated in terms of the positivity of the matrix

� + i�

2J � 0. (3)

Here � is the covariance matrix of the state

� =(

σ (X,ψ)2 σ (X,P,ψ)

σ (P,X,ψ) σ (P,ψ)2

), (4)

where σ (X,P,ψ) = σ (P,X,ψ) = 〈ψ | {�X,�P } |ψ〉 are thecovariance elements for position-momentum correlations, and

J =(

0 1

−1 0

)(5)

is the standard symplectic matrix.This uncertainty principle has several advantages over

the formulation stated in Eq. (2): (i) It is stronger thaninequality (2), in the sense that Eq. (3) implies Eq. (2), butthe converse is not true. (ii) It accounts for the position-momentum correlations, which are relevant in several contexts;e.g., states with strong position-momentum correlations maylead to greater transparency of the Coulomb barrier duringthe interaction of charged particles. This is very importantin the astrophysics of stars and in controlled nuclear fu-sion, where the action of the Coulomb barrier leads to a

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CATARINA BASTOS et al. PHYSICAL REVIEW A 89, 042112 (2014)

very low tunneling probability for low-energy particles [2].(iii) For Gaussian states (which include coherent, squeezed,and thermal states), the RSUP constitutes a necessary andsufficient condition of quantumness. (iv) Moreover, aftera certain reflection transformation [3], it also establishesunequivocally the separable or entangled nature of a two-mode Gaussian state. Experimentally, coherent and squeezedstates play important roles in quantum optics [4], quantumcomputation of continuous variables [5], and investigations ofthe quantum-classical transition [6]. (v) Finally, the RSUP isinvariant under linear symplectic transformations, a propertywhich is not shared by inequality (2). This is important inthe context of semiclassical analysis [6] and in the search fordirections of minimal uncertainty [7].

An experimental violation of the kinematical uncertaintyprinciples (2) and (3) can only be attributed to either thefailure of the Hilbert space formalism to correctly describequantum systems—something which would have profound im-plications on the theoretical edifice of quantum mechanics—oralternatively to some modification of the position-momentumcommutation relations. The latter possibility has been recentlyexplored in Ref. [8]. To the best of our knowledge, no suchexperimental violation has ever been recorded.

In addition to the previous kinematical inequalities, thereare other uncertainty principles (dynamical uncertainty prin-ciples) which try to account for the “unavoidable anduncontrollable disturbance” caused on observable B by ameasurement of observable A. They are now commonly knownas error-disturbance-trade-off relations [1]. In his famousγ -ray thought experiment [9], Heisenberg relates the accuracy(resolution) of an appropriate position measurement to thedisturbance of the particle’s momentum. This is usually calledthe Heisenberg uncertainty principle.

There is a lot of controversy regarding the status andinterpretation of Heisenberg’s formulation [1]. There is nouniversally accepted rigorous formulation for accuracy anddisturbance. Nor is there a consensus on how these quantitiesshould be related. Busch et al. [10] proved a rigorous versionof the Heisenberg uncertainty principle. In their approach,the measures of error and disturbance are figures of meritcharacteristic of the measuring apparatus and hence stateindependent. This is in strong contrast with the preparationuncertainty principles.

Other authors considered state-dependent quantities (seeRef. [11] for a discussion on the merits and drawbacks ofstate-dependent error-trade-off relations). For instance, Ozawaformulated Heisenberg’s uncertainty principle in the followingterms. He defined ε(A,ψ) to be the noise of the A measurementand η(B,ψ) to be the disturbance on B caused by thatmeasurement, when the system is in state ψ . One mightheuristically expect Robertson-like relations

ε(A,ψ)η(B,ψ) � |〈ψ |[A,B]|ψ〉|2

(6)

to bound the error and disturbance of the operators A and B

on state ψ .There have been various accounts of theoretical [12–16]

and experimental [17–19] violations of inequality (6). Thisprompted the search for a universally valid formulation of anuncertainty principle accounting for the noise and disturbance

of the measurement interaction. A famous example of a suchan uncertainty principle was proposed by Ozawa [20] (see alsoRef. [21]).

Ozawa considered a composite system of the object and themeasuring device (the probe), initially prepared in a productstate � = ψ ⊗ ξ , where ψ and ξ describe the object and theprobe, respectively. Working in the Heisenberg picture, heintroduced the noise operator N (A) associated with observableA and the disturbance operator D(B). They are self-adjointoperators, defined by

N (A) = Mout − Ain, D(B) = Bout − B in. (7)

Here Ain = A ⊗ I,B in = B ⊗ I are observables A,B priorto the measurement interaction, Bout = U †(B ⊗ I )U is theobservable B immediately after the measurement and M isthe probe observable. U is the unitary time evolution operatorduring the measuring interaction. Clearly M in = I ⊗ M andMout = U †(I ⊗ M)U . For more details on the measurementinteraction see Ref. [20]. Ozawa then argued that the noiseε(A,ψ) and disturbance η(B; ψ) are defined by

ε(A,ψ)2 = 〈�|N (A)2|�〉, η(B,ψ)2 = 〈�|D(B)2|�〉.(8)

Since M and B are observables in different systems, theycommute [Mout,Bout] = 0. Using this fact, Eq. (8), the triangle,and the Cauchy-Schwartz inequalities, one can then prove theOzawa uncertainty principle (OUP):

ε(A,ψ)η(B,ψ) + 12 |〈[N (A),B in]〉 + 〈[Ain,D(B)]〉|

� 12 |〈ψ |[A,B]|ψ〉. (9)

As in Ref. [20], we use the abbreviated notation 〈C〉 to denote〈�|C|�〉.

If |〈[N (A),B in]〉 + 〈[Ain,D(B)]〉| = 0, then the Heisenbergnoise-disturbance uncertainty relation (6) holds. A particularcase corresponds to [N (A),B in] = [Ain,D(B)] = 0. Ozawadefined such a measuring interaction to be of independentintervention for the pair (A,B).

From the triangle and the Cauchy-Schwartz inequalities onehas

|〈[N (A),B in]〉 + 〈[Ain,D(B)]〉|� 2ε(A,ψ)σ (B,ψ) + 2σ (A,ψ)η(B,ψ). (10)

Upon substitution of Eq. (10) into Eq. (9) one obtains:

ε(A,ψ)η(B,ψ) + ε(A,ψ)σ (B,ψ)

+ σ (A,ψ)η(B,ψ) � |〈ψ |[A,B]|ψ〉2

. (11)

Just as the kinematical uncertainty principle, Eq. (2) doesnot account for the position-momentum correlations, neitherdoes the OUP, Eqs. (9) and (11), account for the noise-disturbance correlations. The purpose of this work is to derivea matrix formulation of the OUP, which encompasses thenoise-disturbance correlations. It is related to the OUP in verymuch the same way as the RSUP relates to Eq. (2): It is strongerthan the latter and admits nicer symmetry properties.

The paper is organized as follows. In the next section,we derive our matrix formulation of the noise-disturbanceuncertainty principle. In Sec. III, we prove that the OUP

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is a consequence of our matrix formulation and we give acounterexample that shows that the converse is not true. InSec. IV, we dwell on the tightness of our matrix inequality,and in Sec. V, we discuss the covariance properties of thenew uncertainty principle under a certain class of unitarytransformations. Finally, in Sec. VI we state our conclusions.

II. MATRIX FORMULATION OF THE OZAWAUNCERTAINTY PRINCIPLE

In the sequel we use Latin indices i,j taking values in theset {1, . . . ,n} and Greek indices α,β in the set {1, . . . ,2n}.

For the sake of generality, we also consider a multidi-mensional system. Using Ozawa’s notation, let Ain

i and B inj ,

i,j = 1, . . . ,n denote some set of self-adjoint operators suchthat [

Aini ,Ain

j

] = [B in

i ,B inj

] = 0,[Ain

i ,B inj

] = iCij , (12)

for i,j = 1, . . . ,n, and where Cij are some self-adjoint opera-tors. If A and B are the particle’s position and momentum, wesimply have Cij = �δij . We may write these collectively as

Zin = (Ain

1 , . . . ,Ainn ,B in

1 , . . . ,B inn

), (13)

satisfying the commutation relations[Zin

α ,Zinβ

] = iGαβ, α,β = 1, . . . ,2n. (14)

G = {Gαβ} is the self-adjoint operator-valued skew-symmetricmatrix

G =(

0 C

−C 0

), (15)

with C = {Cij }. Again, if A and B are the position andmomentum operators, then G reduces to the 2n × 2n standardsymplectic matrix �J.

The noise operators are

N = N (A) = (N1, . . . ,Nn), (16)

whereas

D = D(B) = (D1, . . . ,Dn) (17)

are the disturbance operators. We write these collectively as

K = (N1, . . . ,Nn,D1, . . . ,Dn). (18)

Let

Mout = (Mout

1 , . . . ,Moutn

)(19)

denote the output of the (commuting) probe observables and

Bout = (Bout

1 , . . . ,Boutn

)(20)

denote the output of B. If we write

Zout = (Mout

1 , . . . ,Moutn ,Bout

1 , . . . ,Boutn

), (21)

then we have as before[Zout

α ,Zoutβ

] = 0, α,β = 1, . . . ,2n, (22)

and from Eqs. (7)

Zout = Zin + K. (23)

Let {λα}1�α�2n denote an arbitrary set of complex numbers.We thus have

0 =2n∑

α,β=1

λαλβ

⟨[�Zout

α ,�Zoutβ

]⟩

=2n∑

α,β=1

λαλβ

⟨[�Zin

α + Kα,�Zinβ + Kβ

]⟩

=2n∑

α,β=1

λαλβ

(i〈Gαβ〉 + ⟨[

Zinα ,Kβ

]+ [

Kα,Zinβ

]⟩ + ⟨[Kα,Kβ

]⟩). (24)

Now notice that writing K = ∑α λαKα , we have

2n∑α,β=1

λαλβ〈[Kα,Kβ]〉 = 〈K†K〉 − 〈KK†〉 � 〈K†K〉, (25)

where we have used the fact that KK† is a positive operator.And so

2n∑α,β=1

λαλβ〈[Kα,Kβ]〉 �2n∑

α,β=1

λαλβ〈KαKβ〉

=2n∑

α,β=1

λαλβ

(〈{Kα,Kβ}〉 + 1

2〈[Kα,Kβ]〉

). (26)

And thus2n∑

α,β=1

λαλβ〈[Kα,Kβ]〉 � 22n∑

α,β=1

λαλβ〈{Kα,Kβ}〉. (27)

Upon substitution of Eq. (27) into Eq. (24) we obtain

0 �2n∑

α,β=1

λαλβ

(i〈Gαβ〉 + ⟨[

Zinα ,Kβ

]+ [

Kα,Zinβ

]⟩ + 2〈{Kα,Kβ}〉). (28)

If we define the 2n × 2n real symmetric positive-definitematrix

Kαβ = 〈{Kα,Kβ}〉, (29)

and the 2n × 2n real skew-symmetric matrices G = 〈G〉 and

�αβ = 1

i

⟨[Zin

α ,Kβ

] + [Kα,Zin

β

]⟩, (30)

then we can rewrite Eq. (28) in the matrix form:

K + i

2(� + G) � 0, (31)

which is our matrix version of Ozawa’s uncertainty principle.In Ozawa’s terminology, if the measuring interaction is of

independent intervention, i.e., if

� = 0, (32)

then we obtain

K + i

2G � 0, (33)

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CATARINA BASTOS et al. PHYSICAL REVIEW A 89, 042112 (2014)

which is a matrix generalization of the Heisenberg noise-disturbance relation Eq. (6) based on the γ -ray thoughtexperiment.

III. ON THE CONNECTION WITH THE OZAWAUNCERTAINTY PRINCIPLE

We now argue that our uncertainty principle Eq. (31) isin fact stronger than Ozawa’s uncertainty principle. Indeed,consider for simplicity n = 1. We thus have

K =( 〈N2〉 〈{N,D}〉

〈{D,N}〉 〈D2〉)

, (34)

while

� = 1

i

(0 〈[Ain,D] + [N,B in]〉

〈[D,Ain] + [B in,N ]〉 0

).

(35)

If inequality (31) holds, then the matrix K + i2 (� + G) must

have non-negative determinant, and we obtain

〈N2〉〈D2〉� 〈{N,D}〉2 + 1

4 |〈[Ain,D] + [N,B in]〉 + 〈[Ain,B in]〉|2

� 14 |〈[Ain,D] + [N,B in]〉 + 〈[Ain,B in]〉|2. (36)

Taking the square root yields

〈N2〉1/2〈D2〉1/2 � 12 |〈[Ain,D] + [N,B in]〉 − 〈[B in,Ain]〉|.

(37)

By writing ε(A) = 〈N2〉1/2 and η(B) = 〈D2〉1/2, and using theinequality |a − b| � ||a| − |b||, we obtain

ε(A)η(B) � 12 ||〈[Ain,D] + [N,B in]〉| − |〈[Ain,B in]〉||.

(38)

Thus, in particular

ε(A)η(B) � 12 |〈[Ain,B in]〉| − 1

2 |〈[Ain,D] + [N,B in]〉|,(39)

which is Ozawa’s uncertainty principle Eq. (9).The obvious question is now whether our uncertainty

principle, inequality (31), is equivalent to Ozawa’s or whetherit is, in fact, stronger. The latter is true, as we show inthe following example. The type of measuring interactionconsidered here is called a backaction evading quadratureamplifier [20,22].

In this case, the system is described by a set of quadratureoperators Xa,Ya and the probe by the operators Xb,Yb with

[Xa,Ya] = [Xb,Yb] = i

2. (40)

Then, we have the measuring interaction

Xouta = Xin

a

Xoutb = Xin

b + GXina

Y outa = Y in

a − GY inb

Y outb = Y in

b

, (41)

where G is the gain. The probe observable is then set to M =1G

Xb, and thus

Mout = Xina + 1

GXin

b . (42)

Moreover

N (Xa) = 1

GXin

b , D(Xa) = 0, D(Ya) = −GY inb . (43)

In our previous notation, we have

Ain = Xina , B in = Y in

a , Zin = (Xin

a , Y ina

), (44)

and

N = N (Xa) = 1

GXin

b , D = D(Ya) = −GY inb ,

(45)

K =(

1

GXin

b , − GY inb

).

We conclude that � = 0 and thus the measuring interaction isan independent intervention for the pair (Xa,Ya).

Also from Eq. (40)

G = 12 J. (46)

Now let us consider the state of the probe ξ to havevanishing expectation values 〈ξ |Xin

b |ξ 〉 = 〈ξ |Y inb |ξ 〉 = 0, and

a covariance matrix

�b =(

�b11 �b

12

�b12 �b

22

)=

(1/4 1/2

1/2 1/4

). (47)

Next notice that from Eq. (45)

〈N2〉 = 1

G2

⟨(Xin

b

)2⟩ = 1

G2�b

11 = 1

4G2. (48)

In a similar fashion, we obtain

〈{N,D}〉 = − ⟨{Xin

b ,Y inb

}⟩ = −�b12 = − 1

2 (49)

and

〈D2〉 = G2⟨(Y inb

)2⟩ = G2�b22 = G2

4. (50)

Thus [cf. Eq. (29)]

K =(

14G2 − 1

2

− 12

G2

4

)(51)

We then have from Eqs. (46) and (51)

K + i

2(� + G) = K + i

4J =

(1

4G2 − 12 + i

4

− 12 − i

4G2

4

). (52)

Since det(K + i4 J) = − 1

4 , we conclude that it is not a positivematrix and that our uncertainty principle (31) is violated byany state ξ satisfying 〈ξ |Xin

b |ξ 〉 = 〈ξ |Y inb |ξ 〉 = 0 and with

covariance matrix �b given by Eq. (47). However, sinceε(A) = 〈N2〉1/2 = 1

2Gand η(B) = 〈D2〉1/2 = G

2 , we concludethat ε(A)η(B) = 1

4 and thus such states saturate the Ozawauncertainty principle. This shows that our inequality (31) isindeed stronger than the Ozawa uncertainty principle.

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IV. TIGHTNESS OF THE MATRIX INEQUALITY

Concerning the tightness of our inequality, we remark thatwe obtain an equality in Eq. (28) if and only if we have anequality in Eq. (25). So the uncertainty relation can be saturatedif there is a set of complex numbers {λα}1�α�2n and a state �

such that � lies in the kernel of the operator K:∑1�α�2n

λαKα� = 0. (53)

This may or may not be possible, depending on the commuta-tion relations of the operators {Kα}1�α�2n.

In the case of the backaction-evading quadrature amplifierof the previous section this is not possible. Indeed, we have[cf. Eq. (45)]

[N,D] = −[Xin

b ,Y inb

] = − i

2. (54)

The saturating state �, Eq. (53), would have to satisfy

λ1N� + λ2D� = 0, (55)

for some constants (λ1,λ2) ∈ C2\{(0,0)}. In view of Eq. (54)this has the representation(

λ1

GXin

b + i

2Gλ2

∂

∂Xinb

)� = 0. (56)

There are no square-integrable solutions to Eq. (56), and thusthere is no saturating state.

Alternatively, consider the following noiseless quadraturetransducer [20] with the following input-output relations forthe quadrature operators (40):

Xouta = Xin

a − Xinb ,

Xoutb = Xin

a ,

Y outa = −Y in

b ,

Y outb = Y in

b + Y ina .

(57)

The probe observable is chosen to be M = Xb, and

N (Xa) = 0,

D(Xa) = −Xinb , (58)

D(Ya) = −(Y in

a + Y inb

).

With the previous notation

K = (0, − Y in

a − Y inb

). (59)

And thus, a saturating state satisfies

− λ2(Y in

a + Y inb

)� = 0. (60)

This can be achieved by any state � for λ2 = 0.

V. COMMENTS ON INVARIANCE PROPERTIESOF THE MATRIX FORMULATION

The matrix formulation of the noise-disturbance uncertaintyprinciple (31) is universally applicable. Given the myriadof measurement interactions and apparatuses [23,24], it isvirtually impossible to establish all transformations whichleave (31) unchanged. There are nonetheless instances wherethe inequality Eq. (31) is preserved under a certain type oftransformation, while Eqs. (9) and (11) are not.

Suppose that [Kα,Kβ] = iγ Jαβ , where γ �= 0 is some realconstant and J is the standard symplectic matrix. Moreover, letthe measurement interaction be an independent intervention(� = 0). Then Eq. (31) becomes K + iγ

2 J � 0. Notice thatthis looks formally like the RSUP Eq. (3). Suppose thatthe system undergoes a linear symplectic transformationKα �→ K ′

α = ∑1�β�2n SαβKβ , where S ∈ Sp(2n;R). Then

the noise-disturbance covariance matrix transforms undersimilarity as K �→ K′ = SKST . But given that S−1J(S−1)T =J, we conclude that the matrix uncertainty principle remainsunchanged: K′ + iγ

2 J � 0.But this may not happen for the Ozawa uncertainty relation.

Indeed let us consider again our previous example of thebackaction-evading quadrature amplifier. Remember that, foran independent intervention interaction, the Ozawa inequalitybecomes simply the Heisenberg inequality

εη � 14 . (61)

Suppose that we have now a noise-disturbance covariancematrix K = diag(ε2,η2) and that the probe (and possibly theobject) is subjected to a symplectic transformation such that

K �→ K ′ =√

2

2

(1 1

−1 1

)K. (62)

That is, the noise-disturbance vector K is rotated throughan angle of π

4 . Such a transformation is easily implementedby a certain unitary transformation U (S) generated by anappropriate Hermitian operator, quadratic in the variablesXb,Yb of the probe.

Then the Ozawa uncertainty inequality is modified to

(ε′)2 + (η′)2 �√

1 + 4〈{N ′,D′}〉2, (63)

which is manifestly different from Eq. (61). We also remarkthat the noise-disturbance correlations naturally appear undersuch transformations.

VI. CONCLUSIONS

In this work we have presented a universal matrix formu-lation of the uncertainty principle which is more stringentthan the noise-disturbance relation of Ozawa. Indeed, wehave proved that our formulation implies Ozawa’s one andshowed that is possible to comply with the Ozawa uncertaintyprinciple, while violating our universal form. Our inequalityis also more general in the sense that, unlike Ozawa’s relation,it also accounts for the noise-disturbance correlations.

We recall that recent experimental work performed byRozema et al. [17], using polarized entangled photons, and bySulyok et al. [18], using neutron-spin measurements, provedthe validity of Ozawa’s relation through weak measurementsand an explicit violation of relation (6). It would certainly bean interesting prospect to investigate the experimental validityof our more general formulation with a similar or any otherexperimental setup.

ACKNOWLEDGMENTS

The work of C.B. is supported by the FCT (Portu-gal) Grant No. SFRH/BPD/62861/2009. A.E.B.’s work is

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CATARINA BASTOS et al. PHYSICAL REVIEW A 89, 042112 (2014)

supported by the FAPESP (Brazil) Grant No. 2012/03561-0.The work of O.B. is partially supported by the FCT Project

No. PTDC/FIS/111362/2009. N.C.D. and J.N.P. have beensupported by the FCT Grant No. PTDC/MAT/099880/2008.

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