Wigner-Yanase information on quantum state space: The geometric approach

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Wigner–Yanase information on quantum state space:

The geometric approach

Paolo Gibiliscoa)

Dipartimento di Scienze, Facolta` di Economia, Universita` di Chieti-Pescara ‘‘G. D’Annunzio,’’ Viale Pindaro 42, I–65127 Pescara, Italy

Tommaso Isolab)

Dipartimento di Matematica, Universita` di Roma ‘‘Tor Vergata,’’ Via della Ricerca Scientifica, I–00133 Roma, Italy

共Received 10 April 2003; accepted 3 May 2003兲

In the search of appropriate Riemannian metrics on quantum state space, the con-cept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogs of Fisher information. Although there exists a number of general theorems shedding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the Wigner– Yanase information. Using a well-known approach that mimics the classical pull-back approach to Fisher information, we are able to give explicit formulas for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner–Yanase information. Moreover, we show that this is the only monotone metric for which such an approach is possible. © 2003 American Institute of Phys-ics. 关DOI: 10.1063/1.1598279兴

I. INTRODUCTION

The notion of information proposed by Fisher is fundamental in probability and statistics for a number of reasons; here we mention only the Cramer–Rao inequality and the asymptotic be-havior of maximum lilkelihood estimators for exponential models共one can see Ref. 5 for unex-pected features and applications of Fisher information兲. In classical statistics Rao was the first to point out that Fisher information can be seen as a Riemannian metric on the space of probability densities. This point of view was nicely complemented by the results of Chentsov, saying that共on the simplex of probability vectors兲 Fisher information is the unique Riemannian metric contracting under Markov morphisms. This can be rephrased in a more suggestive way. Markov morphisms, or positive mappings, are the mathematical counterpart of the notion of noise. Now suppose that we want to use a distance to distinguish different states 共probability densities兲 in a statistically relevant way. Then the effect of noise must be that of contracting the metric. Chentsov theorem says therefore that in the classical case there is only one choice, the Fisher information共another argument producing Fisher information can be found in Ref. 43兲.

In the quantum case one deals with density operators instead of density vectors and com-pletely positive mappings play the role of Markov morphisms. As often happens in the quantum counterpart of a classical theory, instead of a uniqueness result, one has a classification theorem, due to Petz. This result states that there is bijection between statistically monotone metrics on quantum state space and the operator monotone functions: we have therefore a rich ‘‘garden’’ of candidates for the role of Fisher information in quantum physics. Among the elements of this family of metrics one can find, in a certain sense, the most relevant Riemannian metrics appeared in the literature.35,37

a兲_{Electronic mail: [email protected]}

b兲_{Electronic mail: [email protected]}

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Despite the existence of general results for the theory13,17,19,26,27,28,30,40,41 a number of open problems resists investigation. For example, it does not exist yet a general formula for the geo-desic path and the geogeo-desic distance associated to an arbitrary monotone metric. For the use of this kind of distances see, for example Ref. 32. Because of the absence of a general formula, inequali-ties共giving bounds for the geodesic distance兲 have been proved.38

In this paper we discuss the Wigner–Yanase skew information. To find the formulas for geodesic path and geodesic distance we mimic the classical approach to Fisher information via sphere geometry共one should note the importance of determining the geodesic path in the study of the 2-Wasserstein metric6兲. Indeed Wigner–Yanase information appears as the pull-back of the square root map.18 Next we prove the formula for the scalar curvature. One proof, due to Ditt-mann, uses the general formula13 and requires a long calculation. The second one just uses the pull-back approach. One should emphasize that, since the scalar curvature determines the asymptotic behavior of the volume共for a Riemannian metric兲 then it has also a statistical meaning in relation to the quantum analog of Jeffrey’s rule for determining prior probability distributions 共see Ref. 35兲. Finally we prove, as a corollary of the results in Refs. 25, 26, and 19 that the Wigner–Yanase information is the only monotone metric that can be seen as a pull-back metric. The paper is organized as follows. In Sec. II we review the geometric approach to Fisher information. Sec. III one finds an introduction to the general theory of statistical monotone met-rics. Sec. IV shows how the Wigner–Yanase information can be seen as a monotone Riemannian metric. In Sec. V we show that the Wigner–Yanase geometry can be seen as the sphere geometry transposed on the space of density matrices; moreover, we characterize it as the unique pull-back metric. Section VI contains some comments on the main results and on some open problems.

II. FISHER INFORMATION AND ITS GEOMETRY

The classical definition of Fisher information for an indexed family of densities p_␪is given by the variance of the score. In the case of a family indexed by only one parameter␪it is the number

I共␪兲⫽E_␪

冋冉

⳵

⳵␪log p␪

冊

2

册

, 共2.1兲

assigned to the parameter␪. For n parameters, say␪⫽(␪1, . . . ,␪n), it is a matrix defined on the parameter manifold given by

I共␪兲i j⫽E␪

冋冉

⳵

⳵␪ilog p␪

冊冉

⳵

⳵␪jlog p␪

冊册

. 共2.2兲

Geometrically this means that I(␪) is a symmetric bilinear form on the tangent spaces of the parameter manifold. In a coordinate free language it reads as

I共␪兲共U,V兲⫽E_␪关U共log p_␪兲 V共log p_␪兲兴, 共2.3兲 where U and V are vectors tangent to the parameter manifold and U(log p_␪) is the derivative of log p_␪ along the direction U, which means U(log p_␪)⫽(d/dt)log p_␪⫹tU

兩t⫽0.

I(␪) is a measure for the statistical distinguishabilty of distribution parameters. Under certain regularity conditions for ␪哫p_␪ the image of this mapping is a manifold of distributions. This manifold is the actual object of interest in information geometry rather than the space of distri-bution parameters and formula共2.3兲 defines a Riemannian metric g on it 共for a general reference see Ref. 1兲. Indeed, a vector u tangent to this manifold is of the form

u⫽ d

dtp␪⫹tU兩t⫽0, and the right hand side of共2.3兲 now reads as

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g共u,v兲ªEp

冋

u p

v

p

册

, 共2.4兲

defining the Fisher metric on the manifold of densities.

We restrict now to Pn傺Rn, the simplex of strictly positive probability vectors, that is,

Pnª兵␳苸Rn:兺i⫽1 n _␳

i⫽1,␳i⬎0,i⫽1,...,n其. An element ␳苸Pn is a density on the n-point set 兵1,...,n其 with ␳(i)⫽␳i. We regard an element u of the tangent space T_␳Pn⬅兵u苸Rn:兺i⫽1

n _u i

⫽0其 as a function u on兵1,...,n其 with u(i)⫽ui.

F is

isometric with an open subset of the sphere of radius 2 inRn. Proof: We consider the mapping␸:Pn→S2

F). From the very

defi-nition of geodesic distance, geodesic path and scalar curvature, one has for S_rn⫺1, with P1, P2 苸Sr n⫺1 , the following: 共1兲 geodesic distance, d共P1, P2兲⫽r•arcos

冉

具

P1, P2

典

r2

冊

; 共2兲 geodesic path connecting P1 and P2,

␥P₁, P₂_共t兲⫽r 共1⫺t兲P1⫹tP2 储共1⫺t兲P1⫹tP2储 共of course, t is not the arc length parameter兲;

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Scal共v兲⫽ 1

r2共n⫺1兲共n⫺2兲, because S_rn⫺1 has constant sectional curvature equal to 1/r2.

Let us denote by dF, ␥F, ScalF, respectively, the corresponding quantities for the Fisher

information. The above considerations give, for␳,␴苸Pn, the following:

共1兲 Bhattacharya distance, dF共␳,␴兲⫽2 arccos

冉

兺

i ␳i 1/2_␴ i 1/2

冊

; 共2兲 geodesic path connecting␳ and␴,

,ⵜ,ⵜ˜), where 具•,•典 is a Riemannian metric onM and ⵜ,ⵜ˜ are affine connections on M such that

,

where X,Y ,Z are vector fields. If Uⵜ,Uⵜ˜ are the parallel transport associated to ⵜ,ⵜ˜ then the above equation is equivalent to

具

Uⵜ共u兲,Uⵜ˜共v兲

F,ⵜm,ⵜe) such that

ⵜ2_⫽1 2共ⵜ

m_⫹ⵜe_兲,

whereⵜm,ⵜeare the mixture and exponential connections. These connections are torsion free and flat: once the representation by scores is used for the tangent spaces, the associated parallel transports are given by

U_␳␴m :T_␳P→T_␴P, U_␳␴m共u兲⫽␳

␴u,

U_␳␴e :T_␳P→T_␴P, U_␳␴e 共u兲⫽u⫺E_␴共u兲.

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III. METRIC CONTRACTION UNDER COARSE GRAINING

In the commutative case a Markov morphism 共or stochastic map兲 is a stochastic matrix T:Rn→Rk. In the noncommutative case a stochastic map is a completely positive and trace preserving operator T: Mn→Mk where Mn denotes the space of n by n complex matrices. We

shall denote byDn the manifold of strictly positive elements of Mn and byDn

1_傺D

n the

submani-fold of density matrices.

In the commutative case a monotone metric is a family of Riemannian metrics g⫽兵gn其 on

兵Pn其, n苸N such that

g_T(m_␳)共TX,TX兲⭐g_␳n共X,X兲

holds for every stochastic mapping T:Rn→Rmand all␳苸Pn and X苸T␳Pn.

In perfect analogy, a monotone metric in the noncommutative case is a family of Riemannian metrics g⫽兵gn其 on兵D_n1其, n苸N such that

g_T(m_␳)共TX,TX兲⭐g_␳n共X,X兲 holds for every stochastic mapping T: Mn→Mmand all␳苸Dn

1

and X苸T_␳D_n1.

Let us recall that a function f :(0,⬁)→R is called operator monotone if for any n苸N, any A, B苸Mnsuch that 0⭐A⭐B, the inequalities 0⭐ f (A)⭐ f (B) hold. An operator monotone function

is said to be symmetric if f (x)ªx f (x⫺1) and normalized if f (1)⫽1. In what follows by operator monotone we mean normalized symmetric operator monotone. With each operator monotone function f one associates also the so-called Chentsov–Morotzova function共see Ref. 8兲,

c_f共x,y兲ª 1 y f

冉

x

y

冊

, for x, y⬎0.

Define L_␳(A)ª␳A, and R_␳(A)ªA␳. Since L_␳,R_␳ commute we may define c(L_␳,R_␳). Now we can state the fundamental theorems about monotone metrics共uniqueness and classification are up to scalars兲.

Theorem 3.1: 共Ref. 7兲 There exists a unique monotone metric on Pn given by the Fisher

information.

Theorem 3.2:共Ref. 34兲 There exists a bijective correspondence between monotone metrics on

Dn

1

and operator monotone functions given by the formula

具

A,B

典

_{␳, f}ªTr„A•c_f共L_␳,R_␳兲共B兲….

The tangent space toD_n1 at ␳ is given by T_␳D_n1⬅兵A苸Mn:A⫽A*,Tr(A)⫽0其, and can be

decomposed as T_␳D_n1⫽(T_␳D_n1)c丣(T_␳D_n1)o, where (T_␳D_n1)cª兵A苸T_␳D_n1:关A,␳兴⫽0其, and (T_␳D_n1)o is the orthogonal complement of (T_␳D_n1)c, with respect to the Hilbert–Schmidt scalar product

具

A,B

典

ªTr(A*B). Each statistically monotone metric has a unique expression共up to a constant兲 given by Tr(␳⫺1A2), for A苸(T_␳D_n1)c. The following result will be used in Sec. V.

Proposition 3.3: 共see Ref. 3兲 Let A苸T_␳D_n1 be decomposed as A⫽Ac⫹i关␳,U兴 where Ac 苸(T␳Dn

1

)cand i关␳,U兴苸(T_␳Dn

1

)o. Suppose␸苸C1(0,⫹⬁). Then 共D␳␸兲共A兲⫽␸

⬘

共␳兲Ac⫹i关␸共␳兲,U兴.

As proved by Lesniewski and Ruskai each monotone metric is the Hessian of a suitable relative entropy; to state this result more precisely, we introduce some notation. In what follows g is an operator convex function defined on (0,⫹⬁) and such that g(1)⫽0. The formula

f共x兲⬅ fg共x兲ª

共x⫺1兲2

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associates a normalized, symmetric operator monotone function f⫽ fg to each g. We denote by

⌬_␴,␳⫽L_␴R_␳⫺1 the relative modular operator. The relative g –entropy of␳ and␴ is defined as Hg共␳,␴兲ªTr„␳1/2g共⌬␴,␳兲共␳1/2兲….

Hg is a divergence onDn in the sense of Refs. 14, and 1. If␳,␴are diagonal, Hg reduces to the

commutative relative g –entropy共see Ref. 9兲.

Theorem 3.4:共Ref. 30兲 Let g be operator convex, g(1)⫽0, f ⫽ f_g and␳苸D_n. Then

⫺_⳵⳵_t _⳵⳵_sHg共␳⫹tA,␳⫹sB兲

冏

t⫽s⫽0

⫽Tr„A•cf共L␳,R␳兲共B兲….

具

•,•

典

_␳f) at ␳ and

Scal1_f(␳) be the scalar curvature of (D_n1,

具

•,•

典

_␳f).

Theorem 3.5:共Ref. 13兲 Let ␴共␳兲 be the spectrum of␳. Then

Scalf共␳兲⫽

兺

x,y ,z苸␴(␳) h共x,y,z兲⫺

兺

x苸␴(␳) h共x,x,x兲, 共3.2兲 Scal_f1共␳兲⫽Scalf共␳兲⫹ 1 4共n 2_⫺1兲共n2_⫺2兲.

IV. WIGNER–YANASE INFORMATION AS A RIEMANNIAN METRIC Let ␳苸Dn

1

be a density matrix and let A be a self-adjoint matrix. The Wigner–Yanase information共or skew information, information content relative to A) was defined as

I共␳,A兲ª⫺Tr共关␳1/2,A兴2兲,

where关•,•兴 denotes the commutator 共see Ref. 42兲. Consider now g(x)ªgwy(x)ª4(1⫺

冑

x). In

this case

Hg共␳,␴兲⫽4„1⫺Tr共␳1/2␴1/2兲….

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f_wy共x兲ª1 4共

冑

x⫹1兲 2_, _c wy共x,y兲ª 1 y fwy

冉

x y

冊

⫽ 4 共

冑

x⫹

冑

y兲2. Let us consider the monotone metric,

具

A,B

典

_␳wyªTr„A cwy共L␳,R␳兲共B兲….

A typical element of (T_␳Dn)o has the form i关␳,A兴, where A is self-adjoint. We have

具

i关␳,A兴,i 关␳,A兴

典

_␳wy⫽Tr„i 关␳,A兴4共L_␳1/2⫹R_␳1/2兲⫺2共i 关␳,A兴兲…

⫽⫺4 Tr„共L␳1/2⫹R␳1/2兲⫺1共关␳,A兴兲共L␳1/2⫹R␳1/2兲⫺1共关␳,A兴兲…

⫽⫺4 Tr„共L␳1/2⫹R␳1/2兲⫺1ⴰ共L␳⫺R␳兲共A兲共L␳1/2⫹R␳1/2兲⫺1ⴰ共L␳⫺R␳兲共A兲…

⫽⫺4 Tr„共L␳1/2⫺R␳1/2兲共A兲共L␳1/2⫺R␳1/2兲共A兲…

⫽⫺4 Tr共关␳1/2_,A_兴2_兲 ⫽4I共␳,A兲,

and this explains why the monotone metric associated with the function 1

4(

冑

x⫹1)

2 _{is called the} Wigner–Yanase monotone metric.

V. THE MAIN RESULT

First of all, we calculate the scalar curvature of Wigner–Yanase information using Theorem 3.5. If fwy(x)ª 1 4(

冑

x⫹1) 2_{, we write Scal} wy 1 for Scalf 1 . Theorem 5.1: Scal_wy1 共␳兲⫽ 1 4共n 2_⫺1兲共n2_⫺2兲.

Proof: Let us calculate the auxiliary functions for cwy(x, y )ª4(

冑

x⫹

冑

冑

z兲. Now one can verify by calculation that the symmetrization of h1⫺

1

2h2 and the symmetriza-tion of 2 h3⫺h4vanish. Hence, by共3.1兲, the symmetrization of h vanishes, too. Since we sum up in formula共3.2兲 over all triples of eigenvalues we may replace h with its symmetrization without changing the summation result. Therefore

Scal_wy共␳兲⫽0 , Scal_wy1 共␳兲⫽1 4共n 2_⫺1兲共n2_{⫺2兲, ᭙}_␳_苸D n 1_. 䊐 Remark 5.2: The fact that Scalwy(␳)⫽0 can be seen by a different approach 共look at the Wigner–

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In what follows we use the pull-back approach to derive共and explain兲 the above formula in a direct way. Furthermore we deduce the geodesic distance and geodesic equation.

Let us denote by S the manifold 兵A苸Mn: Tr A A*⫽4, A⫽A*其. Clearly, since S is the

intersection of the radius 2 sphere inCn⫻n and the subspace of Hermitian matrices, it is isometric with a radius 2 sphere S2

n2⫺1

.

Now, let␸:D_n1→S傺Cn⫻n, ␸(␳)ª2

冑

␳. Then we have the following result共see Refs. 25, 18, 27, and 21兲.

Theorem 5.3: The pull-back by the map␸of the natural metric onS⬅S₂n2⫺1 coincides with the Wigner–Yanase monotone metric.

⫽4 Tr A 共L␳1/2⫹R␳1/2兲⫺2共B兲 ⫽Tr A cwy共L␳,R␳兲共B兲⫽

具

A,B

典

_␳ wy ,

which was to be proved. 䊐

From this result one can deduce the following.

Theorem 5.4: For the geodesic distance, the geodesic path and the scalar curvature of Wigner–Yanase information the following formulas hold:

(1) geodesic distance, dwy共␳,␴兲⫽2 arccos„Tr共␳1/2␴1/2兲…; 共5.1兲 (2) geodesic path, ␥wy␳,␴共t兲⫽2 „共1⫺t兲

冑

␳⫹t

冑

␴…2 Tr共„共1⫺t兲

冑

␳⫹t

冑

␴…2兲; 共5.2兲 (3) scalar curvature Scal_wy1 共␳兲⫽ 14共n 2_⫺1兲共n2_⫺2兲. _共5.3兲

Proof: The formulas are immediate consequences of the preceding theorem and of sphere geometry. Indeed by the pull-back argument the Wigner–Yanase metric looks locally like a sphere of radius 2 of dimension (n2⫺1). But for a sphere of this kind the sectional curvatures are all equal to 14 and therefore the scalar curvature is given by

1 4(n

2_⫺1)(n2_⫺2). _䊐

One may ask if other monotone metrics are the pull-back of some function␸different from the square root. The rest of the section answers this question.

Definition 5.5: A monotone metric

具

•,•

典

_{␳, f} is a pull-back metric if there exists a manifold

S傺Mn and a function␸苸C1(0,⫹⬁) such that the pull-back metric of␸:Dn

1_→S傺M

ncoincides

with

具

•,•

典

_{␳, f}.

Proposition 5.6: Let

具

•,•

典

_{␳, f} be a monotone metric, let c⫽cf be the associated C M -function

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冉

␸共x兲⫺␸共y兲

x⫺y

冊

2

⫽c共x,y兲. 共5.4兲

Proof: Apply the proposition共3.3兲 to tangent vectors in (T_␳D_n1)o. 䊐

Definition 5.7: Let␸,␹苸C1(0,⫹⬁). We say that 共␸,␹兲 is a dual pair if there exists an operator monotone f such that

␸共x兲⫺␸共y兲

x⫺y •

␹共x兲⫺␹共y兲

x⫺y ⫽c共x,y兲, 共5.5兲

where c⫽cf is the C M -function associated with f .

In such a case we say that f 共or cf) is a dual function. If共␸,␸兲 is a dual pair with respect to

f 共or cf) we say that f 共or cf) is self-dual. Obviously one has the following.

Proposition 5.8: To say that

具

•,•

典

_{␳, f} is a pull-back metric by␸it is equivalent to say that f 共or cf) is self-dual with respect to␸.

Definition 5.9: Two dual pairs (␸,␹),(␸˜ ,␹˜ ) are equivalent if there exist constants A1,A2,B1,B2 such that A1A2⫽1,

␸

˜⫽A1␸⫹B1,

␹

˜⫽A2␹⫹B2.

Obviously equivalent pairs define the same C M -function. In what follows we consider dual pairs up to this equivalence relation with the traditional abuse of language. We are ready to state the fundamental result of the theory that classifies dual pairs.

Theorem 5.10:共Refs. 23, 24, 25, 26, 36, and 19兲 Let␸,␹苸C1(0,⫹⬁). Then 共␸,␹兲 is a dual pair if and only if one of the following two possibilities hold:

„␸共x兲,␹共x兲…⫽

冉

x p p , x1⫺p 1⫺p

冊

, p苸关⫺1,2兴⶿兵0,1其, „␸共x兲,␹共x兲…⫽„x,log共x兲….

Corollary 5.11: The function f (x)⫽14(

冑

x⫹1)

2 _{is the only self-dual operator monotone}

func-tion, that is: the Wigner–Yanase metric is the only pull-back metric among statistically monotone metrics.

VI. CONCLUSIONS

Remark 6.1: Note that the formula 共5.1兲 implies dwy(␳,␴)⭐2␲. An analogous inequality holds for the Bures metric 共see Ref. 10, p. 311兲, also known as the SLD-metric: this is the monotone metric associated with f (x)⫽12(1⫹x). Indeed the formula,

dBures共␳,␴兲⫽

冑

2⫺2 Tr共␳1/2␴␳1/2兲1/2, 共6.1兲 seems to be the only other explicit formula for a geodesic distance 共in the family of statistically monotone metrics兲.

Remark 6.2: In general it is difficult to give explicit formulas for geodesic paths of monotone metrics. In the case of the Bures metric these geodesics can be given because they are projections of large circles on a sphere in the purifying space共see Ref. 10, p. 311 and Refs. 12, 4, and 39兲. For a discussion of geodesics for␣-connections see Refs. 27, 28.

Remark 6.3: A classical theorem classifies the spaces of costant curvature.29It is not known at the moment if there are other monotone metrics of costant sectional and scalar curvature.

Remark 6.4: We have seen in the commutative case that for the Levi–Civita connection of the pull-back of the square root the decomposition is available,

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ⵜ2_⫽1 2共ⵜ

m_⫹ⵜe_兲.

In the noncommutative case an analogous decomposition for the pull-back of the square root no longer holds. Indeed, on one hand, the use of Umegaki relative entropy H(␳,␴)⫽Tr„␳(log␳ ⫺log␴)… produces a similar decomposition, but for the Bogoliubov–Kubo–Mori metric.31,1,22,33 On the other hand, if one uses Hwy(␳,␴)⫽4„1⫺Tr(␳1/2␴1/2)… as a divergence on Dn

1

and con-structs the associated dualistic structure (

具

•,•

典

Hwy_,ⵜHwy_,ⵜHw y₎共again following the lines of Refs. 14 and 1兲, then the construction is trivial, namely, the dual connections both coincide with the Levi–Civita connection of the Wigner–Yanase information. This is easily seen on Pn where

H_g(␳,␴) reduces to Csiszar relative g-entropy: it is known that such an entropy induces the

␣-geometry where ␣ is given by the formula ␣⫽3⫹2g

⵮

(1)/g

⬙

(1) 共see Ref. 1, p. 57兲. For g ⫽4(1⫺

冑

x) this gives␣⫽0, that is, the Fisher information case 共see also Ref. 21兲.

ACKNOWLEDGMENTS

This research has been supported by the italian MIUR program ‘‘Quantum Probability and Infinite Dimensional Analysis’’ 2001–2002. It is a pleasure to thank J. Dittmann for a number of valuable conversations on this subject and for permitting us to reproduce in this paper the proof of Theorem 5.1.

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