The Narcowich-Wigner spectrum of a pure state

Vol. 63 (2009) REPORTS ON MATHEMATICAL PHYSICS No.1

THE NARCOWICH-WIGNER SPECTRUM OF A PURE STATE

NUNO COSTA DIAS and JOAo NUNO PRATA

Departamento de Maternatica, Universidade Lus6fona de Humanidades e Tecnologias,Av. Campo Grande, 376, 1749-024 Lisboa, Portugal

and Grupo de Fisica Matematica, Universidade de Lisboa,Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Potugal(e-mail: [email protected]@mail.telepac.pt)

(Received February 11, 2008 - Revised October 27, 2008)

We completely characterize the Narcowich-Wigner spectrum of Wigner functions associatedwith pure states.Keywords: Weyl-Wigner quantum mechanics, Wigner functions, Narcowich-Wigner spectrum.

1. Introduction

In the context of the Weyl-Wigner formulation of quantum mechanics, thederivation of criteria characterizing the phase space functions in terms of theirquantum mechanical properties is a long standing problem. Such criteria should, atleast, provide a simple and practical answer to the following questions: i) Whichsquare-integrable phase-space functions are Wigner functions? ii) Which Wignerfunctions are associated with pure states? iii) Which Wigner functions are associatedwith mixed states? and iv) Which Wigner functions are nonnegative?

Partial answers to these questions have been provided in the literature: Hudson[1] proved that Wigner functions of pure one-dimensional states are everywherenonnegative iff the state is coherent. Soto and Claverie [2] generalized this resultfor higher-dimensional systems. However, the analogous characterization of positiveWigner functions associated with mixed states remains an open question. Some results in this direction were obtained in [3] where the authors modelled the result ofa measurement process on a system initially in a state described by a Wigner function F, in terms of the (nonnegative) phase space distribution obtained by smoothing(convoluting) F with another Wigner function Fa. For fixed Fa, this can be regardedas quantum dynamical map (a linear map that takes Wigner functions to Wignerfunctions) as long as the convolution Fa * F is a Wigner function for any Wignerfunction F. The authors then proved that this will happen provided Fa is point-wisenonnegative. The procedure generates a large set of nonnegative Wigner functions.On the other hand, in [4, 5] the necessary and sufficient conditions were derived fora Gaussian phase space function to be a Wigner function. Gaussians are obviously

[43]

44 N. C. DIAS and J. N. PRATA

important for their role in optics [6] and because they are bona fide probabilitymeasures. Moreover, they provide kernels Fo for quantum dynamical maps in theabove sense. Strictly speaking, Gracia-Bondia and Varilly [4] generalized the resultof [3] by proving that: (i) if Fo is any point-wise nonnegative phase space function(Wigner function or not), then Fo * F is a Wigner function for any Wigner functionF, and (ii) if Fo and F are Wigner functions, then Fo* F is always nonnegative(albeit possibly not a Wigner function). Using these results, they constructed explicitexamples that preclude the Hudson, Soto, Claverie theorem for mixed states; i.e,there are non-Gaussian positive Wigner functions associated with mixed states.

Other authors considered whether certain criteria (namely the uncertainty principle)would be useful in assessing whether a phase space function is a Wigner function.In [7, 8] it was shown that the uncertainty principle does not determine the quantumstate: Conditions for Gaussians in phase space to be Wigner distributions wereexpressed in terms of the so-called symplectic capacity of the associated Wignerellipsoid [9]. In [10], the multidimensional Hardy uncertainty principle was extendedand expressed in the context of Wigner quasi-distributions.

The concept of the Narcowich-Wigner (NW) spectrum was introduced andexplored in [11, 12]. Its main purpose was to provide a criterion to classifycertain phase space functions and a framework suitable to generate Wigner functionsdisplaying specific properties (namely being nonnegative). With each square integrablephase space function F , we may associate its NW spectrum, denoted as W (F), whichis a compact subset of R If this subset contains Planck's constant Ii, then the phasespace function F is a Wigner function and represents some quantum mechanicalstate. If, on the other hand, 0 E W(F), then F is a point-wise nonnegative function,that is a classical probability measure. We see that, quite straightforwardly, the NWspectrum provides a criterion for questions (i) and (iv) above.

On the other hand, the results of [3, 5] were all reformulated and unifiedin the context of NW spectra in [12]. Various authors [11, 13] also speculatedabout the possibility of using the NW spectrum to generate the entire subset ofWigner functions which are everywhere nonnegative (i.e. contain 0 in their NWspectra). While this has still not been fully achieved, by resorting to the NWspectrum, Narcowich [II] ·proved that one can generate a large set of positiveWigner functions of mixed states by convoluting suitable Wigner functions. Thiswas basically a generalization of the previous results of [3] and [4]. Other potentiallyinteresting applications of the NW spectrum are in the field of signal processing[14] and in the study of the classical limit of quantum mechanics [15].

An unresolved problem is that of using the NW spectrum to characterize pureand mixed states and thus to answer the questions (ii) and (iii) above. In its generalform this is still an open issue. For Gaussian states, however, the spectrum is ofthe form W(F) = [-110 , 110] (Eq. (20» and it is known that 110 = Ii iff the state ispure and 110 > Ii iff it is mixed (110 < Ii iff F is not a Wigner function).

In this work we will prove an important result towards the solution of the aboveproblem: we will complete the characterization of the NW spectrum of pure states.In fact the purpose of this paper is to prove the following theorem.

THE NARCOWICH-WIGNER SPECTRUM OF A PURE STATE 45

MAIN THEOREM. Let F E L 2(l~2d, d~) be the Wigner function of a pure state.

(i) If F is a Gaussian, then W(F) = [-Ii, Ii].(ii) If F is non-Gaussian, then W(F) = {-Ii, Ii}.

Notice that, as mentioned before, the result (i) in the Main Theorem is alreadyknown [11]. We include it here for completeness.

We will also prove an auxiliary result concerning the zeros of the so-calledHusimi function [16]. This result is stated in Proposition 2.4.

The Main Theorem is of course consistent with the Hudson, Soto, Claverietheorem: Gaussian states are the only pure states that contain zero in their NWspectrum. Unfortunately, the analogous characterization of the NW spectrum ofmixed states remains an open issue. As shown in [13], mixed states can have a veryintricate Wigner spectrum: intervals, sequences, or combinations of both.

This paper is organized as follows: Section 2 introduces the notation, the maindefinitions and reviews the fundamental properties of the NW spectrum. In Section 3we prove our main results concerning the NW spectrum of pure states. We start byproving some auxiliary results for the Husimi function and review some properties ofBargmann transforms in Section 3.1, and in Section 3.2 we prove the Main Theorem.Finally, in Section 4 we discuss some generalizations and possible applications ofthese results.

2. The Narcowich-Wigner spectrum

Let us first settle the preliminaries. We shall consider a d-dimensional systemon a flat phase space T* M ~ (]Rd)* X ]Rd ~ R2d with a global Darboux chart~ = (q, p) E R2d in terms of which the symplectic form reads

Q(~, ~') = ~TJ~' = e p' _ e a'. J - (OdXd IdXd) (1)- - Idxd °dxd .

Here the superscript "T" denotes matrix transposition. We shall use the standard

norms in real, !Ix lIJR.n = Jxf + ... +x;;, and in complex vector spaces IIzllcn =

Jlzl12 + ... + IZn12. To avoid a proliferation of subscripts, we shall just write IIx IIor Ilzl!. as it will be clear from the context the vector space we are referring toas well as its dimensionality. As usual L 2 (Rn, dx) denotes the space of squareintegrable complex-valued functions on Rn with respect to the Lebesgue measure.We may define in L 2 (Rn , dx) the inner product

(1/JIC/» = [ 1/J(x)c/>(x)dx ,JJR.n

(2)

We shall also consider the vector space T of Hilbert-Schmidt operators actingon the Hilbert space 'H = L 2 (1~d , dq) of our system, and which admit a kernel

46

representation of the form

N. C. DIAS and J. N. PRATA

(3)

(4)

(5)

A:H ~ H ,

1{f(q)~ (A1{f)(q) = f IRd A (q, q' )1jt (q' )dq ' ,

with A(q , q') E L2(IR2d , dqdq') . The Weyl-Wigner transform is the isomorphism (17)

Wq : T ~ L2 (IR2d , d~),

A~ Wq(A)(~) = 17d f JRd e-ip·yA (q + TIf, q - TIf) dy,

where 17 is some real, positive constant. The Weyl-Wigner transform may be extendedto other spaces, but this is all that will be necessary in this work. For a state1jt(q) E H, we define the corresponding density matrix to be the operator P1/I E T ,

(P1/IifJ) (q) = [ r 1/1 (q, q')ifJ(q')dq',JIRd

with r",(q, q') = 1jt(q)1jt(q'). The 17-Wigner function associated with 1jt is [18]:

1Wq(y" 1/f)(~) = (27t17)d Wq(P1/1 )(~)

= 1 [e-2ip.y/q1/f(q _ y )1/f (q + y)dy. (6)(27t'l)d J JRd

If 'I is equal to Planck's constant h, we shall denote Wh(1/f, y,)(~) simply by Wignerfunction or Wigner quasi-distribution in accordance with the literature. Moreover, itis straightforward to prove the following useful identity, also known as the Moyalidentity [l9) ,

L2d Wq(1/f . y,)(~)Wq(ifJ , ifJ)(~)d~ = (27t\)d !(y,lifJ)12, (7)

where Wq(1/f, 1/f), Wq(ifJ , ifJ) are n-Wigner functions associated with states 1/f, ifJ E

L 2 (IRd, dq) .We may consider statistical mixtures of pure states:

P = LPexP1/1a, Pex ~ 0, LPex = 1, (8)ex ex

which yield, via the Weyl-Wigner transform, the mixed state 'I-Wigner functions

(9)ex

The Wigner functions thus represent the states in the quantum phase space. Animportant issue is the identification of the necessary and sufficient conditions fora phase space function to be an n-Wigner function [20). A well-known necessarycondition is stated in the following proposition [20).


PROPOSITION 1.1. If F'1 is an 1]-Wigner function, then

( IF1J(~)12d~ ~ 1 . (10)JR2d (21f1])d

In addition, the equality holds if and only if F1J is associated with a pure state.

The integral on the left-hand side of the previous inequality is called the purityof the system.

The concept of NW spectrum was introduced in [11]. It appears naturally if onewants to formulate the necessary and sufficient conditions (the KLM conditions) fora phase space function to be a Wigner function or, more generally, an 77-Wignerfunction. In addition, we are able to (partially) characterize different types of Wignerfunctions in terms of their NW spectra, namely the nonnegative Wigner functions,the Gaussians, and pure or mixed states [12]. For this purpose it is useful to definethe symplectic Fourier transform. Let G(~) E L2(JR2d,d~). We define its symplecticFourier transform according to

F: L 2(JR2d) -+ L 2(JR2d): G(~) 1-+ G(a) == (FG)(a) = ( G(~)exp(iQ(~,a»d~.JR2d

(11)The inverse transform is given by the formula

1 ~ 1 i ~G(~) == (F- G)(~) = 2 )2d G(a)exp(-iQ(~,a»da.

( n R2d(12)

DEFINITION 1.1. The symplectic Fourier transform F(a) is said to be of thea-positive type (a E JR) if the m x m matrix with entries

~ (ia )Mjk = Fta, - ak) exp 2"Q(aj, ak) (13)

is hermitian and nonnegative for any positive integer m and any set of m pointsai, ... ,am in the dual of the phase space. By abuse of language, we shall oftensay that a phase space function F(~) is of the a-positive type, by which wemean that its symplectic Fourier transform F(a) is of the a-positive type. TheNarcowich-Wigner spectrum of a phase space function F(~) E L2(JR2d,d~) is theset

W(F) = {a E JR IF(a) is of the a-positive type} . (14)

With this definition we may now state the KLM (Kastler, Loupias, Miracle-Sole[21-23]) conditions.

THEOREM 1.1. The phase space function F(~) is an 1]-Wigner function (1] > 0),iff its symplectic Fourier transform F(a) satisfies the KLM conditions:

(i) F(O) = 1,(ii) F(a) is continuous and 77 E W(F).

(15)(16)


(17)

This theorem is a twisted generalization of Bochner's theorem, which states thatnonnegative functions (i.e. classical probability densities) are those of O-positive type.To proceed we introduce the convolution of F, G E L 2(IR2d, ds) which is definedby

(F * G)(s) = [ F(s - s')G(s')ds',J~2d

and is continuous, bounded and vanishes for lis II ---+ 00. Moreover, the symplecticFourier transform of the convolution is tantamount to point-wise multiplication

(:F(F * G»(a) = (:F(F»(a) . (:F(G»)(a). (18)

(19)

Let us now recapitulate some of the properties of the NW spectrum.

PROPOSITION 1.2. Let F, GEL2(IR2d , ds).

(i) If f is continuous, then W(F) is bounded and closed,(ii) a E W(F) {} (-a) E W(F),(iii) {a +a'i a E W(F), a' E W(G)} S; W(F * G), where (F * G) is the

convolution in (17).

The proof of these properties can be found in [13]. Counter-examples to thereciprocal of property (iii) are also given in [13]. In view of property (ii) ofproposition 1.4, we may assume without loss of generality that a ::: o. The resultsof [4], [5] for Gaussian states can be expressed in terms of the NW spectrum asfollows (see [11] for a proof):

LEMMA 1.1. Let F be a Gaussian

JdetA TF(s) = d exp [-(s - so) A(t; -[;0)] ,

n

where A is a real, symmetric, positive definite 2d x 2d matrix and [;0 E IR2d. Thenthe NW spectrum of F is of the form

W(F) = [-T/o, 1]0] , (20)

where 1]0 = max(W(F». That this maximum exists is guaranteed by property (i) ofProposition 1.2.

The following lemma states the well-known results of [3, 4] mentioned in theintroduction. We include it here to shown the usefulness of the NW spectra. Indeed,using the previous results, the proof is trivial, as shown in [12].

LEMMA 1.2. Let F, Fo be Wigner functions.

(i) The convolution Fo * F is point-wise nonnegative.(ii) If Fo is such that 0 E W(Fo) or 2li E W(Fo), then the convolution Fo* F

is a Wigner function.


Proof: If F, Fo are Wigner functions, then {-Ii, Ii} C W(Fo) n W(F) and thus,from Proposition 1.2 (iii) °E W(Fo * F), which means that Fo * F is pointwise nonnegative. On the other hand, if Fo is such that {O, ±Ii} C W(Fo) or{±Ii, ±21i} C W(Fo), then from Proposition 1.4 (iii), {-Ii, 0, Ii} c W(Fo * F), andthus Fo* F is a Wigner function and point-wise nonnegative. 0

This is a powerful way of generating positive Wigner functions, notwithstandingthe fact that the method does not exhaust the entire set of positive Wigner functions.

3. Main results

3.1. Husimi function and Bargmann transform

We start by deriving some results which will be useful to prove the MainTheorem. The following restricted version of Hadamard's theorem for functions ofseveral complex variables was proved in [2].

THEOREM 2.1 (Hadamard, Soto, Claverie). If F(z) is an entire function on en,with order of growth p and without zeros, we have

F(z) = exp (P(z) ,

where P(z) is a polynomial of degree r ::s p.

We recall that the order of growth p of F(z) is given by

. log (1ogM(R))p = lim ,M(R) = sup IF(z)l.

R-HX' log R IIzll=R

Let us now consider the coherent state

(21)

(22)

(23)d (lIq 112

Ii )ljIz(q) = (rrli)-"4 exp -2i: + z . q - "2 II Rezll 2,

Moreover, let Wll(ljIz, ljIz) and Wll(ljI, ljI) denote the Wigner functions associatedwith ljIz and ljI E L 2(JFf,d,dq), respectively.

DEFINITION 3.1. Let ljI E L 2(lRd, dq). The convolution of the form

(24)

is called a Husimi function. The Bargmann transform of ljI is defined by [17], [24]:

Ii 2 1 i (11q1l2 )-F(z) == (ljIlljIz)e2I1Rezll = -- exp --- + z· q ljI(q)dq,(rr Ii)d IRd 21i

The following lemma provides an important characterization of the Bargmanntransform.

LEMMA 3.1. Let ljI E L2(lRd, dq) and let F(z) be its Bargmann transform. Then

F(z) is an entire function on Cd. Moreover, the order of growth p of F(z) is atmost 2.


(27 )

(26)

o

Proof: Since the integral in (25) converges uniformly in any compact subset of<cd, we conclude that F (z) is an entire function on Cd. On the other hand , sincet/J , t/Jz are both normalized, we have from the Cauchy-Schwarz inequality,

IF(z)1 = e~ IIRez \12\(t/JIt/Jz) 1 ~ e~ \IRez !12.

h R2Consequently, M (R ) ~ e1 and thus p ~ 2.

The following result will be useful to prove the Main Theorem.

PROPOSITION 3.1. The Hu simi function Q (~) has no zeros iff the state t/J E

L 2(IRd, dq ) is a Gaussian.

Proof: If we compute WIl(t/Jz, t/Jz) explicitly (cf. (6), (23» , we obtain

1 (I 2)WIl(t/Jz, t/Jz )(~ ) = (rrli)d exp -h"~ - {II

where{ = (x, k ) = (Ii Re( z) , Ii Imt zj ). (28)

The convolution (24) thus reads:

Q (~ ) = (rr ~)d lu exp ( -*Il~-{-~J Il2) W/l (t/J , t/J )(~ /)d~J. (29)

We conclude that if t/J is a Gaussian, then W/l (t/J , t/J ) is a Gaussian and so is Q.Then , evidently, Q has no zeros.

Conversely, let us assume that t/J is not a Gaussian, but that it has no zeros .This means that the Bargmann transform F (z) also has no zeros. Indeed, we have

IF (z) \2 = ehllRezll2 I(t/JI t/Jz)12 = (2rr li)dehIl RezIl2 [ W/l(t/Jz, t/Jz )(~ )WIl ( t/J , t/J ) (~ )d~J JRU

= 2dehilRezf [ exp (-~II{ - (11 2) Wh(t/J, 1/J ) (~ / )d~ 'J JR2d n

= (2rrli)deh IIReZI12Q (2{ ) , (30)

where we used (7). This means that Q has zeros iff F (z) has zeros. From theHadamard, Soto Claverie theorem (Theorem 2.1), since F (z) is an entire functionon c-, with order of growth p ~ 2 and, by hypothesis, without zeros, then it mustbe a Gaussian. We then have, in particular, that

F(iy) = _1_d

[ exp ( _ llqIl2+ i y · q)t/J (q )dq , y E ]Rd , (31)(rr h)4 l JRd 21i

IIqU2__is a Gaussian. As F (i y ) is the Fourier transform of e- 2Jlt/J(q), this is only possible

IlqIl2__if e-2Jl t/J(q) is also a Gaussian . Hence t/J(q) is a Gaussian, which contradicts ourassumption. 0

THE NARCOWICH-WIGNER SPECTRUM OF A PURE STATE

3.2. Proof of the Main Theorem

The following lemma restricts the NW spectra of pure states.

LEMMA 3.2. Let F be the Wigner function of a pure state. Then

W(F) S; [-Ii, Ii] .

51

(32)

(33)

Proof: From Proposition 1.1 we know that, since F is a Wigner function, theequality in (10) holds for 1) = h, Moreover, if 1] > 0 is some other element inW(F), then again from (10), we know that

1 [ 2 1(21TIi)d = lJR2d IF(~)I d~ ::: (21T1])d'

It follows that 0 < 1] ::: h, The lemma is then an immediate consequence of property(ii) of Proposition 1.2. 0

The main theorem of this work refines the previous lemma.

Proof of the Main Theorem:(i) This is well-known. We prove it here for completeness. Let F be a Gaussian(19). From Lemma 1.5, we know that for F to be a Wigner function (pure ormixed), then the inclusion must hold

[-Ii, Ii] S; [-1]0, 1]0] = W(F). (34)

But from Lemma 2.5, if F is associated with a pure state, then the inverse inclusionmust also hold. We conclude that

W(F) = [-Ii, Ii] . (35)

(ii) Let us now assume that F(n is the Wigner function of a pure state, but thatit is not a Gaussian. We already know from Lemma 3.2, that its NW spectrum iscontained in [-Ii, Ii]. Let us then assume that there exists 1] E ] 0, Ii [ n W(F). Letus consider the family of Gaussians

Ga(~) = (1T~)d exp(_~1I~112). ex > O. (36)

The NW spectrum is W (G a) = [-ex, ex] [11]. The convolution of two such Gaussiansis again a Gaussian in this family. In fact, the convolution satisfies the semi-grouplaw

(G a *Gp)(n = Ga+f3(~)'

Let us then define the function

ex,{3 > O. (37)

Q(~) == (Gh-71 * G 7I * F)(~) = (Gh * F)(~) = (Gh-71 * Qo)(~), (38)

where Qo(;) = (G 7I * f)(;). The Wigner spectrum of G7I is W(G 7I ) = [-1],1]],

whereas that of F contains the set {-h, -1], 1], Ii} by assumption. From property(iii) of Proposition 1.2, we conclude that {-Ii, 0, Ii} c W(Qo). This means that


Qo is both a Wigner function and, according to Bochner's theorem, everywherenonnegative. Since Q(~) = (Gil * F)(~) is the convolution of a Gaussian of widthh with a Wigner function, it is a Husimi function. Moreover, since F is nota Gaussian, then Q(~) has at least one zero ~o (Proposition 3.1),

Q(~o) = O. (39)

From (38), we then have

0= ( exp(_II~o-~1I2)Qo(~)d~. (40)J'R.2d h - Tf

This is only possible, if Qo(~) has a negative part. However, that is contradictorywith the fact that Qo is of O-positive type. This proves that there can be no1] E ] 0, h [nW(F). Altogether, this means that the NW spectrum of a non-Gaussianpure state is contained in {-h, 0, h}. However, since non-Gaussian pure states cannothave nonnegative Wigner functions (by the Hudson, Soto, Claverie theorem), weconclude that 0 fJ. W(F), and thus W(F) = {-Ii, h}. 0

4. Concluding remarks

A few remarks are now in order.

• The Main Theorem admits a trivial generalization. We can easily replace h byany 1] > O. And thus, if 1/f is a Gaussian, then the 1]-Wigner function W7j(1{t, 1{t)(~)

has Wigner spectrum [-1], 1]], otherwise its Wigner spectrum is {-1], 1]}.

• The converse result of the Main Theorem remains as an open question. That is:suppose F E L 2 (1!~2d , d~) is such that Fca) is continuous and has NW spectrumW(F) = {-h, h}. Does that mean that F is the Wigner function associated witha non-Gaussian pure state? Or could it be a mixed state? If we could prove thisresult then pure and mixed states could be distinguished exclusively in terms oftheir NW spectra.

• In [11] Narcowich stated the following conjecture.

The convolution of a fixed Wigner function Fg with any other Wigner function FIl

is again a Wigner function iff the NW spectrum of F~ contains 0 or ±2h.

As we mentioned before, the sufficiency of the conditions is trivial to prove (seeLemma 1.2). Narcowich argued that the necessity is proved, if we assume that (i)W(F * G) = W(F) + W(G) == {a +a'ia E W(F), a' E W(G)} for any F, G E

L 2(]R2d, d~); and (ii) there exists a family of Wigner functions F: (~) defined for allsufficiently small E > 0 and satisfying W(FD S; [-h - E, -h + E] U [h - E, h + E].

Let us reproduce here his argument for completeness. If Fg is a Wigner functionand Fg * FIl is yet another Wigner function for any Wigner function FIl' thenh E W(Fg * FIl). From hypothesis (i), it follows that Ii E W(Fg) + W(FD.


If we take E ~ 0, we get Ii E W(F~) + {-Ii, Ii}, which is possible if andonly if either 0 or ±21i belong to W(F~). The problem with this argument isthat, as mentioned before, hypothesis (i) is not true in general [13]. However,that does not mean that the conjectured theorem is wrong. To the best of ourknowledge no counter-example has ever been found. The problem might be that thehypotheses suggested may be too restrictive. Let us just assume that for the fixedWigner function F~, we can always find a non-Gaussian pure state 1/J such that

W(F~ * W/i(o/, 1/J)) = W(F~) +W(W/i(1/J, 0/)). Notice that we do not require this tobe true for all Wigner functions. We only need this to be true for one state W/i(1/J, 1/J).We then have, from the Main Theorem: Ii E W(F~*W/i(1/J, 0/)) = W(F~)+{-li, +Ii}.And the necessity would thus be proved. The main result of this paper shows thatat least hypothesis (ii) is easily met.

• Our Lemma 3.2 has an important consequence in the context of open (dissipativequantum systems). In this framework one usually regards the purity (10) of thesystem interacting with an external environment [25] as an indication of dissipativeeffects. In fact it tends to decrease. In most cases if F/i (~) is the Wigner functionof the system at the initial time, then the Wigner function will usually take theform Gt(~) * F/i(~) at a later time t (up to composition with a linear symplectictransformation, which leaves the NW spectrum unchanged [11]). Here Gt(~) isthe Green function of the master equation. If the Narcowich conjecture mentionedabove is correct, then Gt should be such that ±21i E W (G t) or, alternatively thatoE W(G t ) . Moreover, if F/i is associated with a pure state (F/i = W/i(1/J, 0/)), thenfrom Lemma 3.2, we infer that G t * F/i must be associated with a mixed state,whenever there exists 11 ¥= 0 in W (G t ).

Acknowledgements

This work was partially supported by the grants POCTII020S/2003 and PTDCIMAT/6963512006 of the Portuguese Science Foundation.

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The Narcowich-Wigner spectrum of a pure state

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