Localization of Eigenstates & Mean Wehrl Entropy

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Karol Zyczkowski˙ 1,2 1Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagielloński, ul. Reymonta 4, 30-059 Kraków, Poland 2Centrum Fizyki Teoretycznej PAN Al. Lotników 32/44, 02-668 Warszawa, Poland (October 22, 1999)

In this paper we advocate an alternative method of Dynamics of a periodically time dependent quantum sys- solving the problems with arbitrariness of the choice of tem is reflected in the features of the eigenstates of the Flo- the expansion basis. Instead of working in a finite dis- quet operator. Of the special importance are their localiza- crete basis, we shall use the coherent states expansion tion properties quantitatively characterized by the eigenvec- of the eigenstates of F . For several examples of compact tor entropy, the inverse participation ratio or the eigenvector classical phase spaces one may construct a canonical fam- statistics. Since these quantities depend on the choice of the eigenbasis, we suggest to use the overcomplete basis of coher- ily of the generalized coherent states [12]. Localization ent states, uniquely determined by the classical phase space. properties of any pure quantum state may be character- In this way we define the mean Wehrl entropy of eigenvectors ized by the Wehrl entropy, equal to the average log of of the Floquet operator and demonstrate that this quantity its overlap with a coherent state [13,14]. We propose to is useful to describe quantum chaotic systems. describe the structure of a given Floquet operator F by the mean Wehrl entropy of its eigenstates. This quantity, explicitly defined without any arbitrariness, is shown to be a useful indicator of quantum chaos. This paper is organized as follows. In section II we review the definition of the Husimi distribution, stellar e-mail: [email protected] representation, and the Wehrl entropy. For concreteness we work with the SU(2) vector coherent states, linked to the algebra of the angular momentum operator and corresponding to the classical phase space isomorphic with the I. INTRODUCTION sphere. In section III we define the mean Wehrl entropy of eigenstates and present analytical results obtained for Analysis of quantum chaotic systems is often based on low dimensional Hilbert spaces. Exemplary application the statistical properties of the spectrum of the Hamilto- of this quantity to the analysis of the quantum map de- nian H (in the case of autonomous systems) or the Flo- scribing the model of the periodically kicked top is pro- quet operator F (in the case of periodically perturbed videdinsectionIV. systems). In general, quantized analogous of classically chaotic systems display spectral fluctuations conforming to the predictions of random matrices. Depending on the II. HUSIMI DISTRIBUTION AND STELLAR geometrical properties of the system one uses orthogonal, REPRESENTATION unitary or symplectic ensemble [1,2]. Another line of research deals with eigenstates of the Consider a compact classical phase space Ω, a classi- analyzed quantum system. One is interested in their cal area preserving map Θ : Ω Ω and a correspond- localization properties, which can be characterized by ing quantum map F acting in an→N-dimensional Hilbert the eigenvector distribution [3–6] the entropic localiza- space N . A link between classical and quantum me- tion length [7] or the inverse participation ratio [8]. All chanicsH can be established via a family of generalized this quantities, however, are based on the expansion of coherent states α . For several examples of the classical ~ an eigenstate in a given basis bi , which may be chosen phase spaces there| i exist a canonical family of coherent { } ~ arbitrarily. If one chooses (with a bad will), bi as the states. It forms an overcomplete basis and allows for an eigenbasis of F , all these quantities carry no information{ } identity resolution Ω α α dα = 1. Any mixed quantum whatsoever. One may ask, therefore, to what extend the state, described by a density| ih | matrix ρ can be represented quantitative analysis based on the eigenvector statistics by the generalizedR Husimi distribution [15], (Q-function) is reliable. Let G denote a unitary operator, such that ~bi is its Hρ(α):= αρα. (2.1) eigenbasis. We showed [9–11] that the eigenvector{ } statis- h | | i tics of a quantum map F conforms to the prediction of The standard normalization of the coherent states, random matrices, if operators F and G are relatively ran- α α = 1, assures that Hρ(α)dα = 1. For a pure h | i Ω dom, i.e., their commutators are sufficiently large. quantum state ψ the Husimi distribution is equal to | i R

1 2 the overlap with a coherent state Hψ(α):= ψ α .Let sphere. Some of these zeros may be degenerated, just as us note that the Husimi distribution was successfully|h | i| ap- in the case of a coherent state. This method of charac- plied to study dynamical properties of quantized chaotic terizing a pure quantum state is called the stellar repre- systems [16,17]. sentation [21,22]. Consider a discrete partition of the unity into n terms; In the analyzed case of the classical phase space iso- n n 2 i=1 pi = 1. The Shannon entropy Sd = i=1 pi ln pi morphic with the sphere S the Wehrl entropy (2.2) of a characterizes uniformity of this partition.− In an analo- state ρ equals Pgous way one defines the Werhl entropy ofP a quantum state ρ [13] 2j +1 π 2π Sρ = sin ϑdϑ dϕHρ(ϑ, ϕ)ln Hρ(ϑ, ϕ) , − 4π 0 0 Z Z h i Sρ = Hρ(α)ln[Hρ(α)]dα, (2.2) (2.4) − ZΩ in which the summation is replaced by the integration since the measure element dα is equal to (2j + over the classical space Ω. This quantity characterizes 1) sin ϑdϑdϕ/4π. Under this normalization the entropy the localization properties of a quantum state in the clas- of the maximally mixed state ρ equals to ln N. The Husimi distribution of an∗ eigenstate j, m may be sical phase space. It is small for coherent states localized | i in the classical phase space Ω and large for the delocalized computed directly from the definition (2.3). Due to the states. The maximal Wehrl entropy corresponds to the rotational symmetry it does not depend on the azimuthal maximally mixed state ρ , proportional to the identity angle ϕ matrix, for which the Husimi∗ distribution is uniform. 2(j m) 2(j+m) 2j Although the notions of the generalized coherent H j,m (ϑ)=sin − (ϑ/2) cos (ϑ/2) , | i j m states, the Husimi distribution, and the Wehrl entropy − are well defined for several classical compact phase (2.5) spaces, in this work we analyze in detail only the most important case Ω = S2. This phase space is typical to which simplifies the computation of the Wehrl entropy. physical problems involving spins, due to the algebraic Simple integration gives for the reference state j, j | i properties of the angular momentum operator J.Inthis case one uses the family of spin coherent states lo- N 1 2j ϑ, ϕ Scoh = − = . (2.6) calized at points (ϑ, ϕ) of the sphere S2. These| states,i N 2j +1 also called SU(2) vector coherent states, were introduced Due to the rotational invariance the Wehrl entropy is the by Radcliffe [18] and Arecchi et al. [19] and are an exam- sameforanyothercoherentstate. ple of the general group theoretic construction of Perelo- mov [12]. ¯ N j m Sj,m SJz | i Consider an N =2j+ 1 dimensional representation of 2 1/2 1/2 1/2 1/2=0.5 the angular momentum operator J. For a reference state 3 1 1 2/3 1 ln 2 0.769 one usually takes the maximal eigenstate j, j of the com- 3 | i 0 5/3 ln 2 − ≈ ponent Jz. This state, pointing toward the ”north pole” − of the sphere, enjoys the minimal uncertainty. The vec- 4 3/2 3/2 3/4 3 ln 3 0.951 2 − 2 ≈ tor coherent state represents the reference state rotated 1/2 9/4 ln 3 − by the angles ϑ and ϕ. Its expansion in the basis j, m , 2 4/5 | i m = j,...,+j reads [20] 5 2 1 79/30 ln 4 2 ln 96 1.087 − 5 0 47/15 − ln 6 − ≈ m=j − j m ϑ j+m ϑ 5/2 5/6 ϑ, ϕ = sin − ( )cos ( ) | i 2 2 × 6 5/2 3/2 35/12 ln 5 5 1 ln 50 1.196 m= j 2 3 X− 1/2 15/4 −ln 10 − ≈ 2j 1/2 − exp i(j m)ϕ j, m , (2.3) Table 1. Wehrl entropy S j,m for the eigenstates of − j m | i ¯ | i h − i Jz and its mean SJz for N =2,3,4,5,6. Due to the 2π π geometrical symmetry S = S . where dϕ sin ϑdϑ ϑ, ϕ ϑ, ϕ (2j +1)/4π=1. j,m j, m 0 0 | ih | | i | − i For the SU(2) coherent state the distribution (2.1) The Wehrl entropies for other eigenstates of Jz are col- R R 2 equals in this case Hψ(ϑ, ϕ):= ψ ϑ, ϕ .Twodifferent lected in Tab. 1 for some lower values of N. These results spin coherent states overlap unless|h | theyi| are directed into may be also obtained from the general formulae provided two antipodal points on the sphere. The Husimi repre- by Lee [23] for the Wehrl entropy of the pure states in the sentation of a spin coherent state has thus one zero (de- stellar representation. Eigenstate j, m is represented by generated N 1 times) localized at the antipodal point. j +m zeros at the south pole and j| mizeros at the north − In general, any pure quantum state can be uniquely de- poles. For j =1/2(N= 2) all the− states are SU(2) co- scribed by the set of N 1 points distributed over the herent, so their entropies are equal. For j =1(N=3) −

2 the coherent state 1, 1 is characterized by the smallest for larger values of N, and to study the dependence Smax entropy, while the state| i 1, 0 by the largest (among the of N. pure states). The larger |N, thei more place for a various Let us emphasize that for N>>1 the pure states ex- behaviour of pure states, measured by the values of S. hibiting small Wehrl entropy, (of the order of Smin), are The axis of the Wehrl entropy is drawn schematically in not typical. In the stellar representation coherent states Fig.1. correspond to coalescence of all N 1 zeros of Husimi distribution in one point. In a typical− situation the den- pure sity of the zeros is close to uniform on the sphere, and the mixed a) N=2 Wehrl entropy of such delocalized pure states is large. A S random state can be generated according to the natural uniform masure on the space of pure states by taking any 0 Smin ln2 vector of a N N random matrix distributed according × pure to the Haar measure on U(N). Averaging over this mea- mixed sure one may compute the mean Wehrl entropy S of b) N=3 N 1/2 1 S the pure states belonging to the N dimensionalh Hilberti space. Such integration was performed in [3,27,28] in a 0 Smin 3 ln3 S S Jz max slightly different context leading to pure mixed N 1 c) N=4 S =Ψ(N+1) Ψ(2)= , (2.7) 1/2 1 S h iN − n n=2 0 Smin ln4 X S 4 S Jz max where Ψ denotes the digamma function. Note that an- . . other normalization of the coherent states used in Ref. [28], leads to results shifted by a constant ln N. Such pure mixed a normalization allows one for a direct comparison− be- d) N>>1 1/2 1 S tween the entropies describing the states of various N. In the asymptotic limit N the mean entropy S lnN →∞ h iN 0 Smin=1-1/N N behaves as ln N + γ 1 ln N 0.42278, which is close Smax=? − ∼ − 1- to the maximal possible Wehrl entropy for mixed states Sρ =lnN. This result is schematically marked in Fig 1d.∗ FIG. 1. Axis of Wehrl entropy for pure states of N dimensional Hilbert space; a) N =2forwhichS¯min = S¯max =1/2; b) N =3forwhichS¯min =2/3, S¯J 0.77, S 3 0.83, III. MEAN WEHRL ENTROPY OF z ≈ h i ≈ Smax 0.973; c) N =4forwhichS¯min =3/4, S¯J 0.95, EIGENSTATES OF QUANTUM MAP ≈ z ≈ S 4 1.08, Smax 1.24; and d) N>>1, where h i ≈ ≈ S¯min =1 1/N while S N ln N 0.423. − h i ≈ − Consider a quantum pure state in the N-dimensional Hilbert space. Its Wehrl entropy computed in the vector It has been conjectured by Lieb [24] that vector coher- coherent states representation may vary from 1 1/N ,for ent states are characterized by the minimal value of the a coherent state, to the number of order of ln N−,forthe Wehrl entropy Smin = Scoh, the minimum taken over all typical delocalized state. This difference suggests a sim- mixed states. For partial results in the direction to prove ple measure of localization of eigenstates of a quantum this conjecture see [25,23,26]. It was also conjectured map F . Denoting its eigenstates by ψi ; i =1,...,N we [23] that the states with possibly regular distribution of define the mean Wehrl entropy of eigenstates| i all N 1 zeros of the Husimi function on the sphere are characterized− by the largest possible Wehrl entropy N ¯ 1 among all pure states Smax. Such a distribution of zeros SF = S ψ . (3.1) N | ii is easy to specify for (N 1) = 4, 6, 8, 12, 20, which cor- i=1 X respond to the Platonian− polyhedra. For N = 2 all pure This quantity may be straightforwardly computed nu- states are coherent, so Smin = Smax =1/2. For N =3 merically for an arbitrary quantum map F . For quan- the maximal Wehrl entropy Smax =5/3 ln 2 0.97 is achieved for the state 1, 0 , for which the− two≈ zeros of tum analogues of classically chaotic systems exhibiting | i no time reversal symmetry all eigenstates are delocalized. Husimi function are localized at the opposite poles of the ¯ sphere. For N = 4 the state with three zeros located In this case the mean Wehrl entropy of eigenvectors SF at the equilateral triangle inscribed in a great circle is fluctuates around S N ln N. In the oppositeh casei ∼ of an integrable dynamics the characterized by Smax =21/8 2ln2 1.24. It will be interesting to find such maximally− delocalized≈ pure states eigenstates are, at least partially, localized. A simple example is provided by any Hamiltonian diagonal in the

3 Jz basis (or the basis of any other component of J). The IV. MEAN WEHRL ENTROPY FOR THE ¯ mean Werhl entropy SJz is given in table 1 for some val- KICKED TOP ues of N. Further analysis shows [31] that for larger N 1 the mean entropy behaves as 2 ln N. This result has a In order to demonstrate usefulness of the mean Wehrl simple interpretation. Let us divide the surface of the entropy in the analysis of quantum chaotic systems we 1 sphere into N h¯− cells. A typical eigenstate of Jz is present numerical results obtained for the periodically ∼ localized in a longitudinal strip of a constant polar angle kicked top. This model is very suitable for investigation θ and covers √N of the cells, so its entropy is of the order of quantum chaos [35,1]. Classical dynamics takes place of ln √N. on the sphere, while the quantum map is defined in terms ¯ The quantity S is well–defined in a generic case of op- of the components of the angular momentum operator J. erators F with a nondegenerate spectrum. In the case The size of the Hilbert space is determined by the quan- of the degeneracy, there exists a freedom of choosing the tum number j and equals N =2j+ 1. One step evo- eigenvectors; to cure this lack of uniqueness we define S¯ 2 F lution operator reads Fo =exp( ipJz)exp( ikJx/2j). as the minimum over all possible sets of eigenvectors of For p =1.7 the classical system becomes− fully− chaotic for F . Having a general definition of the mean Wehrl entropy the kicking strength k 6 [35]. This system possesses a of eigenvectors of an arbitrary unitary operator, one may generalized antiunitary≈ symmetry and can be described pose a question, for which operators Fmin (Fmax)ofa by the orthogonal ensemble. The time reversal symme- fixed N with a nondegenerate spectrum this quantity is try may be broken by an additional kick [1]. The system the smallest (the largest). It is clear that S¯ is larger 2 Fmin Fu = Fo exp( ik0Jy /2j) pertains to CUE and will be than Scoh (for N>2), since the set of any N coher- called the unitary− top. ent states does not form an orthogonal basis. On the ¯ a) b) other hand, the minimum is smaller than SJz , as explic- t t itly demonstrated in Appendix for N = 3. The value ¯ 0.5 0.5 SFmax is larger than the average over the random unitary matrices S U(N) = S N and smaller than Sρ =lnN. 0 0 h i h i ∗ ¯ The mean Werhl entropy of the eigenstates SF may be related with the eigenvector statistics of the oper- −0.5 −0.5 ator F . Let us expand a given coherent state in the N 1 2 3 4 5 6 1 2 3 4 5 6 eigenbasis of the Floquet operator, α = i=1 ci(α) ψi . φ φ The dynamical properties of a quantum| i system are char-| i P c) d) acterized locally [29] by the Shannon entropy Ss(α):= t t N 2 2 i=1 ci(α) ln ci(α) . The mean Wehrl entropy may 0.5 0.5 −be thus| written| as| an average| over the phase space P 0 0 1 S¯ = S (α)dα. (3.2) F N s −0.5 −0.5 ZΩ 1 2 3 4 5 6 1 2 3 4 5 6 This link is particularly useful to analyze the influence φ φ of the time reversal symmetry. In presence of any generalized antiunitary symmetry the operator F may be described by the circular orthogonal ensemble (COE). FIG. 2. Husimi distribution of exemplary eigenstates of the There exists a symmetry line in the phase space and the Floquet operator of the orthogonal kicked top for N =62 coherent states located along this line display eigenvec- in the dominantly regular regime (k =0.5), a) and b), and tor statistics typical of COE [32]. This symmetry is also chaotic regime (k =8.0), c) and d). The sphere is represented visible in the stellar representation of the eigenstates and in a rectangular projection with t =cosϑ. manifests itself by a clustering of zeros of Husimi func- tions [33,34]. However, a typical coherent state does not Fig. 2 presents the Husimi distributions of two eigen- exhibit such a symmetry and its eigenvector statistics states of Fo for p =1.7 in the regime of regular motion is typical to the circular unitary ensemble (CUE). Thus (k =0.5) and two, for which the classical dynamics is for a system with the time-reversal symmetry the mean chaotic (k =8.0). In the quasiregular case the eigen- Wehrl entropy will be slightly smaller than for the analo- states are localized close to parallel strips, covered uni- gous system with the time reversal symmetry broken, but formly by the eigenstates of Jz. On the other hand, the much larger than the Shannon entropy of real eigenvec- eigenstates of the chaotic map are delocalized at the en- tors of matrices pertaining to the orthogonal ensemble. tire sphere. These differences are well characterized by the values of the Wehrl entropies, equal correspondingly: a) 2.77, b) 2.66; and c) 3.72, d) 3.80. The data, obtained for N = 62, may be compared with the mean entropy of

4 ¯ the unperturbed system, SJz 2.465, the mean Wehrl V. CONCLUDING REMARKS entropy of chaotic system without≈ time reversal symetry, S 3.712, and the maximal entropy of the mixed h i62 ≈ The Wehrl entropy of a given state characterizes its state, Sρ =ln62 4.1271. localization in the classical phase space. We have shown The above∗ eigenstates≈ are typical for both systems, and ¯ that the mean Wehrl entropy SF of eigenstates of a given the other 60 states display a similar character. The prop- evolution operator F may serve as a useful indicator of erties of all eigenstates are thus described by the mean quantum chaos. Let us emphasize that this quantity, ¯ Wehrl entropy of eigenstates SF . The dependence of this linked to the classical phase space by a family of coher- quantity on the kicking strength k is presented in Fig. 3. ent states, does not depend on the choice of basis. This To show a relative difference between the entropies typi- contrasts the others quantities, like eigenvector statis- cal to the regular dynamics we use the scaled coefficient tics, localization entropy, inverse participation ratio, often used to study the properties of eigenvectors. It will S¯ S¯ F Jz be interesting to find the unitary operators (or rather the µ(F ):= − ¯ . (4.1) S N SJz repers) for which ¯ is the smallest or the largest. h i − SF The mean Wehrl entropy of eigenstates enables one Per definition µ is equal to zero, if F is diagonal in the to detect the transition from regular motion to chaotic Jz basis, which corresponds to the integrability. In the dynamics. On the other hand, it is not related to the chaotic regime F is well described by CUE and µ is close classical Kolmogorov–Sinai entropy (or to the Lapunov to unity. This is indeed the case for the unitary top exponent), so it cannot be used to measure the degree of with k0 = k/2andk>6. The growth of µ is bounded chaos in quantum systems. Such a link with the classical and therefore it cannot, in general, follow the increase of dynamics is established for the coherent states dynam- the classical Kolmogorov–Sinai entropy Λ (the Lapunov ical entropy of a given quantum map [36,28], but this exponent averaged over the phase space), which for the quantity is much more difficult to calculate. Both these classical system grows with the kicking strength k [11]. quantities characterize the global dynamical properties of The data for the orthogonal top fluctuate below unity, a quantum system, in contrast to the entropy of Mirbach due to existence of the symmetry line. The difference and Korsch [37,38], which describes the local features. between the coefficients µ obtained for both models does Mean Wehrl entropy characterizes the structure of not depend on the kicking strength, but decreases with eigenvectors of F , and is not related at all to the spec- N and vanish in the classical limit N . →∞ trum of this operator. Thus it is possible to construct a unitary operator with a Poissonian spectrum and the delocalized eigenvectors. Or conversely, one may find an operator with spectrum typical to CUE and all eigenstates localized. This shows that the relevant informa- 1.0 1 tion concerning the dynamical properties of a quantum system described by an unitary evolution operator F is contained as well in its spectrum and in its eigenstates. I am indebted to W. S lomczyński for fruitful discus- 0.5 sions and a constant interest in the progress of this research. I am also thankful to M. Ku´sandP.Pakoński for helpful remarks. It is a pleasure to thank Bernhard Mehlig for the invitation to Dresden and the Center for 0.0 0 Complex Systems for a support during the workshop. Fi- 0246810 nancial support from Polski Komitet BadańNaukowych k in Warsaw under the grant no 2P-03B/00915 is gratefully acknowledged.

FIG. 3. Scaled mean Wehrl entropy µ of the eigenvectors of the Floquet operator for the kicked top as a function of APPENDIX A: MINIMAL MEAN WERHL the kicking strength k for N = 62. The data are obtained ENTROPY FOR N =3 for two models: unitary top ( ) and orthogonal top ( ). The • crosses denote values of the classical Kolmogorov-Sinai en- In the case j =1(N= 3) any pure state may be tropy Λ, which characterize the transition to chaos in the described by the position of two points on the sphere - classical analogue of the orthogonal top. zeros of the corresponding Husimi distribution. In the stellar representation the eigenstates m of Jz are described by 1 + m zeros at the south pole| i of the sphere, and 1 m zeros at the north pole, with m = 1, 0, 1. − − The mean Werhl entropy of eigenstates of Jz is equal to

5 1 (ln 2)/3 0.769. We find another set of three or- it does not solve the problem of finding the global mini- thogonal− pure≈ states, characterized by a smaller value of mum. One can expect, that this minimum will occur for S¯, by allowing to move one of the zeros of the Husimi asetofNmutually orthogonal states, each of them as representation along the meridian ϕ =0(andϕ=π). In close to the coherent state, as possible. other words, let us define two orthogonal states

χ := cos χ 1 +sinχ0 , | +i | i | i χ := sin χ 1 +cosχ0 . (A1) | −i − | i | i Keeping the state 1 fixed we define thus a one param- |− i eter family of orthogonal basis Oχ = 1 , χ , χ+ . [1] F. Haake, Quantum Signatures of Chaos, Springer, The Husimi distributions of the state{|χ − ihas| − twoi | zerosi} Berlin, 1991. | −i at the polar angles θ ,π , while the zeros of χ+ are [2] G. Casati and B. Chirikov (eds), Quantum Chaos. Be- { − } 2 | 2 i located at π, 2π θ+ .Hereθ =(2 c )/(2 + c )and tween Order and Disorder, Cambridge University Press, 2 { −2 } − − Cambridge, 1995. θ+ =(2c 1)/(2c + 1) with c =tanχ. The angle θ be- longs to [0−, 2π)andtheθ=πrepresents the south pole, [3] M. Ku´s, J. Mostowski, and F. Haake, J. Phys. A 21 while θ =0(orθ=2π) denotes the north pole. The (1998) L1073. location of zeros of Husimi representation of eigenstates [4] O. Bohigas, in Chaos and Quantum Physics, eds. M.- of J and O is presented in Fig.4. J. Giannoni, A. Voros, and J. Zinn-Justin, Les Houches z π/4 Summer School, Session LII, North-Holland, Amster- a) N N b) dam, 1991. * [5] F. Izrailev, Phys. Rep. 129 (1990) 299. [6] F. Haake and K. Zyczkowski,˙ Phys. Rev. A 42 (1990) 1013. [7] G. Casati, I. Guarneri, F. Izrailev, and R. Scharf, Phys. * Rev. Lett. 64 (1990) 5. - [8] E. Heller, Phys. Rev. A 35 (1987) 1360. [9] M. Ku´sandK.Zyczkowski,˙ Phys. Rev. A 44 (1991) 956. [10] K. Zyczkowski,˙ Operator Theory: Adv. & Appl. 70 = =0 (1994) 277. [11] K Zyczkowski,˙ Acta Phys. Pol. B 24 (1993) 967. [12] A. Perelomov, Generalized Coherent States and their Ap- * * plications, Springer, Berlin, 1986. S S [13] A. Wehrl, Rev. Mod. Phys. 50 (1978) 221. [14] A. Wehrl, Rep. Math. Phys. 30 (1991) 119. [15] K. Husimi, Proc. Phys. Math. Soc. Jpn. 22 (1940) 264. [16] K. Takahashi, J. Phys. Soc. Jap. 55 (1986) 762. [17] K. Zyczkowski,˙ Phys. Rev. A 35 (1987) 3546. FIG. 4. Cross-section of the Bloch sphere along the merid- [18] J. M. Radcliffe, J. Phys. A 4 (1971) 313. ian φ =0(φ=π). a) zeros of the eigenstates of the operator [19] F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Jz: 1 ( ), 0 ( ), and 1 ( ); b) zeros of the eigenstates Phys. Rev. A 6 (1972) 2211. |− i • | i ∗ | i × of Oπ/4: 1 ( ), χ ( ), and χ+ ( ). [20] V. R. Vieira and P. D. Sacramento, Ann. Phys. (N.Y.) |− i • | −i ∗ | i × 242 (1995) 188. In the paper of Lee [23] one finds the Wehrl entropy [21] R. Penrose, Emperor’s New Mind, Oxford University of a N = 3 state expressed as a function of the angle ω Press, Oxford, 1989. between both zeros of the Husimi distribution: [22] P. Lebœuf and A. Voros, J. Phys. A 23 (1990) 1765. [23] C-T. Lee, J. Phys. A 21 (1988) 3749. 2/3+σ/6 σ S = +ln 1 , (A2) [24] E. H. Lieb, Comm. Math. Phys. 62 (1978) 35. 1 σ/2 − 2 [25] H. Scutaru, preprint FT-180 (1979) and Lanl e-print − math-ph/9909024. where σ =sin2(ω/2). The equivalent result was ear- [26] P. Schupp, Lanl e-print math-ph/9902017. lier established by Scutaru [25], S =2/3+[a ln(1 + a)], [27] K. R. W. Jones, J. Phys. A 24 (1991) 121. where a = σ/(2 σ). Substituting for ω the angles− π θ [28] W. Slomczy´ nski and K. Zyczkowski,˙ Phys. Rev. Lett. 80 we get explicit− formulae for the Wehrl entropy of− each± (1998) 1880. state χ , which allow us to write the mean entropy [29] K. Zyczkowski,˙ J. Phys. A 23 (1990) 4427. ¯ | ±i SO as a function of the parameter χ. The minimum is [30] W. K. Wootters, Found. Phys. 20 (1990) 1365. χ ˙ achieved for χ = π/4(soc= 1 and cos θ cos θ+ =1/3) [31] W. Slomczy´ nski, J. Kwapień, and K. Zyczkowski, unpub- ¯ − lished. and reads SO =1 [ln(9/4)]/3 0.730. This results π/4 − ≈ [32] K. Zyczkowski,˙ in Quantum Chaos, ed. H. Cerdeira, shows that the eigenbasis of Jz does not provide the reper characterized by the minimal mean Wehrl entropy, but World Scientific, Singapore, 1991.

6 [33] E. Bogomolny, O. Bohigas, and P. Lebœuf, Phys. Rev. Lett. 68 (1992) 2726. [34] D. Braun, M. Ku´s, and K. Zyczkowski,˙ J. Phys. A 30 (1997) L117. [35] F. Haake, M. Ku´s, and R. Scharf, Z. Phys. B 65 (1987) 381. [36] W. Slomczy´ nski and K. Zyczkowski,˙ J. Math. Phys. 35 (1994) 5674; 36 (1995) 5201(E). [37] B. Mirbach and H. J. Korsch, Phys. Rev. Lett. 75 (1995) 362. [38] B. Mirbach and H. J. Korsch, Ann. Phys. (N.Y.) 265 (1998) 80.