Entropy of entanglement and multifractal exponents for random states

Entropy of entanglement and multifractal exponents for random states

O. Giraud,^1,2J. Martin,^1,2,3and B. Georgeot^1,2

1Université de Toulouse; UPS; Laboratoire de Physique Théorique (IRSAMC); F-31062 Toulouse, France

2CNRS; LPT (IRSAMC); F-31062 Toulouse, France

3Institut de Physique Nucléaire, Atomique et de Spectroscopie, Université de Liège, 4000 Liège, Belgium 共Received 23 July 2008; published 9 March 2009兲

We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal dimen- sions. Expanding the von Neumann entropy around its maximum we show that the first order only depends on the participation ratio, while higher orders involve other multifractal exponents. These results can be applied to entanglement behavior near the Anderson transition.

DOI:10.1103/PhysRevA.79.032308 PACS number共s兲: 03.67.Mn, 64.60.al, 05.30.⫺d, 71.30.⫹h

I. INTRODUCTION

Entanglement is an important characteristic of quantum systems, which has been much studied in the past few years due to its relevance to quantum information and computa- tion. It is a feature that is absent from classical information processing and a crucial ingredient in many quantum proto- cols. In the field of quantum computing, it has been shown that a process involving pure states with small enough en- tanglement can always be simulated efficiently classically 关1兴. Thus a quantum algorithm exponentially faster than clas- sical ones requires a minimal amount of entanglement 共at least for pure states兲. Conversely, it is possible to take advan- tage of the weak entanglement in certain quantum many- body systems to devise efficient classical algorithms to simu- late them关2兴. All these reasons make it important to estimate the amount of entanglement present in different types of physical systems and relate it to other properties of the sys- tem. However, in many cases the features specific to a sys- tem obscure its generic behavior. One way to circumvent this problem and to extract generic properties is to construct en- sembles of systems which after averaging over random real- izations can give analytic formulas. Such an approach has proven successful, e.g., in the quantum chaos field, where random matrix theory 共RMT兲 can describe many properties of complex quantum systems.

One of the interesting questions which has been addressed in many studies 共see, e.g.,关3兴and references therein兲is the behavior of entanglement near phase transitions. It has been shown that the entanglement of the ground state changes close to phase transitions. For example, in theXXZ andXY spin chain models, the entanglement between a block of spins and the rest of the system diverges logarithmically with the block size at the transition point 关4兴, making classical simulations harder. However, such results cannot be applied directly to systems where the transition concerns one-particle states, for which entanglement has to be suitably defined.

A famous example is the Anderson transition of electrons in a disordered potential, which separates localized from ex- tended states, with multifractal states at the transition point 关5兴. In the regime of localization the electron wave function is exponentially localized with envelope ⌿共x兲⬃exp共−兩x

−x₀兩/l兲, where l is the localization length. At the transition

point the wave function has a multifractal structure, namely, the square of the amplitude of the wave function is distrib- uted according to a multifractal law. Previous works关6兴have described the lattice on which the particle evolves as a spin chain and studied entanglement in this framework. However, the lattice can alternatively be described in terms of quantum computation with a much smaller number of two-level sys- tems 关7兴.

In this paper, we study entanglement of random vectors which can be localized, extended, or multifractal in Hilbert space. We then test the applicability of this approach to more realistic quantum states. We consider entanglement between blocks of qubits. In the case of the Anderson transition, this amounts to directly relate entanglement to the quantum simulation of the system on anr-qubit system, the number of lattice sites being 2^nrrather thann_ras in关6兴. Entanglement of random pure states was mainly studied in the case of col- umns of random matrices 关8兴. Such random matrices have been studied in the framework of RMT, where one considers various ensembles of matrices with random coefficients whose joint distribution depends on symmetry properties im- posed on the matrices. In the case of extended random vec- tors the relevant ensemble is the circular unitary ensemble 共CUE兲 关9兴, which is the ensemble of random unitary matrices with unitarily invariant Haar measure. However, such ex- tended vectors cannot describe systems with various amounts of localization, from genuine localization to multifractality.

Recently it was shown in 关10,11兴that for localized random vectors, the linear entanglement entropy 共first order of the von Neumann entropy兲of one qubit with all the others can be related to the localization properties through the inverse participation ratio 共IPR兲 ␰. For a wave function 兩⌿典 on a N-dimensional space, ␰⁼共兺i兩⌿i兩²兲/共兺i兩⌿i兩⁴兲 measures the number of principal components of 兩⌿典. Indeed if 兩⌿典 has onlyM equal nonzero components the IPR is␰^{=M. Here we} develop the approach of关11兴to obtain a general description of bipartite entanglement in terms of certain global proper- ties for random vectors both extended and localized. First we show that for any bipartition, the linear entropy can be writ- ten in terms of the IPR. We then show that higher-order terms also depend on higher moments of the wave function.

In particular, for multifractal systems they are controlled by the multifractal exponents. We then show that these formulas are relevant also when applied to physical systems.

1050-2947/2009/79共3兲/032308共5兲 032308-1 ©2009 The American Physical Society

Bipartite entanglement of a pure state兩␺典 belonging to a Hilbert space HA丢HB is measured through the entropy of entanglement, which has been shown to be a unique en- tanglement measure 关12兴. Let␳A be the density matrix ob- tained by tracing subsystemBout of␳⁼兩␺典具␺兩. The entropy of entanglement of the state with respect to the bipartition 共A,B兲 is the von Neumann entropy of ␳A, that is S= −tr共␳Alog₂␳A兲. It is convenient to define the linear en- tropy as S_L=_d−1^d 共1 − tr␳A

2兲, whered= dimHAⱕdimHB. The scaling factor ensures that SL varies in关0,1兴. We will show that the average value ofSover a set of random states can be expressed only in terms of averages of the moments of the wave function

pq=

兺

i=1 N

兩␺i兩^2q 共1兲

provided some natural assumptions are made. Here we con- sider ensembles of random vectors of size N⬅2^nr with the following two properties: 共i兲 the phases of the vector com- ponents are independent uniformly distributed random vari- ables and共ii兲the joint distributionP共x1, . . . ,xN兲of the modu- lus squared of the vector components is such that all marginal distributionsP共xi兲,P共xi,xj兲fori⫽j,P共xi,xj,xk兲for i⫽j⫽kand so on, do not depend on the indices. As a con- sequence, all correlators具兩␺i₁兩^2s1兩␺i₂兩^2s2. . .典of the components of 兩␺典 are independent of the indices i₁⫽i₂⫽. . . involved.

Random vectors realized as columns of CUE matrices are instances of vectors having such properties.

II. LINEAR ENTROPY

In this section, we show that the mean entropy of en- tanglement can be expressed at first order as a function of the second momentp₂of the distribution of the components␺i, which is related to the IPR byp₂= 1/␰. This is expressed by Eq. 共5兲 below, which is valid for any bipartition of the sys- tem into two subsystems.

Let us first consider the simplest case of entanglement of one qubit with respect to the others. Thend= 2 and the linear entropySLis simply the tangle␶, or the square of the gen- eralized concurrence 关13兴. It is given by␶^{= 4 det}␳A. If we consider a vector兩␺典of sizeN, the bipartition with respect to qubitisplits the components␺jof兩␺典into two sets, accord- ing to the value of theith bit of the binary decomposition of j. If兩␺^共0兲典 and兩␺^共1兲典 are the two corresponding vectors, the linear entropy is

␶= 4共具␺^共0兲兩␺^共0兲典具␺^共1兲兩␺^共1兲典−兩具␺^共0兲兩␺^共1兲典兩²兲. 共2兲

After averaging ␶ over random phases, only the diagonal terms survive in the scalar product 兩具␺^共0兲兩␺^共1兲典兩². Since it is assumed that two-point correlators of the vector 兩␺典 do not depend on indices, their average can be expressed solely in terms of the mean moments, as 具兩␺i兩²兩␺j兩²典=^具p1

2典−具p₂典

N共N−1兲 fori⫽j.

Normalization of 兩␺典 implies p₁= 1. As vectors 兩␺^共0兲典 and 兩␺^共1兲典 always contain components of兩␺典 with different indi- ces, we get

具␶典=N− 2

N− 1共1 −具p₂典兲. 共3兲 Since p₂= 1/␰^where␰is the IPR, Eq.共3兲is exactly the Eq.

共3兲 of Ref. 关11兴. Let us now turn to the general case and consider the entropy of entanglement of␯ qubits withnr−␯ others关共␯^,nr−␯兲bipartition兴. The vector兩␺典is now split into vectors兩␺^共j兲典, 0ⱕjⱕ2^␯− 1, depending on the values of the␯ qubits. The reduced density matrix ␳A then appears as the Gram matrix of the兩␺^共j兲典, and the linear entropy is

SL= 2^␯

2^␯− 1

冉

^{1 −i,j=02}

^兺

^{␯−1兩具␺共i兲兩␺共j兲典兩2}

冊

^{. 共4兲}

When averaging over random vectors, each term in Eq. 共4兲 withi⫽jyields 2^nr−␯ two-point correlators, while each term with i=j yields 2^nr−␯共2^nr−␯− 1兲 two-point correlators and 2^nr−␯ terms of the form 具兩␺i兩⁴典. Inserting these expressions into Eq.共4兲gives具SL典=共N− 2^␯兲共1 −具p2典兲/共N− 1兲, which gen- eralizes Eq.共3兲. The first-order series expansion of the mean von Neumann entropy around its maximum can be expressed as

具S典 ⯝␯^{−2␯− 1}

2 ln 2

冉

^{1 −NN− 2− 1␯共1 −具p2典兲}

冊

^{, 共5兲}

withp₂= 1/␰^{. Equation}共5兲shows that for any partition of the system into two subsystems, the average bipartite entangle- ment of random states only depends at first order on the localization properties of the states, through the mean partici- pation ratio. For CUE vectors, formula 共5兲 reduces to the expression for the mean entanglement derived earlier in关14兴.

More interestingly, this formula also applies to multifractal quantum states. There, the asymptotic behavior of the IPR is governed by the fractal exponentD₂, where one defines gen- eralized fractal dimensionsD_qthrough the scaling of the mo- mentspq⬀N^{−Dq共q−1兲}. Thus the linear entropy is only sensitive to a single fractal dimension. These results imply that en- tanglement grows more slowly with the system size for mul- tifractal systems.

To test the relevance of Eq. 共5兲for describing entangle- ment in realistic settings, we consider eigenvectors of N⫻Nunitary matrices of the form

Ukl=e^i␾k N

1 −e^2i␲N␥

1 −e^{2i␲共k−l+N␥兲/N}, 共6兲

where ␾k are independent random variables uniformly dis- tributed in关0 , 2␲关. These random matrices display interme- diate statistical properties关15兴and possess eigenvectors that are multifractal 关16兴, both features being tuned through the value of the real parameter␥. We also illustrate Eq.共5兲with eigenstates of a many-body Hamiltonian with disorder and interactionH=兺i⌫i␴i

z+兺i⬍jJij␴i x␴j

x. This system can describe a quantum computer in presence of static disorder关17兴. Here the ␴i are the Pauli matrices for qubiti, energy spacing be- tween the two states of qubit iis given by ⌫irandomly and

uniformly distributed in the interval关⌬0−␦/2 ,⌬0+␦/2兴, and theJijuniformly distributed in the interval关−J,J兴represent a random static interaction. For largeJand␦⬇⌬0, eigenstates are delocalized in the basis of register states, but without multifractality. They display properties of quantum chaos, with eigenvalues statistics close to the ones of RMT 关17兴.

For both systems 共unitary matrices and many-body Hamil- tonian兲, components of the eigenvectors have been shuffled in order to reduce correlations, but leaving the peculiarities of the distribution itself unaltered. Figure 1 plots the first- order expansion 共5兲as a function of the mean IPR for three different bipartitions, showing remarkable agreement with the exact 具S典, both for multifractal 共Fig. 1, left panel兲 and nonfractal 共right panel兲 states, and even for moderately en- tangled states. The agreement is better for the nonfractal sys- tem than for the multifractal one. This can be understood from the study of higher-order terms in the entropy.

III. ENTROPY OF ENTANGLEMENT

While the linear entropy does not depend on other fractal dimensions than D₂, the entropy of entanglement does. In this section, we calculate the mean value of each term in the expansion of the entropy of entanglement in powers of 1 −␶ for a bipartition 共1 ,nr− 1兲. Again, the average is taken over ensembles of random vectors. In particular we work out explicitly关see Eq.共11兲兴the second-order term as a function of the first moments of the wave function, which are linked to multifractal exponents.

In the case of a 共1 ,nr− 1兲 bipartition of the system, the entropy of entanglement can be expressed in a simple way as a function of␶^as

S共␶兲=h

冉

^{1 +}

^冑

^{21 −␶}

冊

^{, 共7兲}

where h共x兲= −xlog₂x−共1 −x兲log2共1 −x兲. The series expan- sion of S共␶兲up to ordermin共1 −␶兲reads

S_m共␶兲= 1 − 1 ln 2

兺

n=1

m 共1 −␶兲ⁿ

2n共2n− 1兲. 共8兲 The tangle␶corresponds, up to a linear transformation, to S₁共␶兲. Let us now calculate the average of higher orders in this expansion. The second-order expansion ofS共␶兲involves calculating the mean value of

兩具␺^共0兲兩␺^共1兲典兩⁴=

兺

i,j,k,l=1 N/2

u_i^ⴱuju_k^ⴱulviv^ⴱ_jvkv_l^ⴱ, 共9兲

where the star denotes complex conjugation andu_iandv_iare the components of 兩␺^共0兲典 and兩␺^共1兲典, respectively. Under the assumption of random phases, only terms whose phases can- cel survive in Eq.共9兲. Since the phases of all components of 兩␺典are independent, cancellation of the phase can only occur if the sets兵i,k其and兵j,l其 are equal. Thus

具兩具␺^共0兲兩␺^共1兲典兩⁴典= 2

兺

i,k=1 i⫽k N/2

具兩u_iu_kv_iv_k兩²典+

兺

i=1 N/2

具兩u_iv_i兩⁴典. 共10兲

The correlators in Eq.共10兲can be expressed as a function of the moments as follows. Using standard notations 关18兴, we will denote by ␭⊢n a partition ␭=共␭1,␭2, . . .兲 of n, with

␭1ⱖ␭2ⱖ. . .. For any partition␭⊢n, we definep_␭=p_␭

1p_␭

2. . ., where p_␭

i are given by Eq. 共1兲. The monomial symmetric polynomials are defined asm_␭=兺具兩␺1兩^2␭1兩␺2兩^2␭2. . .典 共the sum runs over all P_␭ permutations of the ␭i兲, and we set c_␭=m_␭/P_␭. The p_␭ andm_␭ are related by the simple linear relation p_␭=兺_␮L_␭␮m_␮, where L_␭␮ is an invertible integer lower-triangular matrix 共关18兴, p.103兲. Upon our assumption 共ii兲, any correlator of the form具兩␺i₁兩^2s1兩␺i₂兩^2s2. . .典is equal to a c_␭for some partition ␭ofn, and thus can be expressed as a function of the moments. For instance the two correlators in Eq. 共10兲 are respectively equal to c₁₁₁₁ and c₂₂. Treating similarly all terms involved in ␶²gives

具␶²典=N共N− 2兲共N²− 6N+ 16兲c₁₁₁₁+ 4N共N− 2兲共N− 4兲c₂₁₁

+ 4N共N− 2兲c₂₂. 共11兲

This term involves the calculation of three correlators. Using the relation between thec_␭andp_␭and the fact that the vec- tors are normalized to one we get

c₂₂=具p₂²典−具p₄典

N共N− 1兲 , c₂₁₁=具p₂典−具p₂²典− 2具p₃典+ 2具p₄典 N共N− 1兲共N− 2兲 ,

c₁₁₁₁=1 − 6具p₂典+ 8具p₃典+ 3具p₂²典− 6具p₄典

N共N− 1兲共N− 2兲共N− 3兲 . 共12兲 The calculation of the general term具␶ⁿ典can be performed along the same lines. Expanding Eq. 共2兲we get

1/ξ⁻¹

160 120 80 40 0

1/ξ⁻¹

S

60 50 40 30 20 10 0 5 4 3 2 1 0

FIG. 1. 共Color online兲Mean entropy of entanglement as a func- tion of the mean IPR. Left: eigenvectors of Eq.共6兲with␥= 1/3; the average is taken over 10⁶eigenvectors. Right: eigenvectors of the Hamiltonian H 共see text兲 with␦⁼⌬0 and J/␦= 1.5; average over N/16 central eigenstates, with a total number of vectors⬇3⫻10⁵. Triangles correspond to ␯= 1, squares to ␯= 2 and circles to

␯=n_r/2, with n_r= 4 – 10 共bipartition of the ␯ first qubits with the n_r−␯others兲. Black symbols are the theoretical predictions for具S典 at first order关Eq.共5兲兴and green共gray兲symbols are the computed mean values of the exact具S典.

␶ⁿ= 4ⁿ

兺

k=0

n

冉

^nk

冊

^{共− 1兲n−k}

⫻

冉 ^兺

^{i=1N/2兩ui兩2}

冊

^k

冉 ^兺

^{i=1N/2兩vi兩2}

冊

^k

冉 ^兺

^{i,j uiⴱviujvⴱj}

冊

^{n−k. 共13兲}

The expansion of 共兺i,ju_i^ⴱv_iu_jv^ⴱ_j兲^t contains products of the form共ui₁. . .ui_t兲^ⴱuj₁. . .uj_t. Only terms where the phases com- ing from the u_i

k

ⴱ compensate those coming from theuj_ksur- vive when averaging over random phases. Thus we keep only terms where兵j1, . . . ,jt其is a permutation of兵i1, . . . ,it其. If PKis the number of permutations of a setK, the average of Eq. 共13兲over random vectors reads

具␶ⁿ典=

冬

⁴ⁿ

^兺

^k=0n

^冉

^nk

^冊

^{共− 1兲n−kpq11,. . .,p,. . .,q}

^兺

^kk

^兿

^{j=1k 兩upj兩2兩vqj兩2}

⫻_i

兺

1,. . .,i_n−k

P_兵i₁,. . .,i_n−k其兩ui₁vi₁兩². . .兩ui_n−kvi_n−k兩²

冭

^.

共14兲 Terms with the same correlator can be grouped together.

Each correlator in Eq. 共14兲 is somec_␭艛␭⬘, with ␭,␭

⬘

^parti- tions of n. For ␭⊢n and ␮^⊢k we define the coefficient A_␭␮=_␮!^k!兺^共n−k_共s−k^兲!_兲!, where the sum runs over all vectors s=共s1, . . . ,sN/2兲 which are permutations of ␭, and k=共␮1, . . . ,␮N/2兲. We have used the notations a=共a1,a₂, . . .兲 anda! =a₁!a₂! . . .. Finally we get

具␶ⁿ典= 4ⁿ

兺

␭,␭⬘^⊢n

冋 ^兺

^k=0n

^冉

^nk

^冊

^{共− 1兲k}

^兺

^{␮⊢kP␮A␭␮A␭⬘␮}

册

^{c␭艛␭⬘,}

共15兲 which provides an expression for the nth order for具S典 as a function of the 具p_␭典. In the special case of CUE random vectors, by resumming the whole series we recover after some algebra the well-known result 关19兴 具S共␶兲典

=_{ln 2}¹ 兺_k=N^N−1_/2+1¹_k. Note that similar expressions can be derived for a general 共␯^,nr−␯兲 bipartition. In this case, the entropy S= −tr共␳Alog₂␳A兲 can be expanded around the maximally mixed state␳0= 1/2^␯, as

S=␯^{+ 1} ln 2

兺

n=1

⬁ 共− 2^␯兲ⁿ

n共n+ 1兲tr关共␳A−␳0兲ⁿ⁺¹兴. 共16兲 After averaging over random vectors, one can check that the traces in Eq. 共16兲can be written as 具tr␳A

k典=兺_␭⊢ka_␭^共k兲c_␭, with a_␭^共k兲some integer combinatorial coefficient. The entropy can thus be written as a linear combination of c_␭ with rational coefficients that can be expressed in terms of thea_␭^共k兲.

In Fig.2 we illustrate the accuracy of higher-order terms in the series expansion of S for multifractal random vectors by comparing the first and second-order expansion for eigen- vectors of the matrices 共6兲. As expected, the second-order expansion is much more accurate than the first order one and gives a much better estimate of the mean entropy of en-

tanglement already for small system sizes. For large N, the dominant term in S₂ is ⬀具p₂²典. Numerically we obtained 具p₂²典⬃N^−0.81for ␥= 1/3, and具p₂²典⬃N^−1.53for ␥= 1/7, which is indeed consistent with the slopes of the linear fit of log₂共1 −具S₁典/具S典兲 共see Fig.2兲. If one replaces具p₂²典appearing in Eq. 共12兲 by 具p2典² 共squares in Fig. 2兲, the second-order expansion is now governed only by three multifractal dimen- sions D₂, D₃, and D₄. Although it becomes less and less accurate with the system size because of the increase of the variance ofp₂, it remains a very good improvement over the first order in the case of moderate multifractality 共Fig. 2, right兲.

IV. CONCLUSION

Our results show that the entanglement of random vectors directly depends on whether they are localized, multifractal or extended. The numerical simulations for different physical examples show that our theory describes well individual sys- tems whose correlations are averaged out. Previous results 关11兴 have shown that Anderson-localized states have en- tanglement going to zero for large system size. The present work shows that multifractal states, such as those appearing at the Anderson transition, approach the maximal value of entanglement in a way controlled by the multifractal expo- nents. Although extended and multifractal states are both close to maximal entanglement, the way multifractal states approach the maximal value for large system size is slower.

ACKNOWLEDGMENTS

The authors thank CalMiP in Toulouse for access to their supercomputers. This work was supported by the Agence Nationale de la Recherche 共ANR project INFOSYSQQ, Contract No. ANR-05-JCJC-0072兲 and the European pro- gram EC Grant No. IST FP6-015708 EuroSQIP.

nr 9 8 7 6 5 4 3 2 nr

S−Sm S

10 9 8 7 6 5 4 3 2 10⁰ 10⁻¹ 10⁻² 10⁻³ 10⁻⁴ 10⁻⁵ 10⁻⁶

FIG. 2. 共Color online兲Relative difference of the entropy of en- tanglement共7兲and its successive approximationsS_m共m= 1 , 2兲with respect to the number of qubits for eigenvectors of Eq.共6兲for共left兲

␥= 1/3 and共right兲␥= 1/7. The average is taken over 10⁷eigenvec- tors, yielding an accuracy ⱗ10⁻⁶on the computed mean values.

Green triangles correspond to the first-order expansion S₁, blue squares and red circles to the second-order expansionS₂. The dif- ference between the latter two is that for blue squares 具p₂²典 appearing in Eq. 共12兲 has been replaced by 具p₂典² yielding a less accurate approximation. Dashed line is a linear fit yielding 1 −具S₁典/具S典⬃N^−0.84for␥= 1/3 andN^−1.58for␥= 1/7.

关1兴R. Jozsa and N. Linden, Proc. R. Soc. London, Ser. A 459, 2011共2003兲.

关2兴G. Vidal, Phys. Rev. Lett. 91, 147902共2003兲; F. Verstraete, D.

Porras, and J. I. Cirac,ibid. 93, 227205共2004兲.

关3兴L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod.

Phys. 80, 517共2008兲.

关4兴G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett.

90, 227902共2003兲.

关5兴A. D. Mirlin, Phys. Rep. 326, 259共2000兲; F. Evers and A. D.

Mirlin, Rev. Mod. Phys. 80, 1355共2008兲.

关6兴L. Gong and P. Tong, Phys. Rev. E 74, 056103共2006兲; X. Jia, A. R. Subramaniam, I. A. Gruzberg, and S. Chakravarty, Phys.

Rev. B 77, 014208 共2008兲; I. Varga and J. A. Mendez- Bermudez, Phys. Status Solidi C 5, 867共2008兲.

关7兴A. A. Pomeransky and D. L. Shepelyansky, Phys. Rev. A 69, 014302共2004兲.

关8兴H.-J. Sommers and K. Zyczkowski, J. Phys. A 37, 8457 共2004兲; O. Giraud, ibid. 40, 2793共2007兲; M. Znidaric,ibid.

40, F105共2007兲.

关9兴M. L. Mehta,Random Matrices共Academic, New York, 1991兲. 关10兴L. Viola and W. G. Brown, J. Phys. A 40, 8109共2007兲; W. G.

Brown, L. F. Santos, D. J. Starling, and L. Viola, Phys. Rev. E 77, 021106共2008兲.

关11兴O. Giraud, J. Martin, and B. Georgeot, Phys. Rev. A 76, 042333共2007兲.

关12兴S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319共1997兲. 关13兴P. Rungta and C. M. Caves, Phys. Rev. A 67, 012307共2003兲. 关14兴A. J. Scott, Phys. Rev. A 69, 052330共2004兲.

关15兴E. Bogomolny and C. Schmit, Phys. Rev. Lett. 93, 254102 共2004兲.

关16兴J. Martin, O. Giraud, and B. Georgeot, Phys. Rev. E 77, 035201共R兲 共2008兲.

关17兴B. Georgeot and D. L. Shepelyansky, Phys. Rev. E 62, 3504 共2000兲; 62, 6366共2000兲.

关18兴I. G. Macdonald,Symmetric Functions and Hall Polynomials, 2nd ed.共Oxford University Press, New York, 1995兲.

关19兴D. N. Page, Phys. Rev. Lett. 71, 1291共1993兲.