Wehrl Entropy, Lieb Conjecture, and Entanglement Monotones

PHYSICAL REVIEW A 69, 022317 ͑2004͒

Wehrl entropy, Lieb conjecture, and entanglement monotones

Florian Mintert Max Planck Institute for the Physics of Complex Systems, No¨thnitzerstrasse 38 01187 Dresden, Germany and Uniwersytet Jagiellon´ski, Instytut Fizyki im. M. Smoluchowskiego, ul. Reymonta 4, 30-059 Krako´w, Poland

Karol Z˙ yczkowski Uniwersytet Jagiellon´ski, Instytut Fizyki im. M. Smoluchowskiego, ul. Reymonta 4, 30-059 Krako´w, Poland and Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotniko´w 32/44, 02-668 Warszawa, Poland ͑Received 2 September 2003; published 24 February 2004͒ We propose to quantify the entanglement of pure states of NϫN bipartite quantum systems by deﬁning its Husimi distribution with respect to SU(N)ϫSU(N) coherent states. The Wehrl entropy is minimal if and only if the analyzed pure state is separable. The excess of the Wehrl entropy is shown to be equal to the subentropy of the mixed state obtained by partial trace of the bipartite pure state. This quantity, as well as the generalized ͑Re´nyi͒ subentropies, are proved to be Schur concave, so they are entanglement monotones and may be used as alternative measures of entanglement.

DOI: 10.1103/PhysRevA.69.022317 PACS number͑s͒: 03.67.Mn, 03.65.Ud, 89.70.ϩc

I. INTRODUCTION original statement and its proof was given in ͓12,13͔.Itis also straightforward to formulate the Lieb conjecture for We- Investigating the properties of a quantum state it is useful hrl entropy computed with respect to the SU(K)-coherent to analyze its phase-space representation. The Husimi distri- states ͓14,15͔, but this conjecture seems not to be simpler bution is often very convenient to work with: it exists for any than the original one. quantum state, is non-negative for any point ␣ of the classi- In this work we consider the Husimi function of an arbi- cal phase space ⍀, and may be normalized as a probability trary mixed quantum state ␳ of size N with respect to distribution ͓1͔. The Husimi distribution can be defined as SU(N)-coherent states and calculate both its statistical mo- the expectation value of the analyzed state % with respect to ments and the Wehrl entropy. The difference between the the coherent state ͉␣͘, localized at the corresponding point latter quantity and the minimal entropy attained for pure ␣෈⍀. states can be considered as a measure of the degree of mix- If the classical phase space is equivalent to the plane, one ing. uses the standard harmonic oscillator coherent states ͑CS’s͒, Analyzing pure states of a bipartite system, it is helpful to but in general one may apply the group-theoretical construc- define coherent states with respect to product groups. The tion of Perelomov ͓2͔. For instance, the group SU(2) leads Wehrl entropy of a given state with respect to the product to spin-coherent states parametrized by the sphere S2ϭCP1 states was first considered in the original paper of Wehrl ͓16͔ ͓3,4͔, while the simplest, degenerated representation of and later used in ͓17͔ to characterize correlations between SU(K) leads to the higher vector coherent states param- both subsystems. Recently Sugita proposed using moments Ϫ etrized by points on the complex projective manifold CPK 1 of the Husimi distribution defined with respect to the ͓5–7͔. One may define the Husimi distribution of the SU(N)ϫSU(N) product-coherent states as a way to charac- N-dimensional pure state ͉␾͘ with respect to SU(K)-CS’s for terize the entanglement of the analyzed state ͓18͔. any KрN, but it is important to note that if KϭN, any pure In this paper we follow his idea and compute explicitly state is by definition SU(N) coherent. the Wehrl entropy and the generalized Reńyi-Wehrl entropies Localization properties of a state under consideration may for any pure state ͉⌿͘ of the NϫN composite system. The be characterized by the Wehrl entropy, defined as the con- Husimi function is computed with respect to SU(N) tinuous Boltzmann-Gibbs entropy of the Husimi function ϫSU(N)-CS’s, so the Wehrl entropies achieve their mini- ͓8͔. The Wehrl entropy admits the smallest values for coher- mum if and only if the state ͉⌿͘ is a product state. Hence the ent states, which are as localized in the phase space, as al- entropy excess defined as the difference with respect to the lowed by the Heisenberg uncertainty relation. Interestingly, minimal value quantifies, to what extend the analyzed state is this property, first conjectured by Wehrl ͓8͔ for the harmonic entangled. The Wehrl entropy excess is shown to be equal to oscillator coherent states, was proved soon afterwords by the subentropy defined by Jozsa et al. ͓19͔. Calculating the Lieb ͓9͔, but an analogous result for the SU(2)-CS’s still excess of the Reńyi-Wehrl entropies we define the Reńyi awaits rigorous proof. This unproven property is known in subentropy—a natural continuous generalization of the sub- the literature as the Lieb conjecture ͓9͔. It is known that the entropy. SU(2)-CS’s provide a local minimum of the Wehrl entropy Several different measures of quantum entanglement in- ͓10͔, while a proof of the global minimum was given for troduced in the literature ͑see, e.g., ͓20–22͔ and references pure states of dimension Nϭ3 and Nϭ4 only ͓11,12͔.A therein͒ satisfy the list of axioms formulated in ͓20͔. In par- generalized Lieb conjecture, concerning the Reńyi-Wehrl en- ticular, quantum entanglement cannot increase under the ac- tropies of integer order qу2, occurred to be easier than the tion of any local operations, and a measure fulfilling this

1050-2947/2004/69͑2͒/022317͑11͒/$22.5069 022317-1 ©2004 The American Physical Society F. MINTERT AND K. Z˙ YCZKOWSKI PHYSICAL REVIEW A 69, 022317 ͑2004͒ property is called an entanglement monotone. To characterize quantitatively the localization of an ana- The nonlocal properties of a pure state of an NϫN system lyzed state % in the phase space we compute its Husimi Ϫ has to be characterized by N 1 independent monotones distribution H% with respect to SU(N)-CS’s and analyze the ͓ ͔ Ϫ 21 . In this work we demonstrate that N 1 statistical mo- moments mq of the distribution: ments of properly defined Husimi functions naturally provide such a set of parameters, since they are Schur-concave func- ͑ ͒ϭ ͵ ␮͑␣͒ ͑ ͑␣͒͒q ͑ ͒ mq % d H% . 4 ⍀ tions of the Schmidt coefficients and thus are nonincreasing N under local operations. Here d␮(␣) denotes the unique, unitarily invariant measure NϪ1 II. HUSIMI FUNCTION AND WEHRL ENTROPY on CP , also called the Fubini-Study measure. The measure d␮ is normalized such that for any state % the first ෈ NϫN Any density matrix % C can be represented by its moment m1 is equal to unity, so the non-negative Husimi ␣ Husimi function H%( ), which is defined as its expectation distribution H% may be regarded as a phase-space probability value with respect to coherent states ͉␣͘: distribution. The definition of the moments m is not restricted to in- ͑␣͒ϭ͗␣͉ ͉␣͘ ͑ ͒ q H% % . 1 teger values of q. However, from a practical point of view it will be easier to perform the integration for integer values of In general one may define the Husimi distribution with q. But once m are known for all integer q, there is a unique р р q respect to SU(K)-coherent states with 2 K N, but most analytic extension to complex ͑and therefore also real͒ q,as often one computes the Husimi distribution with respect to integers are dense at infinity. the SU(2)-coherent states, also called spin-coherent states Another quantity of interest is the Wehrl entropy SW , de- ͓3,4͔. Let us recall that any family of coherent states ͕͉␣͖͘ fined as ͓8͔ has to satisfy the resolution of identity, ͑ ͒ϭϪ͵ ␮͑␣͒ ͑␣͒ ͑␣͒ ͑ ͒ SW % d H% ln H% . 5 ⍀ ͵ d␮͑␣͉͒␣͗͘␣͉ϭ1, ͑2͒ N ⍀ Again, performing the integration might be difficult. But if ␮ ␣ ϭ where d ( ) is a uniform measure on the classical manifold all the moments mq are known in the vicinity of q 1, one ϭ q ץ q ץ ⍀ . In the case of the degenerated representation of can derive SW quite easily. Using H / q H ln H, one gets ͓ ͔ ⍀ SU(K)-coherent states 5–7 this manifold K ͒%͑ mץ ϭ ͓ Ϫ ϫ ͔ϭ KϪ1 U(K)/ U(K 1) U(1) CP is just equivalent to the ͒ϭϪ q ͑ ͒ SW͑% lim . 6 qץ → space of all pure states of size K. This complex projective H q 1 space arises from the K-dimensional Hilbert space K by taking into account normalized states and identifying all el- The moments m of the Husimi function allow us to write H q ements of K which differ by an overall phase only. In the the Reńyi-Wehrl entropy simplest case of SU(2)-coherent states it is just the well- ⍀ ϭ 1ϭ 2 known Bloch sphere 2 CP S . 1 S ͑%͒ ln m ͑%͒, ͑7͒ Analyzing a mixed state of dimensionality N we are going W,q ª1Ϫq q to use SU(N)-coherent states, which will be denoted by ͉␣ ⍀ N͘. With this convention the space is equivalent to the which tends to the Wehrl entropy S for q→1. As a Husimi Ϫ W complex projective space CPN 1, so every pure state is function is related to a selected classical phase space, the SU(N) coherent and their Husimi distributions have the Wehrl entropy is also called classical entropy ͓8,16͔, in con- ⍀ ϭϪ same shape and differ only by the localization in . trast to the von Neumann entropy SN Tr % ln %, which has The SU(N)-coherent states may be defined according to no immediate relation to classical mechanics. the general group-theoretical approach by Perelomov, by a In Sec. III we consider how strongly a given state is set of generators of SU(N) acting on the distinguished ref- mixed and in Sec. IV we discuss the nonlocality of bipartite erence state ͉0͘. We are going to work with the degenerate pure states. For this purpose we pursue an approach inspired representation of SU(N) only, in which a coherent state may by the Lieb conjecture ͓9͔, according to which the Wehrl Ϫ ␥ be parametrized by N 1 complex numbers i : entropy SW of a quantum state is minimal if and only if % is coherent. In order to measure a degree of mixing of a mono- ␥ ␥ ͉␣ ͘ϭ͉␥ ␥ ͘ 1J1 NϪ1JNϪ1͉ ͘ ͑ ͒ N 1 ,..., NϪ1 ªe •••e 0 , 3 partite mixed state of size N we therefore use the SU(N)-coherent state, while for the study of entanglement of where operators Ji may be interpreted as lowering operators a bipartite pure NϫN state we use the coherent states related ͓6͔. In language of the N-level atom they couple the highest to the product group SU(N)ϫSU(N). Nth level with the ith one ͑see, e.g., ͓7͔͒. In the simplest case ͉ ͘ of SU(2)-coherent states the reference state 0 is equal to III. MONOPARTITE SYSTEMS: MIXED STATES the eigenstate ͉j, j͘ of the angular momentum’s z component Jz with maximal eigenvalue, while the lowering operator In this paragraph we will focus on mixed states % ϭ ϭ Ϫ ෈ NϫN H reads J1 JϪ Jx iJy . C acting on an N-dimensional Hilbert space N . Such

022317-2 WEHRL ENTROPY, LIEB CONJECTURE, AND . . . PHYSICAL REVIEW A 69, 022317 ͑2004͒ a state may appear as a reduced density matrix %r , defined As a by-product we find a bound on SW(%). The suben- ⌿ ϭ ͉⌿ ⌿͉ ͉⌿ by the partial trace %r( ) TrA ͗͘ of a pure state ͘ tropy Q(%) is known not to be larger than the von Neumann ෈H H ͓ ͔ ͑ ͒ N N of a bipartite system. One could also consider a entropy SN 19 . Using this fact and Eq. 9 we find that р ϩ system coupled to an environment in which a mixed state is SW(%) SN(%) CN . As it is also known that the von Neu- obtained by tracing over the environmental degrees of free- mann entropy SN is not larger than the Wehrl entropy SW(%) dom. The von Neumann entropy SN quantifies, on the one ͓8͔, we end up with the following upper and lower bounds on hand, the degree of mixing of %r and, on the other, the non- SW(%): local properties of the bipartite state ͉⌿͘. ͑ ͒ϩ у ͑ ͒у ͑ ͒ ͑ ͒ To obtain an alternative measure of mixing of % we are SN % CN SW % SN % . 13 going to investigate its Husimi function defined with respect (1) ␣ ϭ͗␣ ͉ ͉␣ ͘ to the SU(N)-CS’s as H% ( ) N % N . The moments IV. BIPARTITE SYSTEMS: PURE STATES mq of the Husimi distribution can be expressed as functions of the eigenvalues ␭ of %, which in the case of a reduced Let us now focus on bipartite systems described by a Hil- i H state of a bipartite system coincide with the Schmidt coeffi- bert space that can be decomposed into a tensor product HϭH H cients of the pure state ͉⌿͘. The derivation of the explicit A B of two subspaces. With a local unitary transfor- U ϭU U ͉⌿͘෈H result in terms of the Euler gamma function, mation l A B any pure state can be trans- formed to its Schmidt form ͓23͔ N!⌫͑qϩ1͒ ϭ ␮ N mq ⌫͑ ϩ ͒ q,N , q N ͉⌿ ϭ ͱ␭ ͉ ͉ ͑ ͒ ͘ ͚ i i͘A i͘B , 14 iϭ1 with ϭ H H ␭ N ␭qϩNϪ1 where N min(dim A ,dim B) and the real prefactors i ␮ ϭ i ͑ ͒ called Schmidt coefficients are the eigenvalues of the reduced q,N ͚ N , 8 iϭ1 density matrix % . Thanks to the Schmidt decomposition we ͑␭ Ϫ␭ ͒ r ͟ i j can assume dim H ϭdim H ϭN without loss of generality. jϭ1,jÞi A B Following an idea of Sugita ͓18͔ we consider the question is provided in Appendix A. Performing the limit ͑6͒ we find whether the Wehrl entropy of a bipartite state can serve as a ͉␣(M)͘ that the Wehrl entropy SW equals the subentropy Q(%)upto measure of entanglement. He defined a coherent state 2 an additive constant CN : of an M-partite qubit system as the tensor product of M coherent states ͉␣(1)͘ of single-qubit systems. We generalize ͑ ͒ϭ ͑ ͒ϩ ͑ ͒ 2 SW % Q % CN . 9 this approach for bipartite systems, omitting the restriction to qubits, and calculate all moments and the Wehrl entropy of The N-dependent constant can be expressed as the respective Husimi functions. For a bipartite pure state N ͉⌿͘෈H, we use the tensor product of two SU(N)-coherent ϭ⌿ ϩ ͒Ϫ⌿ ͒ϭ ͑ ͒ ͉␣(2)͘ϭ͉␣ ͘ 2෈H CN ͑N 1 ͑2 ͚ 1/k, 10 states N N to define a Husimi function. More kϭ2 ͉␣(2)͘ϭ͉␣ ͘ ͉␣ ͘ ͉␣ ͘ explicitly one can write N N A N B , with N A ෈H and ͉␣ ͘ ෈H ͑we will drop the index referring to where the digamma function is defined by ⌿(x) A N B B the subsystem wherever there is no ambiguity͒. Such bipar- x. The subentropyץ/(ln ⌫(x ץϭ tite coherent states were used already in the original paper of ͓ ͔ N ␭Nln ␭ Wehrl 16 . ͑ ͒ϭϪ i i ϭ ͑␭ជ ͒ ͑ ͒ ͉⌿͘ Q % ͚ N :Q 11 The Husimi function H of a bipartite state is then iϭ1 ␭ Ϫ␭ ͒ given by ͟ ͑ i j jϭ1,jÞi ␣ ␣ ͒ϭ͉ ⌿͉ ͉␣ ͉␣ ͉2 ͑ ͒ H⌿͑ A , B ͗ ͑ N͘A N͘B) . 15 was defined in an information-theoretical context ͓19͔.Itis related to the von Neumann entropy SN(%) in the sense that By definition any product state is a both quantities give the lower and the upper bounds for the SU(N)ϫSU(N)-coherent state. According to the Lieb con- information which may be extracted from the state % ͓19͔. jecture, originally formulated for SU(2)-coherent states, the The subentropy Q(%) takes its minimal value 0 for pure Wehrl entropy is minimal for coherent states ͓9͔. Hence it is states with only one nonvanishing eigenvalue. Therefore we natural to expect that the Wehrl entropy define the entropy excess ⌬S as ͑␺͒ϭϪ͵ ␮ ͑␣͒ ͵ ␮ ͑␣͒ ⌬ ͒ϭ ͒Ϫ ␳ ͒ ͑ ͒ SW d A d B S͑% SW͑% SW͑ ␺ , 12 ⍀ ⍀ N N ␳ ϫ ␣ ␣ ͒ ␣ ␣ ͒ ͑ ͒ where ␺ refers to an arbitrary pure state and the Wehrl H⌿͑ A , B ln H⌿͑ A , B 16 entropy of an arbitrary N-dimensional pure state is given by ␳ ϭ SW( ␺) CN . Hence the entropy excess, equal to the suben- provides a measure of how ‘‘incoherent’’ a given state is. tropy ⌬S(%)ϭQ(%), is non-negative and equal to zero only Due to the equivalence of ‘‘coherence’’ and separability, we for pure states. expect that SW can also serve as an entanglement measure. In

022317-3 F. MINTERT AND K. Z˙ YCZKOWSKI PHYSICAL REVIEW A 69, 022317 ͑2004͒

the following we discuss the properties of SW and show that In order to use the moments mq as entanglement mono- it satisfies all requirements of entanglement monotones tones, it is conventional to rescale them such that they vanish ͓20,21͔. for separable states and are positive for entangled ones: There is a one-to-one correspondence between states and Husimi functions. Concerning entanglement, two different 1 M ϭ ͑␮ Ϫ1͒. ͑22͒ states that are connected by a local unitary transformation are q 1Ϫq q,N considered equivalent. As by construction the moments mq of H are invariant under local unitary transformations, they These quantities are analogous to the Havrda-Charvat en- reflect this equivalence and therefore can be expected to be tropy ͑also called Tsallis entropy͓͒24,25͔ and in the limit q good quantities to characterize the nonlocal properties of a →1 one obtains the subentropy bipartite state. For a pure state ͉⌿͘෈H H , there are N A B ϭ ͑␭ជ ͒ ͑ ͒ Ϫ1 independent moments, determining NϪ1 independent lim M q Q . 23 → Schmidt coefficients ͑one coefficient is determined by the q 1 normalization͒. In our case the moments read V. RE´ NYI SUBENTROPY

ϭ ͵ ␮ ͑␣ ͒ ͵ ␮ ͑␣ ͓͒ ͑␣ ␣ ͔͒q ͑ ͒ As discussed in previous sections, the excess of the Wehrl mq d A A d B B H⌿ A , B , 17 ⍀ ⍀ ⌬ N N entropy S may be used as a measure of the degree of mixing for monopartie states or of the degree of entanglement where the integration is performed over the Cartesian prod- for pure states of bipartite systems. Since the moments of the ⍀ ϫ⍀ uct N N , being the space of all product pure states of Husimi distribution are found, we may extend the above ␮ the bipartite system. The proper normalization of d A and analysis for the Reńyi-Wehrl entropy SW,q defined by Eq. ␮ ϭ Ϫ ͑ ͒ d B assures that m1 1. The required N 1 monotones can 7 . ϭ be provided by mq with q 2,...,N. The moments can be Considering, for instance, the case of mixed states of a ␭ ͑ ͒ expressed as a function of the Schmidt coefficients i , de- monopartite system we use Eq. 8 to find the minimal fined in Eq. ͑14͒, Reńyi-Wehrl entropy attained for pure states %␺ :

2 1 N!⌫͑qϩ1͒ N!⌫͑qϩ1͒ ϭ ͑ ͒ϭ ͑ ͒ m ϭͩ ͪ ␮ , ͑18͒ CN,q SW,q %␺ Ϫ ln ⌫͑ ϩ ͒ , 24 q ⌫͑qϩN͒ q,N 1 q q N which for q→1 reduces to Eq. ͑10͒. In an analogy to Eq. ͑9͒ ͓ ͑ ͔͒ and are related to the monopartite moments Eq. 8 by a we define the Reńyi-Wehrl entropy excess multiplicative factor. The derivation of this result is provided ⌬ ͒ϭ ͒Ϫ ͑ ͒ in Appendix A. Having closed expressions for the moments SW,q͑% SW,q͑% CN,q . 25 that can be extended to real q, one easily finds the bipartite Wehrl entropy Applying Eq. ͑8͒ it is straightforward to obtain result in ␭ terms of the eigenvalues i of the analyzed state %: ͑␺͒ϭϪ͵ ␮ ͑␣ ͒ ͵ ␮ ͑␣ ͒ N ␭qϩNϪ1 SW d 1 A d 2 B 1 i ⍀ ⍀ ⌬ ͑ ͒ϭ ϭ ͑␭ជ ͒ N N SW,q % ln ͚ N :Qq . 1Ϫq iϭ1 ͑␭ Ϫ␭ ͒ ϫH⌿͑␣ ,␣ ͒ln H⌿͑␣ ,␣ ͒. ͑19͒ ͟ i j A B A B jϭ1,jÞi ͑26͒ Albeit arising from a completely different physical context this integral is formally similar to that one in ͓19͔ leading to This result shows that the excess of the Reńyi-Wehrl entropy the definition of subentropy. Performing the limit ͑6͒ one ϭ ␭ជ Qq(%) Qq( ) may be called Reńyi subentropy since for q finds that SW equals the corresponding monopartite quantity →1 it tends to the subentropy ͑11͒. On the one hand, it may up to an additive constant: be treated as a function of an arbitrary quantum state %;on the other, it may be defined for an arbitrary classical prob- ͑⌿͒ϭ ͑␭ជ ͒ϩ ͑ ͒ ជ SW Q 2CN . 20 ability vector ␭. In a sense Qq is a quantity analogous to the Reńyi entropy The subentropy Q(␭) takes its minimal value 0 if and Sq , which may be defined for an arbitrary probability vector: only if a given state is separable—i.e., if all but one Schmidt coefficients vanish. Therefore we define the bipartite entropy 1 N ͑␭ជ ͒ϭ ␭q ͑ ͒ excess ⌬S (⌿), Sq ln͚ i . 27 W 1Ϫq iϭ1 ⌬S ͑⌿͒ϭS ͑⌿͒ϪS ͑␾ ͒ϭQ͑ ͒, ͑21͒ W W W sep %r In the case of quantum states Sq is defined as a function of the qth moment of the respective state: which is vanishing if and only if ͉⌿͘ is separable. Further- ´ (q) 1 more, one can also consider the Renyi-Wehrl entropy SRW ͑ ͒ϭ ͑ q͒ ͑ ͒ ϭ Ϫ Sq % ln Tr % . 28 ͓1/(1 q)͔ln mq . 1Ϫq

022317-4 WEHRL ENTROPY, LIEB CONJECTURE, AND . . . PHYSICAL REVIEW A 69, 022317 ͑2004͒

͑ ͒ →ϱ ␭ជ ϭϪ ␭ v For q one gets limq→ϱQq( ) ln max , where ␭ ␭ជ max is the largest entry of . Hence this limit coincides with →ϱ ␭ជ the limit q of the Reńyi entropy Sq( ). ͑ ͒ vi Qq is expansible; i.e., it does not vary if the probabil- ␭ជ ␭ ␭ ity vector is extended by zero, Qq( 1 ,..., N) ϭ ␭ ␭ Qq( 1 ,..., N,0). A discrete interpolation between the subentropy Q and the Shannon entropy S was recently proposed in ͓26͔. Note that ͑ ͒ a continuous interpolation between these quantities may be FIG. 1. Reńyi entropy Sq dashed lines and Reńyi subentropy ͑ ͒ ´ Qq solid lines as a function of q for two exemplary probability obtained by the Renyi subentropy Qq for q increasing from distributions. We have chosen Nϭ4 probability vectors with power- unity to infinity combined with the Reńyi entropy Sq for q ϰ ␬ ␬ϭ ͑ ͒ ␬ϭ ͓ ͔ law-distributed components p j j with 3/2 in a and 3in decreasing from infinity to unity. It is well known 27 that ͑ ͒ ϭ ͑ ͒ b . Stars at q 1 represent Shannon entropy upper and suben- the Reńyi entropy Sq is both a nonincreasing function of q ͑ ͒ tropy lower . We conjecture that Qq is both nondecreasing and and convex with respect to q. concave with respect to the Reńyi parameter q. We conjecture that the Reńyi subentropy has opposite properties: it is nondecreasing as a function of q and it is ␭ជ In the following we will sketch some properties of Qq( ) concave with respect to q. See Fig. 1 for some exemplary as a function of a classical probability vector. All the consid- cases. Figure 2 presents both quantities for Nϭ2, while Figs. erations are also applicable in the quantum case, particularly 3 and 4 present the curves of isoentropy Qq and equal mo- ϭ if we use Qq as an entanglement monotone using the moments M q obtained for N 3. ments discussed in the preceding section. Let us define two distinguished probability vectors which correspond to ex- VI. SCHUR CONCAVITY treme cases: ␭ជ with ␭*ϭ1/N (iϭ1,...,N) describes the * i So far we defined the rescaled moments, Reńyi entropy ␭ជ ␭(s)ϭ␦ maximal random event, whereas s with i i, j with and Reńyi subentropy. All these quantities are constructed in j෈͓1,...,N͔ describes an event with a certain result. For a such a way that they are vanishing exactly for separable bipartite quantum system a vector of Schmidt coefficients states. In order to legitimately use these quantities as en- given by ␭ជ represents a maximally entangled state, whereas tanglement monotones for pure states, we still have to show * ␭ជ that they are nonincreasing under local operations and clas- s describes a separable state. sical communication or, analogously, that they are Schur The Reńyi subentropy Q (␭ជ ) has the following proper- q concave ͓21,28͔—i.e., ties. ͑ ͒ ␭ជ ϭ i Qq( s) 0 for any q. ␭ជ ՞␰ជ ⇒ ⌬S ͑␭ជ ͒у⌬S ͑␰ជ ͒ ͑29͒ ͑ ͒ W W ii Qq takes its maximal value or equivalently for the other considered quantities. The ex- 1 ⌫͑qϩN͒ 1 pression ␭ជ ՞␰ជ means that ␭ជ is majorized by ␰ជ; i.e., the com- Qmaxϭ ln q 1Ϫq ⌫͑qϩ1͒N! q ponents of both vectors listed in increasing order satisfy N ͚ j ␭ р͚ j ␰ Ͻ р iϭ1 i iϭ1 i for 0 j N. In order to be Schur concave, ⌬S (␭ជ ) has first to be invariant under exchange of any two for ␭ជ . For q→1 one has QmaxϭC Ϫln N. W * q N arguments, which is obviously the case, and second it has to ͑ ͒ (1)ϭ ␭ជ ϭ iii As m0 1 one immediately has limq→0Qq( ) 0 satisfy ͓29͔ for any vector ␭ជ . ⌬ץ ⌬ץ ͑iv͒ For q→1 one obtains the regular subentropy, SW SW ͑␭ Ϫ␭ ͒ͩ Ϫ ͪ р0. ͑30͒ ␭ץ ␭ץ ␭ជ ϭ ␭ជ 1 2 limq→1Qq( ) Q( ). 1 2

͑ ͒ ͑ ͒ ϭ FIG. 2. Reńyi entropy Sq thin line and Reńyi subentropy Qq thick line line for the N 2 probability vectors as a function of the one independent variable x for qϭ1/2 ͑a͒, qϭ1 ͑b͒, qϭ2 ͑c͒, and qϭ10 ͑d͒. The maximum of the Reńyi subentropy is increasing with q whereas →ϱ Ϫ ␭ ␭ the maximum of the Reńyi entropy is independent of q. In the limit q both quantities converge to ln max , where max is the largest component of the vector ␭ជ .

022317-5 F. MINTERT AND K. Z˙ YCZKOWSKI PHYSICAL REVIEW A 69, 022317 ͑2004͒

ϭ ϭ ͑ ͒ ϭ ͑ ͒ FIG. 3. Re´nyi subentropy Qq as a function of two independent variables of the N 3 probability distributions, q 1/2 a , q 1 b , q ϭ ͑ ͒ ϭ ͑ ͒ 2 c , and q 5 d . Gray scale is used to represent the values of subentropy: the higher q, the larger the maximum of Qq at the center ␭ជ ϭ(1/3,1/3,1/3), which refers to maximally entangled states. * ␮ Ͼ ␭ Ͼ␭ ␭ ϩ ␭ Ͼ ␭ For convenience we will ﬁrst consider q,N . Once we know Since x1 x2 and 1 2, one has x1 1 x2 2 x2 1 ␮ ϩ ␭ Ͼ about the Schur convexity or Schur concavity of q,N one x1 2. Therefore for q 1 the integrand is non-negative can easily deduce the Schur concavity of all other discussed and so is the integral. For 0ϽqϽ1 the integrand is nonposi- ␮ ␮ Ͼ quantities. The quantities q,N can be expressed by the rep- tive. Thus q,N is Schur convex for q 1 and Schur concave resentation ͓see Eq. ͑B1͔͒ for 0ϽqϽ1. Using this, one easily concludes that

␮ץ ץ ץ qϩN͒͑⌫ ␮ ϭ ͵ ͑␭ជ ជ ͒q ͑␭ Ϫ␭ ͒ͩ Ϫ ͪ q,N ͯ у ͑ ͒ q,N dx x , 1 2 0. 33 qץ ␭ץ ␭ץ ⌬ qϩ1͒͑⌫ 1 2 qϭ1

ץ ␮ץ ϭϪ ⌬ ⌬ where dx is a shorthand notation for dx1•••dxN and the As SW can be expressed as SW limq→1 q,N / q, integration is performed over the simplex ⌬ containing all Schur concavity of the subentropy can directly inferred from ជ ͚ ϭ the corresponding properties of ␮ . Also the rescaled mo- probability vectors x with ixi 1. Making use of this rela- q,N tion and assuming ␭ Ͼ␭ without loss of generality, we ob- ments ͑22͒ are Schur concave for all values of q,as1Ϫq is 1 2 Ͼ ␮ tain negative for q 1 where q,N is Schur convex. Þ For the Reńyi subentropy Qq with q 1 we have ␮ ⌫͑qϩN͒ץ ␮ץ ץ ץ ͒ q,N Ϫ q,N ϭ ͵ ͑␭ជ ជ ͒qϪ1͑ Ϫ ␭ ⌫ ϩ ͒ dx x x1 x2 . Qq Qqץ ␭ץ 1 2 ͑q 1 ⌬ ͑␭ Ϫ␭ ͒ͩ Ϫ ͪ ␭ץ ␭ץ 2 1 ͑31͒ 1 2 ␮ץ ␮ץ 1 1 ϭ ͑␭ Ϫ␭ ͒ͩ q,N Ϫ q,N ͪ ͑ ͒ ␭ . 34ץ ␭ץ The full measure of the probability simplex ⌬ can be divided Ϫ 1 2 Ͻ 1 q Qq 1 2 into two symmetric parts, the first part specified by x1 x2, Ͼ the second one by x1 x2. After exchanging the labels x1 ␮ Qq is a positive quantity. Using Schur concavity of q,N for and x2 in the second part, the first and the second part coin- Ͻ Ͻ Ͼ cide and one gets 0 q 1 and Schur convexity for q 1, we conclude that Qq is Schur concave for all positive values of q. Since we have shown that the subentropy, rescaled mo- ␮ ⌫͑qϩN͒ץ ␮ץ q,N Ϫ q,N ϭ ͵ ͑ Ϫ ͒͑␭ Ϫ␭ ͒ ments, and Reńyi subentropy vanish if and only if the con- ␭ ⌫͑ ϩ ͒ dx x1 x2 1 2ץ ␭ץ 1 2 q 1 ⌬Ͼ sidered state is separable and that these quantities are Schur

qϪ1 concave, these quantities are entanglement monotones and ϫͫͩ ␭ x ϩ␭ x ϩ͚ ␭ x ͪ may serve as legitimate measures of quantum entanglement. 1 1 2 2 i i iϾ2 Let us emphasize that the monotones found in this work ͓ ͔ Ϫ differ from the Re´nyi entropies 30 and the elementary sym- q 1 ͓ ͔ Ϫͩ ␭ x ϩ␭ x ϩ͚ ␭ x ͪ ͬ. ͑32͒ metric polynomials of the Schmidt coefﬁcients 31 and can- 1 2 2 1 i i iϾ2 not be represented as a functions of one of these quantities.

ϭ ϭ ͑ ͒ ϭ ͑ ͒ ϭ ͑ ͒ FIG. 4. Rescaled moments M q as a function of two independent variables for N 3 probability vectors, q 1/2 a , q 1 b , q 2 c , ϭ ͑ ͒ and q 5 d . Labels at corners identify pure separable states. Note that in general M q is not a monotonically increasing function of q.

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TABLE I. Wehrl entropy excess in various setups. This quantity measures the ‘‘distance’’ of a given mixed state from the subset of pure states, the distance of a given pure state from the subset of all SU(K)-coherent states, and the distance of a given pure state of a composite system from the subset of all separable ͑product͒ states.

States Mixed Pure Pure Pure Pure Systems simple simple simple bipartite multipartite Coherent states SU(N)-CS SU(2)-CS SU(K)-CS SU(N) SU(N)-CS SU(N) M-CS ͉␣ ෈ NϪ1 ͉␣ ෈ 1 ͉␣ ෈ KϪ1 ͉␣ 2෈ NϪ1 ϫ2 ͉␣ M෈ NϪ1 ϫM N͘ CP 2͘ CP K͘ CP N͘ (CP ) N͘ (CP ) ϭ ͉␣ ϭ ͉␣ ϭ ͉␣ ϭ ͉␣ ϭ ͉␣ Minimal Wehrl entropy Smin S( N͘) Smin S( 2͘) Smin S( K͘) Smin 2S( N͘) Smin MS( N͘) ␳ H →H ͉␺͘෈ NϪ1 ͉␺͘෈ NϪ1 N2Ϫ1 NMϪ1 Generic states : N N CP CP ͉⌿͘෈CP ͉⌿͘෈CP ⌬ ␳ Ϫ ͉␺ Ϫ ͉␺ Ϫ ͉⌿ Ϫ ͉⌿ Ϫ Wehrl entropy excess SS( ) CN S( ͘) C2 S( ͘) CK S( ͘) 2CN S( ͘) MCN Measures degree of mixing non-SU(2) non-SU(K) bipartite non-M-partite coherence coherence entanglement separability

VII. OUTLOOK ferent in the cases discussed. For the space of N-dimensional pure states analyzed by Husimi functions computed with re- Defining the Husimi function of a given pure state of a spect to SU(2)-coherent states this statement became famous bipartite NϫN system with respect to coherent states related ͓ ͔ ϫ as the Lieb conjecture 9 . Although it was proved in some to SU(N) SU(N) product groups and computing the Wehrl special cases of low-dimensional systems ͓12,13,15͔ and is entropy allows us to establish a link between the phase-space widely believed to be true for arbitrary dimensions, this con- approach to quantum mechanics and the theory of quantum jecture still awaits a formal proof. On the other hand, it is entanglement. easy to see that the Wehrl entropy of a mixed state is mini- This approach may be considered as an example of a mal if and only if the state is pure or the Wehrl entropy of a more general method of measuring the closeness of an ana- bipartite pure state is minimal if and only if the state is sepa- lyzed state to a family of distinguished states. This idea, ͓ ͔ rable. This is due to the fact that in both cases the entropy inspired by the work of Sugita 18 , may be outlined in the excess is equal to the subentropy which is non-negative and following setup: ͓ ͔ ͑ ͒ ͑ ͒ is equal to zero only if the state is pure 19 . Our results a Consider a set of pure or mixed states you wish to provide thus a new physical interpretation of the subentropy analyze. Q of a mixed state ␳ acting on H . Consider a purification ͑ ͒ ͑ ͒ ͉␣ N b Select a family of distinguished coherent states ͘, ͉⌿ ෈H H ෈H ␳ ͘ N N of % N . The subentropy Q( ) provides a related to a given symmetry group and parametrized by a measure of the localization of ͉⌿͘ in the Cartesian product point on a certain manifold ⍀. This family of states has to Ϫ of the manifolds of monopartite pure states, CPN 1 satisfy the identity resolution, ͐⍀͉␣͗͘␣͉d⍀ϭI. ϫ NϪ1 ͑ ͒ CP . c Define the Husimi distribution with respect to the ‘‘co- One of the main advantages of our approach is that it can herent’’ states and find the Wehrl entropy for a coherent state, ϭ ͉␣͘ easily be generalized to the problem of pure states of multi- Smin S( ). partite systems—see the last column of Table I. It is clear ͑d͒ Calculate the Wehrl entropy for the analyzed state ͉⌿͘ ͉⌿͘ ͉␺͘ that the entropy excess of is equal to zero if is a S( ). product state, but the reverse statement requires a formal ͑ ͒ ⌬ ϭ ͉␺͘ Ϫ e Compute the entropy excess S S( ) Smin , proof. Moreover, the Schmidt decomposition does not work which characterizes quantitatively to what extent the ana- for a three- ͑or many-͒ partite case. Thus in order to compute ͉␺͘ lyzed state is not ‘‘coherent.’’ explicitly the entropy excess in these cases, one has to con- For instance, analyzing the space of pure states of size N sider by far more terms than in the bipartite case. In the we may distinguish the spin SU(2)-coherent states or, in special case of three qubits any pure state may be character- general, the SU(K)-coherent states ͑with KϽN) param- ͓ ͔ 1 KϪ1 ized by a set of five parameters 32,33 , and it would be etrized by a point on CP and on CP , respectively. In the interesting to express the entropy excess of an arbitrary ϫ case of the N L composite quantum system we may distin- three-qubit pure states as a function of these parameters. The guish the SU(N) SU(L)-coherent states. They form the set second moment for this system was already calculated ͓18͔ ͑ ͒ of product separable states and are labeled by a point on but expressions for other moments or for the Wehrl entropy NϪ1ϫ LϪ1 CP CP . For mixed states of size N we may select are still missing. SU(N)-coherent states—i.e., all pure states. In all these three cases we define the same quantity, entropy excess ⌬S, which ACKNOWLEDGMENTS has entirely different physical meaning. It quantifies the degree of non-SU(K) coherence, the degree of entanglement, We are indebted to Andreas Buchleitner, Marek Kus´, Ber- and the degree of mixing, respectively, as listed in Table I. nard Lavenda, and Prot Pakon´ski for fruitful discussions, Describing the items ͑a͒–͑e͒ of the above procedure we comments and remarks and acknowledge fruitful correspon- have implicitly assumed that the Wehrl entropy is minimal if dence with Ayumu Sugita. Financial support by Volkswagen and only if the state is ‘‘coherent.’’ This important point re- Stiftung and Polish Ministry of Scientific Research under quires a comment, since the status of this assumption is dif- Grant No. PBZ-Min-008/P03/03 is gratefully acknowledged.

022317-7 F. MINTERT AND K. Z˙ YCZKOWSKI PHYSICAL REVIEW A 69, 022317 ͑2004͒

APPENDIX A: MOMENTS OF THE HUSIMI FUNCTION NϪ1 N! x(max) ϭ ͵ i f q Ϫ ͟ dxi As already mentioned, every pure state belonging to an ͑2␲͒N 1 iϭ1 0 H N-dimensional Hilbert space N is a SU(N)-coherent state 2␲ 1 and vice versa. Therefore, following Eq. ͑3͒, we can param- N ϪM ␸ ϫ ͵ ␸ i( i i) i 2 N ϩM d ie xi ( i i) etrize all SU(N)-coherent states by 0

Ϫ Ϫ ͚NϪ1N ϩM NϪ1 1/2 NϪ1 N 1 q 1/2 ␹ϭ1 ␹ ␹ ␸ ͱ i i ϫͩ Ϫ ͪ ͑ ͒ ͉␣͘ϭͩ Ϫ ͪ ͉ ͘ϩ ͉ ͘ ͑ ͒ 1 ͚ x j , A5 1 ͚ xi 0 ͚ xie i , A1 ϭ iϭ1 iϭ1 j 1

where the integrations over xi are performed in increasing with (max) order of i and the upper integration limit is given by xi NϪ1 ϭ1Ϫ͚ ϭ ϩ x . It is useful to perform the ␸ integrations NϪ1 j i 1 j i ﬁrst. They lead to ␦N M terms, so that one gets р р Ϫ i , i 0 xi 1 ͚ x j jϭiϩ1 Ϫ Ϫ Ϫ͚NϪ1N N 1 N 1 q ␹ϭ1 ␹ x(max) N ϭ ͩ ͵ i i␦ ͪͩ Ϫ ͪ f q N! ͟ dxixi N ,M 1 ͚ x j . and iϭ1 0 i i jϭ1 ͑A6͒ NϪ1 ␮ϭ ␲͒NϪ1 ␸ d ͓N!/͑2 ͔ ͟ dxid i . In order to perform the integrations over xi we need to deﬁne iϭ1 an auxiliary function

Ϫ Ϫ Ϫ͚NϪ1N ͉ ͘ ϭ Ϫ N 1 N 1 q ␹ϭ1 ␹ The states i , i 0,...,N 1, form an orthonormal basis, ˜x(max)(␤) N ͉ ͘ ͑␤ ͒ϭͩ ͵ i i ͪͩ ␤Ϫ ͪ while 0 is the reference state. The Husimi function H⌿ of gq ,N ͟ dxix ͚ x j , ϭ i ϭ ͉⌿͘ i 1 0 j 1 a pure state with respect to SU(N) SU(N)-coherent ͑ ͒ ͉␣(2)͘ϭ͉␣ ͘ ͉␣ ͘ A7 states N N A N B reads (max) NϪ1 with ˜x (␤)ϭ␤Ϫ͚ ϭ ϩ x . In the following we will N i j i 1 j ϭ ͱ␭ ␭ ␣ ͉␯ ␮͉␣ show that, for any integer q, H⌿ ͚ ␯ ␮A͗ N ͘AA͗ N͘A ␯,␮ϭ1 NϪ1 NϪ1 ϫ ͗␣ ͉␯͘ ͗␮͉␣ ͘ ͑ ͒ ͩ N ͪͩ Ϫ N ͪ B N BB N B , A2 ͟ i! q ͚ i ! iϭ1 iϭ1 g ͑␤,N͒ϭ␤(qϩNϪ1) ͑A8͒ q ͑qϩNϪ1͒! where ͉⌿͘ is represented in its Schmidt basis, ͉⌿͘ ϭ͚ ͱ␭ ͉␯͘ ͉␯͘ ␯ ␯ A B . The qth moment is then given by ͑ ͒ ␤ holds. According to the deﬁnition A7 , gq( ,N) satisﬁes the relation N ϭ ͱ␭ ␭ ␭ ␭ 2 ␤ mq ͚ ␯ ␯ ␮ ␮ f q N ␯ ␯ ϭ1 1••• q 1••• q ͑␤ ͒ϭ ͵ NϪ1 ͑␤Ϫ Ϫ ͒ 1••• q g ,N dx Ϫ x g ϪN x Ϫ ,N 1 . ␮ ␮ ϭ q N 1 NϪ1 q NϪ1 N 1 1••• q 1 0 ͑ ͒ ϫ N N M M ͒ ͑ ͒ A9 ͑ 1 ,..., N , 1 ,..., N , A3 Making use of with ␤ N ϪN ͑␤ ͒ϭ ͵ 1͑␤Ϫ ͒q 1 gq ,1 dx1 x1 x1 0 ϭ ͵ ␮͑␣͒͗␣͉␯ ͗͘␮ ͉␣͗͘␣͉␯ ͗͘␮ ͉␣͘ ͗␣͉␯ ͘ f q d 1 1 2 2 ••• q N !͑qϪN ͒! ϭ␤qϩ1 1 1 ͑ ͒ ϫ ␮ ͉␣ ͑ ͒ ϩ , A10 ͗ q ͘. A4 ͑q 1͒!

N M ͉␯ we immediately have Eq. ͑A8͒ for Nϭ2: The integers i and i count how often the states i͘ and ͗␮ ͉ i appear in f q . Note that one has to integrate separately ␤ N ϪN over both subsystems. Due to the symmetry of the Schmidt ͑␤ ͒ϭ ͵ 1͑␤Ϫ ͒q 1 ͑ ͒ gq ,2 dx1 x1 x1 . A11 decomposition with respect to the two subsystems, both in- 0 tegrations lead to the same result and it is just the square of ͑ ͒ ␤ Ϫ the integral over f q that is entering mq . Using the parametri- Assuming that Eq. A8 holds true for gq( ,N 1) and mak- ͑ ͒ ͑ ͒ zation A1 f q can be expressed as ing use of Eq. A9 one gets

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Ϫ ␤ Ϫ␯ Ϫ͚N 2N N ϩ Ϫ ϪN q q 1 q iϭ1 i ͑␤ ͒ϭ ͵ NϪ1͑␤Ϫ ͒q N 2 NϪ1 N N N gq ,N dxNϪ1xNϪ1 xNϪ1 ␮ ϭ ␭ 1␭ 2 ␭ NϪ1 0 q,N ͚ ͚ ͚ Ϫ N ϭ N ϭ ••• N ϭ 1 2 ••• N 1 1 0 2 0 NϪ1 0 NϪ2 NϪ2 Ϫ qϪ͚N 1N ͩ N ͪͩ ϪN Ϫ N ͪ ϫ␭ iϭ1 i ͑ ͒ ͟ Ϫ ͚ i! q N 1 i ! N . A17 iϭ1 iϭ1 ϫ ϩ Ϫ ϪN ͒ . ͑q N 2 NϪ1 ! As a starting point for this line of reasoning we are going to demonstrate that ␮Ϫ ϭ0. For Nϭ2 this equality can be ͑A12͒ 1,N checked by direct calculation. For NϾ2 we have Using further Eq. ͑A10͒ one has N ␭NϪ3 ␭ ␮ ϭ i i N ͑ ϩ Ϫ ϪN ͒ Ϫ1,N ͚ NϪ1! q N 2 NϪ1 ! ϭ Þ ␭ Ϫ␭ g ͑␤,N͒ϭ␤qϩNϪ1 i 1,i j ͑␭ Ϫ␭ ͒ i j q ͑ ϩ Ϫ ͒ ͟ i k q N 1 ! kϭ1,kÞi,kÞ j Ϫ Ϫ N 2 N 1 ␭NϪ2 ͩ N ͪͩ Ϫ N ͪ j ͟ i! q ͚ i ! ϩ . ͑A18͒ iϭ1 iϭ1 ϫ ͑ ͒ ͑␭ Ϫ␭ ͒ ϩ Ϫ ϪN ͒ . A13 ͟ j k ͑q N 2 NϪ1 ! kϭ1,kÞ j ␭ ␭ Ϫ␭ ϭ ϩ␭ ␭ Ϫ␭ Finally one ends up with Since i /( i j) 1 ␣ /( i j), we can make use of ͚N ␭NϪ3 ͟ ␭ Ϫ␭ NϪ1 NϪ1 the relation iϭ1,iÞ j i / kϭ1,kÞi,kÞ j( i k) ϭ␮ ␭ ␭ ␭ ␭ ͩ N ͪͩ Ϫ N ͪ q,NϪ1( 1 ,..., jϪ1 , jϩ1 ,..., N), in which the right- ͟ i! q ͚ i ! iϭ1 iϭ1 hand side vanishes by assumption. Therefore we end up with ͑␤ ͒ϭ␤qϩNϪ1 gq ,N ͑ ϩ Ϫ ͒ ; q N 1 ! N ␭NϪ3 ͑A14͒ ␮ ϭ␭ i ͑ ͒ Ϫ1,N j ͚ N . A19 iϭ1 ͑ ͒ ␤ ␤ϭ ␭ Ϫ␭ ͒ i.e., Eq. A8 also holds for gq( ,N). Finally, for 1, we ͟ ͑ i k get kϭ1,kÞi

NϪ1 NϪ1 We still have the freedom to choose the index j. As the result ͫ ͑N ͒ͬͩ Ϫ N ͪ does not depend on the choice of j, both sides of the equality ͟ i! q ͚ i !NϪ1 iϭ1 iϭ1 have to be zero. Now we can come back to the proof of Eq. ϭ ␦ f q N! ͟ N ,M . ͑ ͒ ϭ ͑qϩNϪ1͒! jϭ1 j j A17 . For N 2 it is just a straightforward calculation to ͑A15͒ show that ϩ ϩ N 2 q ␭q 1 ␭q 1 Now we have m ϭ͚␯ ␯ ϭ ␭␯ ␭␯ f , where the ␯ qϪ␯ 1 2 q 1 q 1 1••• q q ␭ ␭ ϭ ϩ ϭ␮ ͑ ͒ ••• ͚ 1 2 q,N , A20 ␦ ␯ϭ0 ␭ Ϫ␭ ␭ Ϫ␭ n␯ ,m␯ terms assure one that there are only integer powers of 1 2 2 1 ␭ N the ␯ . Since f q depends only on the integer powers i of ␭ i which can be checked by multiplying both sides with 1 the ␭␯ , we need to count how many terms with ﬁxed powers Ϫ␭ ͑ ͒ Ϫ i 2. Assuming that Eq. A17 is true for N 1 we get occur. Using simple combinatorics one can see that there are q ␯ 2 ϭ ␮ ␭ ͑ ͒ q! mq,N ͚ qϪ␯,NϪ1 N . A21 ␯ϭ0 ͩ NϪ1 NϪ1 ͪ ͩ ͑N ͒ ͪͩ Ϫ N ͪ ͑ ͒ ͟ i ! q ͚ i ! Making use of the assumption A17 one obtains iϭ1 iϭ1 NϪ1 ␭NϪ2 ␭qϩ1Ϫ␭qϩ1 terms containing the expression ϭ i i N mq,N ͚ NϪ1 ␭ Ϫ␭ iϭ1 i N NϪ1 ͑␭ Ϫ␭ ͒ N N N Ϫ qϪ͚ ϭ N ͟ i j ␭ 1␭ 2 ␭ N 1␭ i 1 i . jϭ1,jÞi 1 2 ••• NϪ1 N N ␭qϩNϪ1 Thus ﬁnally we get ϭ i ͚ N Ϫ iϭ1 Ϫ␯ Ϫ͚N 2N 2 q q 1 q iϭ1 i ͟ ͑␭ Ϫ␭ ͒ N!⌫͑qϩ1͒ N N i j ϭͩ ͪ ␭ 1␭ 2 jϭ1,jÞi mq ͚ ͚ ͚ 1 2 ⌫͑qϩN͒ N ϭ0 N ϭ0 ••• N ϭ0 ••• 1 2 NϪ1 N ␭NϪ2 ␭qϩ1Ϫ␭qϩ1 Ϫ qϩ1 i i N N qϪ͚N 1N Ϫ␭ ͚ . ϫ␭ NϪ1␭ iϭ1 i ͑ ͒ N N ␭ Ϫ␭ Ϫ . A16 iϭ1 i N N 1 N ␭ Ϫ␭ ͒ ͟ ͑ i j ϭ Þ To end, we will show by induction that for any positive j 1,j i ␮ integer q the quantity q,N can be written as ͑A22͒

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␮ ϭ NϪ2 Now one just has to make use of Ϫ1,N 0 and one imme- ⌫͑qϩN͒ 1 1Ϫa ͑ ͒ ϭ ͵ diately gets that the assumption A17 holds true for N. This hq,N ͚ dxNϪ1 ⌫͑qϩ1͒ ϭ result together with Eq. ͑A16͒ completes the proof of Eq. ͑8͒ i 1 ␭ Ϫ␭ ͒ 0 ͟ ͑ i j for positive integers q. Since integers are dense at ϱ,we jÞi conclude that there is a unique generalization to real q. Thus qϩNϪ2 ϫ͓͑1ϪaϪx Ϫ ͒␭ Ϫ ϩbϩx Ϫ ␭ Ϫ ͔ , Eq. ͑8͒ is also valid for real q. Equation ͑18͒ for the moments N 1 N 1 N 1 N 1 of the Husimi function of a monopartite system differs by a ͑B4͒ proportionality constant only and its proof is analogous. where we were using the assumption for hq,NϪ1(a ϩ ϩ ␭ xnϪ1 ,b xnϪ1 nϪ1). Performing the integration one gets APPENDIX B: INTEGRAL REPRESENTATION Ϫ ⌫͑qϩN͒ N 2 ͓͑1Ϫa͒␭ ϩb͔qϩnϪ1 OF THE MOMENTS ϭ i hq,N ⌫͑ ϩ ͒ ͚ q 1 ͩ iϭ1 ␭ Ϫ␭ ͒ In order to prove that ͟ ͑ i j jÞi NϪ2 Ϫ ͒␭ ϩ qϩnϪ1 ⌫͑qϩN͒ ͓͑1 a NϪ1 b͔ ␮ ϭ ͵ ͑␭ជ ជ ͒q ͑ ͒ ϩ ͚ . ͑B5͒ q,N dx x , B1 ⌫͑qϩ1͒ ⌬ iϭ1 ͪ ␭ Ϫ␭ ͒ ␭ Ϫ␭ ͒ ͑ NϪ1 i ͟ ͑ i j jÞi where the integration is performed over the probability sim- We are done if we manage to show that plex ⌬, we consider the function h (a,b) of two real vari- q,N NϪ2 ables a and b deﬁned as 1 1 ϭ ͚ NϪ2 NϪ1 iϭ1 Ϫ Ϫ Ϫ ͑␭ Ϫ Ϫ␭ ͒ ͟ ͑␭ Ϫ␭ ͒ ͟ ͑␭ Ϫ Ϫ␭ ͒ ⌫͑qϩN͒ 1 a 1 a iϾ1xi N 1 i i j N 1 j ϭ ͵ ͵ jÞi jϭ1 hq,N dxnϪ1 dx1 ⌫͑qϩ1͒ 0 ••• 0 ͑B6͒ nϪ1 nϪ1 q or, equivalently, ϫͫ ␭ ϩͩ Ϫ Ϫ ͪ ␭ ϩ ͬ ͑ ͒ Ϫ Ϫ ͚ xi i 1 a ͚ xi n b . B2 N 2 N 2 ϭ ϭ ␭ Ϫ␭ i 1 i 1 ϭ NϪ1 j ϭ ͑ ͒ p ͚ ͟ ␭ Ϫ␭ 1. B7 iϭ1 jÞi i j Now we are going to show by induction that h can be ex- Ϫ ␭ This is a polynomial of (N 2)nd order in NϪ1.Ifweﬁnd pressed as Ϫ ␭ N 1 different values for NϪ1 such that p equals 1, the ␭ polynomial p has to be identically equal to unity. Inserting k n ϩ Ϫ ϭ Ϫ ͓͑1Ϫa͒␭ ϩb͔q n 1 (k 1,...,N 1) one gets ͒ϭ i ͑ ͒ hq,N͑a,b ͚ . B3 ϭ ␭ Ϫ␭ i 1 ͟ ͑␭ Ϫ␭ ͒ ϭ k j ϭ ͑ ͒ i j p ͟ ␭ Ϫ␭ 1. B8 jÞi jÞk k j Thus pϵ1, which completes the proof of Eq. ͑B3͒. Setting For Nϭ2 it is straightforward to check that the assumption aϭbϭ1 in Eqs. ͑B2͒ and ͑B3͒, we immediately get Eq. ͑B3͒ holds. For NϾ2 we obtain ͑B1͒.

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