arXiv:quant-ph/0307169v2 4 Nov 2003 ftegoa iiu a ie o uesae fdi- of states pure for proof given a was while minimum mension [10], global entropy the Wehrl of the of minimum local fitgrorder entropies R´enyi integer -Wehrl the of concerning conjecture, Lieb the by characterized be may definition | rvnpoet skoni h ieauea the as literature the in known conjecture is property proven h uiidsrbto fthe of distribution Husimi the oeetsae aaerzdb onso h complex the on manifold points by projective parametrized states coherent onatrod yLe 9,btteaaoosrsl for result analogous proved the was but [9], states the Lieb coherent by afterwords oscillator soon harmonic [8] Wehrl the phase by for first the relation. conjectured in uncertainty property Heisenberg this localized Interestingly, the as by allowed values are as smallest which space, admits states, entropy coherent Wehrl for The Husimi the [8]. of function entropy Boltzmann–Gibbs continuous the motn ont htif that note to important tdrpeetto of representation ated a edfie steepcainvleo h analyzed the of value expectation the as state distribution defined Husimi be The can [1]. distribution probability point a any as for non-negative is state, α quantum any The for ists representation. space phase distribution its analyze to osrcino eeoo 2.Frisac,tegroup the sphere instance, the For [2]. 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Nauk, Akademia Polska Teoretycznej, Fizyki Centrum S 9.I skonta the that known is It [9]. and 3 = sotnvr ovnett okwt:i ex- it with: work to convenient very often is iatt uesae hsqatt,a ela h generali the sta se as mixed is well the state as of quantity, subentropy pure This the analyzed to state. the equal pure if be bipartite only to shown and is if entropy minimal is entropy ASnmes 36 n 89.70.+c, Mn, 03.67 numbers: PACS ma and monotones entanglement are entanglement. they so Schur-concave, be 2 ydfiigisHsm itiuinwt epc to respect with distribution Husimi its defining by ( = N epooet uniyteetnlmn fpr ttsof states pure of entanglement the quantify to propose We 1 a lnkIsiuefrtepyiso ope ytm N¨ot systems complex of physics the for Institute Planck Max .INTRODUCTION I. )–coherent. 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SU of product entropy to Wehrl respect with The states coherent define to degree the of measure a mixing. as for considered of attained be entropy can minimal states, the pure and between quantity difference latter The the entropy. Wehrl the and moments SU esr ufiln hspoet scle entanglement a called and is operations, property local monotone. any this of fulfilling in- action and measure cannot the 21] entanglement formulated under quantum 20, axioms particular, crease [19, of In list [19]. e.g. the in satisfy (see therein) literature references the in introduced subentropy. the of generalization continuous natural a endb os ta.[8.Cluaigteecs fthe of subentropy excess the the Calculating to the [18]. equal define R´enyi–Wehrl we al. be entropies et Jozsa to Wehrl by shown The defined is entangled. to is dif- quantifies, excess state value the entropy analyzed minimal as the the defined extend to what excess state respect entropy the with the if ference Hence only state. and uct if minimum their rirr ie unu state quantum mixed arbitrary one. to original not seems the conjecture than this simpler but be 15], [14, states coherent ytm h uiifnto scmue ihrespect with computed is function to Husimi The system. SU nti ae eflo i daadcmueexplic- compute and idea his follow we paper this In nlzn uesae fabpriesse ti helpful is it system bipartite a of states pure Analyzing h o-oa rpriso uesaeo an of state pure a of properties non-local The entanglement quantum of measures different Several nti okw osdrteHsm ucino an of function Husimi the consider we work this In SU ( ( ( N N N ( ) oeetsae,adcluaebt t statistical its both calculate and states, coherent ) ) e Rey)sbnrpe,aepoe to proved are (R´enyi) subentropies, zed Zyczkowski ˙ N × × i´ w3/4 268Wrzw,Poland Warszawa, 02-668 nik´ow 32/44, ) SU SU eue satraiemaue of measures alternative as used be y × N aal.Teecs fteWehrl the of excess The parable. eotie ypriltaeo the of trace partial by obtained te ntesr 8017Dresden 01187 38 hnitzerstr. SU ( ( × N N N oeetsae.TeWehrl The states. coherent ) –Swsfis osdrdi h original the in considered first was )–CS ( N 2 iatt unu systems quantum bipartite , –S oteWhletoisachieve entropies Wehrl the so )–CS, 3 | Ψ i ρ fthe of fsize of R´enyi subentropy N N × | ihrsetto respect with Ψ N i saprod- a is composite SU N ( × K N )– – 2 system has to be characterized by N 1 independent momentum’s z-component J with maximal eigenvalue, − z monotones [20]. In this work we demonstrate that N 1 while the lowering operator reads J1 = J− = Jx iJy. statistical moments of properly defined Husimi functions− To characterize quantitatively the localization of− an an- naturally provide such a set of parameters, since they are alyzed state ̺ in the phase space we compute its Husimi Schur–concave functions of the Schmidt coefficients and distribution H̺ with respect to SU(N)–CS and analyze thus are non-increasing under local operations. the moments mq of the distribution
II. HUSIMI FUNCTION AND WEHRL q mq(̺)= dµ(α) (H̺(α)) . (4) ENTROPY ZΩN
N×N Here dµ(α) denotes the unique, unitarily invariant mea- Any density matrix ̺ C can be represented by N−1 ∈ sure on CP , also called Fubini–Study measure. The its Husimi function H̺(α), that is defined as its expec- measure dµ is normalized such that for any state ̺ the tation value with respect to coherent states α , | i first moment m1 is equal to unity, so the non–negative Husimi distribution H may be regarded as a phase-space H (α)= α ̺ α (1) ̺ ̺ h | | i probability distribution. In general one may define the Husimi distribution with The definition of the moments mq is not restricted to respect to SU(K)-coherent states with 2 K N, but integer values of q. However, from a practical point of most often one computes the Husimi distribution≤ ≤ with view it will be easier to perform the integration for integer respect to the SU(2) coherent states, also called spin co- values of q. But once mq are known for all integer q, there herent states [3, 4]. Let us recall that any family of coher- is a unique analytic extension to complex (and therefore ent states α has to satisfy the resolution of identity also real) q, as integers are dense at infinity. {| i} Another quantity of interest is the Wehrl entropy SW , defined as [8]
dµ(α) α α = ½ , (2) | ih | ZΩ SW (̺)= dµ(α) H̺(α) ln H̺(α) . (5) where dµ(α) is a uniform measure on the classical mani- − ΩN fold Ω. In the case of the degenerated representation Z Again, performing the integration might be difficult. of SU(K) coherent states [5, 6, 7] this manifold, ΩK = K−1 But if all the moments m are known in the vicin- U(K)/[U(K 1) U(1)] = CP , is just equivalent q to the space− of all× pure states of size K. This complex ity of q = 1, one can derive SW quite easily. Using q q projective space arises from the K–dimensional Hilbert ∂H /∂q = H ln H, one gets space K by taking into account normalized states, and H ∂mq(̺) identifying all elements of K , which differ by an overall SW (̺)= lim . (6) phase only. In the simplestH case of SU(2) coherent states − q→1 ∂q 1 2 it is just the well known Bloch sphere, Ω2 = CP = S . The moments mq of the Husimi function allow us to write Analyzing a mixed state of dimensionality N we are go- the R´enyi–Wehrl entropy ing to use SU(N) coherent states, which will be denoted by αN . With this convention the space Ω is equivalent 1 | i N−1 SW,q(̺) := ln mq(̺) , (7) to the complex projective space CP , so every pure 1 q state is SU(N)–coherent and their Husimi distributions − which tends to the Wehrl entropy S for q 1. As a have the same shape and differ only by the localization W → in Ω. Husimi function is related to a selected classical phase classical en- The SU(N) coherent states may be defined according space, the Wehrl entropy is also called tropy von Neumann entropy to the general group-theoretical approach by Perelomov, [8, 16], in contrast to the S = Tr̺ ln ̺, that has no immediate relation to clas- by a set of generators of SU(N) acting on the distin- N − guished reference state 0 . We are going to work with sical mechanics. the degenerate representation| i of SU(N) only, in which a In section III we consider how strongly a given state coherent state may be parametrized by N 1 complex is mixed and in section IV we discuss the non-locality numbers γ , − of bipartite pure states. For this purpose we pursue an i approach inspired by the Lieb conjecture [9], according γ1J1 γN−1JN−1 αN = γ1,...,γN−1 := e e 0 , (3) to which the Wehrl entropy SW of a quantum state is | i | i ··· | i minimal if and only if ̺ is coherent. In order to measure where operators Ji may be interpreted as lowering oper- a degree of mixing of a monopartite mixed state of size N ators [6]. In language of the N–level atom they couple we therefore use the SU(N) coherent state, while for the the highest N-the level with the i-th one (see e.g. [7]). In study of entanglement of a bipartite pure N N state the simplest case of SU(2) coherent states the reference we use the coherent states related to the product× group state 0 is equal to the eigenstate j, j of the angular SU(N) SU(N). | i | i × 3
III. MONOPARTITE SYSTEMS: MIXED eigenvalue. Therefore we define the entropy excess ∆S STATES as
∆S(̺)= S (̺) S (ρ ) , (12) In this paragraph we will focus on mixed states ̺ W − W ψ N×N ∈ C acting on an N–dimensional Hilbert space . N where ρ refers to an arbitrary pure state and the Wehrl Such a state may appear as a reduced density matrixH ψ entropy of an arbitrary N-dimensional pure state is given ̺ , defined by the partial trace ̺ (Ψ) = Tr Ψ Ψ of a r r A by S (ρ )= C . Hence the entropy excess, equal to the pure state Ψ of a bipartite system.| ih | One W ψ N N N subentropy ∆S(̺) = Q(̺), is non negative and equal to could also consider| i ∈ H a system⊗ H coupled to an environment, zero only for pure states. in which a mixed state is obtained by tracing over the As a byproduct we find a bound on S (̺). The suben- environmental degrees of freedom. The von Neumann W tropy Q(̺) is known not to be larger than the von Neu- entropy S quantifies, on one hand, the degree of mixing N mann entropy S [18]. Using this fact and (9) we find of ̺ , and on the other, the nonlocal properties of the N r that S (̺) S (̺)+ C . As it is also known that the bipartite state Ψ . W N N von Neumann≤ entropy S is not larger than the Wehrl To obtain an| alternativei measure of mixing of ̺ we N entropy S (̺) [8], we end up with the following upper are going to investigate its Husimi function defined with W (1) and lower bound on SW (̺) respect to the SU(N)–CS as H̺ (α)= α ̺ α . The h N | | N i moments mq of the Husimi distribution can be expressed S (̺)+ C S (̺) S (̺) . (13) N N ≥ W ≥ N as functions of the eigenvalues λi of ̺, which in the case of a reduced state of a bipartite system coincide with the Schmidt coefficients of the pure state Ψ . The deriva- IV. BIPARTITE SYSTEMS: PURE STATES tion of the explicit result in terms of the| i Euler Gamma function Let us now focus on bipartite systems described by a N!Γ(q + 1) Hilbert space that can be decomposed into a tensor m = µ with product = H of two sub-spaces. With a local q Γ(q + N) q,N H HA ⊗ HB unitary transformation l = A B any pure state Ψ can be transformedU toU its ⊗Schmidt U form [22] N q+N−1 (8) | i ∈ H λi µ = N q,N N i=1 Ψ = λi i A i B , (14) X (λi λj ) | i | i ⊗ | i − i=1 6 j=1Y,j=i X p where N =min(dim A,dim B) and the real prefac- is provided in Appendix A. Performing the limit (6) we H H tors λi called Schmidt coefficients are the eigenvalues of find that the Wehrl entropy SW equals the subentropy the reduced density matrix ̺r. Thanks to the Schmidt Q(̺) up to an additive constant CN , decomposition we can assume dim A = dim B = N without loss of generality. FollowingH an ideaH of Sugita S (̺)= Q(̺)+ C . (9) W N [17] we consider the question whether the Wehrl entropy of a bipartite state can serve as a measure of entan- The N-dependent constant can be expressed as glement. He defined a coherent state α(M) of an M- | 2 i N partite qubit system as the tensor product of M coher- (1) CN = Ψ(N + 1) Ψ(2) = 1/k , (10) ent states α of single qubit systems. We general- − | 2 i k=2 ize this approach for bipartite systems, omitting the re- X striction to qubits and calculate all moments and the where the digamma function is defined by Ψ(x) = Wehrl entropy of the respective Husimi functions. For ∂ ln Γ(x) ∂x . The subentropy a bipartite pure state Ψ , we use the tensor prod- | i ∈ H (2) ⊗2 uct of two SU(N)-coherent states αN = αN N N | i | i ∈ H λi log λi to define a Husimi function. More explicitly one can Q(̺)= =: Q(~λ) (11) (2) − N write α = αN A αN B , with αN A A and i=1 | N i | i ⊗ | i | i ∈ H X (λi λj ) α (we will drop the index referring to the − | N iB ∈ HB j=1,j=6 i subsystem, wherever there is no ambiguity). Such bi- Y partite coherent states were used already in the original was defined in an information theoretical context [18]. paper of Wehrl [16]. It is related to the von Neumann entropy SN (̺) in the The Husimi function H of a bipartite state Ψ is then sense that both quantities give the lower and the upper given by | i bounds for the information which may be extracted from the state ̺ [18]. The subentropy Q(̺) takes its mini- 2 mal value 0 for pure states with only one non vanishing H (α , α )= Ψ ( α α ) . (15) Ψ A B h | | N iA ⊗ | N iB
4
By definition any product state is a SU(N) SU(N) The subentropy Q(λ) takes its minimal value 0 if and coherent state. According to the Lieb conjecture,× orig- only if a given state is separable, i.e. if all but one inally formulated for SU(2) coherent states, the Wehrl Schmidt coefficients vanish. Therefore we define the bi- entropy is minimal for coherent states [9]. Hence it is partite entropy excess ∆SW (Ψ) natural to expect that the Wehrl entropy ∆S (Ψ) = S (Ψ) S (φ )= Q(̺ ) (21) W W − W sep r S (ψ)= dµ (α) dµ (α) W A B that is vanishing if and only if Ψ is separable. Fur- − ΩN ΩN Z Z thermore, one can also consider the| R´enyi–Wehrli entropy H (α , α ) ln H (α , α ) (16) Ψ A B Ψ A B (q) 1 SRW = 1−q ln mq. provides a measure of how “incoherent” a given state is. In order to use the moments mq as entanglement mono- Due to the equivalence of “coherence” and separability tones, it is conventional to rescale them such that they we expect that SW can also serve as an entanglement vanish for separable states and are positive for entangled measure. In the following we discuss properties of SW ones and show that it satisfies all requirements of entangle- 1 ment monotones [19, 20]. M = (µ 1) . (22) q 1 q q,N − There is a one-to-one correspondence between states − and Husimi functions. Concerning entanglement, two dif- These quantities are analogous to the Havrda-Charvat ferent states that are connected by a local unitary trans- entropy (also called Tsallis entropy) [23, 24] and in the formation are considered equivalent. As by construction limit q 1 one obtains the subentropy the moments mq of H are invariant under local unitary → transformations, they reflect this equivalence and there- lim Mq = Q(~λ) . (23) fore can be expected to be good quantities to character- q→1 ize the nonlocal properties of a bipartite state. For a pure state Ψ A B, there are N 1 indepen- dent moments,| i determining∈ H ⊗ H N 1 independent− Schmidt V. RENYI´ SUBENTROPY coefficients (one coefficient is determined− by the normal- ization). In our case the moments read As discussed in previous sections, the excess of the
q Wehrl entropy ∆S may be used as a measure of the de- mq = dµA(αA) dµB(αB ) HΨ(αA, αB ) , (17) gree of mixing for monopartie states, or degree of entan- ΩN ΩN glement for pure states of bipartite systems. Since the Z Z h i moments of the Husimi distribution are found, we may where the integration is performed over the Cartesian extend the above analyzis for the R´enyi–Wehrl entropy product ΩN ΩN , being the space of all product pure × SW,q defined by (7). states of the bipartite system. The proper normaliza- Considering, for instance, the case of mixed states of tion of dµA and dµB assures that m1 = 1. The re- a monopartite system we use (8) to find the minimal quired N 1 monotones can be provided by mq with R´enyi–Wehrl entropy attained for pure states ̺ , q =2,...,N− . The moments can be expressed as a func- ψ tion of the Schmidt coefficients λi, defined in eq (14) 1 N!Γ(q + 1) C = S (̺ )= ln , (24) N,q W,q ψ 1 q Γ(q + N) N!Γ(q + 1) 2 − mq = µq,N (18) Γ(q + N) which for q 1 reduces to (10). In an analogy to (9) we define the R´enyi–Wehrl→ entropy excess and are related to the monopartite moments (eq. 8) by a multiplicative factor. The derivation of this result is ∆SW,q(̺)= SW,q(̺) CN,q . (25) provided in Appendix A. Having closed expressions for − the moments that can be extended to real q, one easily Applying (8) it is straightforward to obtain result in finds the bipartite Wehrl entropy terms of the eigenvalues λi of the analyzed state ̺
N 1 λq+N−1 SW (ψ)= dµ1(αA) dµ2(αB) i ~ − ∆SW,q(̺)= ln =: Qq(λ) . ΩN ΩN 1 q N Z Z i=1 H (α , α ) ln H (α , α ) . (19) − X (λ λ ) Ψ A B Ψ A B i − j 6 j=1Y,j=i Performing the limit (6) one finds that SW equals the (26) corresponding monopartite quantity up to an additive This result shows that the excess of the R´enyi Wehrl constant entropy Qq(̺)= Qq(~λ) may be called R´enyi subentropy since for q 1 it tends to the subentropy (11). On ~ SW (Ψ) = Q(λ)+2CN . (20) one hand it→ may be treated as a function of an arbitrary 5
a) b) c) d) Sq Sq Sq Sq ln2 ln2 ln2 ln2 0.6 0.6 0.6 0.6
0.4 0.4 0.4 0.4
0.2 0.2 0.2 0.2
0 0 0 0 0 0.5 x 1 0 0.5 x 1 0 0.5 x 1 0 0.5 x 1
FIG. 1: R´enyi entropy Sq (thin line) and R´enyi subentropy Qq (thick line) line for the N = 2 probability vectors as a function of the one independent variable x for q = 1/2 (a), q = 1 (b), q = 2 (c) and q = 10 (d). The maximum of the R´enyi subentropy is increasing with q whereas the maximum of the R´enyi entropy is independent of q. In the limit q →∞ both quantities converge to − ln λmax, where λmax is the largest component of the vector ~λ.
(001) (001) (001) (001) a) b) c) d)
* * * *
(100) (010) (100) (010) (100) (010) (100) (010)
FIG. 2: R´enyi subentropy Qq as a function of two independent variables of the N = 3 probability distributions, q = 1/2 (a), q = 1 (b), q = 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 ~λ∗ = (1/3, 1/3, 1/3), which refers to maximally entangled states. (001) (001) (001) (001) a) b) c) d)
* * * *
(100) (010) (100) (010) (100) (010) (100) (010)
FIG. 3: Rescaled moments Mq as a function of two independent variables for N = 3 probability vectors, q = 1/2 (a), q = 1 (b), q = 2 (c) and q = 5 (d). Labels at corners identify pure separable states. Note that in general Mq is not a monotonically increasing function of q. quantum state ̺, on the other it may be defined for an of the q-th moment of the respective state arbitrary classical probability vector ~λ. 1 In a sense Q is a quantity analogous to the R´enyi S (̺)= ln(Tr̺q). (28) q q 1 q entropy Sq, which may be defined for an arbitrary prob- − ability vector In the following we will sketch some properties of Qq(~λ) as a function of a classical propability vector. All the con- N 1 siderations are also applicable in the quantum case, par- S (~λ)= ln λq. (27) q 1 q i ticularly if we use Qq as an entanglement monotone using i=1 − X the moments discussed in the preceding section. Let us define two distinguished propability vectors which corre- ~ ∗ 1 In the case of quantum states Sq is defined as a function spond to extreme cases: λ∗ with λi = N (i = 1,...,N) 6 describes the maximal random event, whereas ~λs with or equivalently for the other considered quantities. The λ(s) = δ with j [1,...,N] describes an event with a expression ~λ ξ~ means that ~λ is majorized by ξ~, i.e. i i,j ≺ certain result. For∈ a bipartite quantum system a vector the components of both vectors listed in increasing order j j of Schmidt coefficients given by ~λ represents a maxi- satisfy λi ξi for 0 < j N. In order to be ∗ i=1 ≤ i=1 ≤ mally entangled state, whereas ~λ describes a separable Schur concave, ∆S (~λ) has first to be invariant under s P PW state. exchange of any two arguments, which is obviously the The R´enyi subentropy Qq(~λ) has the following proper- case, and second it has to satisfy [28] ties ∂∆SW ∂∆SW ~ (λ1 λ2) 0 . (30) i.) Qq(λs)=0 for any q. − ∂λ − ∂λ ≤ 1 2 max ii.) Qq takes its maximal value Qq = For convenience we will first consider µq,N . Once we 1 Γ(q+N) 1 ~ know about Schur convexity or Schur concavity of µq,N 1−q ln Γ(q+1)N! N q for λ∗. For q 1 one has max → one can easily deduce Schur concavity of all the other Qq = CN ln N. − discussed quantities. The quantities µq,N can be ex- pressed by the following representation (see eq. B1) iii.) As m(1) = 1 one immediately has lim Q (~λ)=0 0 q→0 q µ = Γ(q+N) dx(~λ~x)q, where dx is a short hand no- for any vector λ. q,N Γ(q+1) ∆ tation for dx . . . dx and the integration is performed 1 R N iv.) For q 1 one obtains the regular subentropy, over the simplex ∆ containing all propability vectors ~x → limq→1 Qq(~λ)= Q(~λ). with i xi = 1. Making use of this relation and assuming λ1 > λ2 without loss of generality, we obtain P v.) For q one gets limq→∞ Qq(~λ) = ln λmax, → ∞ − ∂µq,N ∂µq,N Γ(q + N) q−1 where λmax is the largest entry of λ. Hence this = dx (~λ~x) (x1 x2) . limit coincides with the limit q of the R´enyi ∂λ1 − ∂λ2 Γ(q + 1) ∆ − → ∞ Z (31) entropy S (~λ). q The full measure of the propability simplex ∆ can be divided into two symmetric parts, the first part specified vi.) Qq is expansibile, i.e. it does not vary if by x < x , the second one by x > x . After exchanging the probability vector ~λ is extended by zero, 1 2 1 2 the labels x and x in the second part, the first and the Q (λ ,...,λ )= Q (λ ,...,λ , 0). 1 2 q 1 N q 1 N second part coincide and one gets A discrete interpolation between the subentropy Q and ∂µq,N ∂µq,N Γ(q + N) the Shannon entropy S was recently proposed in [25]. = dx(x x )(λ λ ) ∂λ − ∂λ Γ(q + 1) 1 − 2 1 − 2 Note that a continuous interpolation between these quan- 1 2 Z∆> tities may be obtained by the R´enyi subentropy Qq for q q−1 q−1 λ x +λ x + λ x λ x +λ x + λ x (32). increasing from unity to infinity combined with the R´enyi 1 1 2 2 i i − 1 2 2 1 i i i>2 i>2 ! entropy Sq for q decreasing from infinity to unity. It is X X well known [26] that the R´enyi entropy S is both a non- q Since x1 > x2 and λ1 > λ2, one has x1λ1 +x2λ2 > x2λ1 + increasing function of q and convex with respect to q. x1λ2. Therfore for q > 1 the integrand is non-negative We conjecture that the R´enyi subentropy has opposite and so is the integral. For 0 ∂µq,N VI. SCHUR CONCAVITY As ∆S can be expressed as ∆S = lim → , W W − q 1 ∂q Schur concavity of the subentropy can directly inferred So far we defined rescaled moments, R´enyi entropy and from the corresponding properties of µq,N . Also the R´enyi subentropy. All these quantities are constructed in rescaled moments (22) are Schur concave for all values such a way that they are vanishing exactly for separable of q, as 1 q is negative for q > 1 where µq,N is Schur − states. In order to legitimately use these quantities as en- convex. tanglement monotones, we still have to show, that they For the R´enyi subentropy Qq with q = 1 we have 6 are non increasing under local operations and classical ∂Q ∂Q communication, or analogously that they are Schur con- (λ λ ) q q = 1 − 2 ∂λ − ∂λ cave [20, 27], i.e. 1 2 1 1 ∂µq,N ∂µq,N ~ ~ ~ ~ (λ1 λ2) . (34) λ ξ ∆SW (λ) ∆SW (ξ) , (29) 1 q Q − ∂λ − ∂λ ≺ ⇒ ≥ − q 1 2 7 d) Calculate the Wehrl entropy for the analyzed state, S q a) S q b) S( ψ ) 1 1 | i e) Compute the entropy excess ∆S = S( ψ ) S , | i − min 0.5 0.5 which characterizes quantitatively to what extend the an- Q Q q q alyzed state ψ is not ’coherent’. | i 0 0 0 2 4 6 0 2 4 6 For instance, analyzing the space of pure states of size q q N we may distinguish the spin SU(2) coherent states, or, in general the SU(K) coherent states (with K States Mixed Pure Pure Pure Pure Systems simple simple simple bipartite multipartite Coherent SU(N) CS SU(2) CS SU(K) CS SU(N) ⊗ SU(N) CS SU(N)⊗M CS states N−1 1 K−1 ⊗2 N2−1 ⊗M NM−1 |αN i ∈ CP |α2i ∈ CP |αK i ∈ CP |αN i ∈ CP |αN i ∈ CP Minimal Wehrl entropy Smin = S(|αN i) Smin = S(|α2i) Smin = S(|αK i) Smin = 2S(|αN i) Smin = MS(|αN i) N−1 N−1 N−1 ×2 N−1 ×M Generic states ρ : HN → HN |ψi ∈ CP |ψi ∈ CP |Ψi ∈ (CP ) |Ψi ∈ (CP ) Wehrl entropy S(ρ) − CN S(|ψi) − C2 S(|ψi) − CK S(|Ψi) − 2CN S(|Ψi) − MCN excess ∆S measures non SU(2) non SU(K) bi-partite non-M-partite mixing degree of coherence coherence entanglement separability TABLE I: Wehrl entropy excess in various set-ups. 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; the distance of a given pure state of a composite system from the subset of all separable (product) states. VIII. ACKNOWLEDGMENT with f = dµ(α) α ν µ α α ν µ α ... α ν µ α We are indebted to Andreas Buchleitner, Marek Ku´s, q h | 1ih 1| ih | 2ih 2| i h | q ih q | i Bernard Lavenda and Prot Pako´nski for fruitful discus- Z (A4) sions, comments and remarks and acknowledge fruitful The integers and count how often the states ν Ni Mi | ii correspondence with Ayumu Sugita. Financial support and µi appear in fq. Note that one has to integrate by Volkswagen Stiftung and a research grant by Komitet separatelyh | over both sub-systems. Due to the symmetry Bada´nNarodowych is gratefully acknowledged. of the Schmidt decomposition with respect to the two subsystems, both integrations lead to the same result and it is just the square of the integral over fq that is entering APPENDIX A: MOMENTS OF THE HUSIMI mq. Using the parametrization (A1) fq can be expressed FUNCTION as N−1 x(max) 2π N! i As already mentioned, every pure state belonging to an f = dx dϕ ei(Ni−Mi)ϕi q (2π)N−1 i i N–dimensional Hilbert space N is a SU(N) coherent i=1 0 0 H Y Z Z state and vice versa. Therefore, following (3), we can N−1 1 N− (Ni+Mi) q− 1 1 N +M parametrize all SU(N) coherent states by x 2 (1 x ) 2 χ=1 χ χ ,(A5) i − j j=1 P − − N 1 1/2 N 1 X iϕ α = 1 x 0 + √x e i i , (A1) where the integrations over xi are performed in increasing | i − i | i i | i i=1 i=1 order of i and the upper integration limit is given by X X (max) N−1 x = 1 xj . It is useful to perform the ϕi N−1 i − j=i+1 with 0 xi 1 j=i+1 xj and dµ = integrations first. They lead to δNi,Mi terms, so that one N≤−1 ≤ − P N! dx dϕ . The states i , i =0,...,N 1 gets (2π)N−1 i=1 i i P | i − N−1 (max) form an orthonormal basis, while 0 is the reference xi Q | i Ni state. The Husimi function H of a pure state Ψ fq = N! dxi x δN ,M Ψ | i i i i (2) i=1 Z0 ! with respect to SU(N) SU(N)-coherent states αN = Y α α reads ⊗ | i N−1 N A N B q− N−1 N | i ⊗ | i (1 x ) χ=1 χ . (A6) − j N j=1 P X HΨ = λν λµ αN ν A A µ αN A Ah | i h | i In order to perform the integrations over xi we need to ν,µ=1 X p define an auxiliary function B αN ν B B µ αN B , (A2) N−1 (max) N−1 h | i h | i x˜i (β) N q− N−1 N g (β,N)= dx x i (β x ) χ=1 χ , where Ψ is represented in its Schmidt basis, Ψ = q i i − j | i | i i=1 Z0 ! j=1 P ν √λν ν A ν B . The q-th moment is than given by Y X | i ⊗ | i (A7) P N withx ˜(max)(β)= β N−1 x . In the following we will i − j=i+1 j mq = λν1 ...λνq λµ1 ...λµq show that for any integer q ν ...ν =1 P 1 q N−1 N−1 µ1...µXq =1 q (q+N−1) ( i=1 i!)(q i=1 i)! 2 gq(β,N)= β N − N (A8) fq ( 1, ..., N , 1, ..., N ) , (A3) (q + N 1)! N N M M Q −P 9 holds. According to the definition (A7), gq(β,N) satisfies To end, we will show by induction that for any positive the following relation integer q the quantity µq,N can be written as β N−2 NN−1 q q−ν1 q− i=1 Ni gq(β,N)= dxN−1 x gq−N (β xN−1,N 1) N−1 N−1 N−1 N1 N2 NN−1 q− i=1 Ni 0 − − µq,N = ...P λ1 λ2 ...λN−1 λN . Z P (A9) N N N X1=0 X2=0 NX−1=0 Making use of (A17) β As a starting point for this line of reasoning we are going N −N g (β, 1) = dx x 1 (β x )q 1 to demonstrate that µ−1,N = 0. For N = 2 this equality q 1 1 − 1 Z0 can be checked by direct calculation. For N > 2 we have q+1 1!(q 1)! = β N − N , (A10) N (q + 1)! λN−3 λ µ = i i + −1,N (λ λ ) λ λ 6 k=1,k=6 i,k=6 j i k i j we immediately have (A8) for N =2 i=1X,i=j − − N−2 β λQj N1 q−N1 . (A18) gq(β, 2) = dx1 x (β x1) . (A11) (λ λ ) 1 − k=1,k=6 j j k Z0 − λi Q λα Assuming that (A8) holds true for gq(β,N 1) and mak- Since = 1+ , we can make use of − λi−λj λi−λj ing use of (A9) one gets λN−3 the following relation N i = i=1,i=6 j k=1,k=6 i,k=6 j (λi−λk) β NN−1 q+N−2−NN− g (β,N) = dx − x (β x − ) 1 µq,N−1(λ1,...,λj−1, λj+1,...,λN ), in which the right q 0 N 1 N−1 − N 1 P Q ( N−2 N !)(q−N − N−2 N )! hand side vanishes by assumption. Therefore we end up R i=1 i N−1 i=1 i . (A12) (q+N−2−NN− )! with Q P1 Using further (A10) one has N N−3 λi µ−1,N = λj N . (A19) q+N−1 N−1!(q + N 2 N−1)! i=1 k=1,k=6 i(λi λk) gq(β,N) = β N − − N X − (q + N 1)! − We still have the freedomQ to choose the index j. As the ( N−2 !)(q N−1 )! i=1 Ni − i=1 Ni . (A13) result does not depend on the choice of j, both sides of (q + N 2 − )! the equality have to be zero. Now we can come back to Q − −P NN 1 the proof of (A17). For N = 2 it is just a straight forward Finally one ends up with calculation to show that N−1 N−1 ( !)(q )! q q+1 q+1 q+N−1 i=1 i i=1 i λ λ gq(β,N)= β N − N , λν λq−ν = 1 + 2 = µ , (A20) (q + N 1)! 1 2 λ λ λ λ q,N Q −P ν=0 1 2 2 1 (A14) X − − i.e. (A8) also holds for g (β,N). Finally for β = 1, we q which can be checked by multiplying both sides with λ get 1 λ . Assuming that (A17) is true for N 1 we get − 2 − N−1 N−1 ( !) q ! N−1 q i=1 Ni − i=1 Ni fq = N! δN ,M . ν (q +N 1)! j j mq,N = µq−ν,N−1λN . (A21) Q P j=1 − Y ν=0 (A15) X Now we have m = N λ ...λ f 2, where the Making use of the assumption (A17) one obtains q ν1...νq =1 ν1 νq q δnν ,mν -terms assure that there are only integer powers − P N 1 λN−2 λq+1 λq+1 of the λνi . Since fq depends only on the integer pow- i i N mq,N = N−1 − ers i of the λν , we need to count how many terms λ λ i i=1 j=1,j=6 i(λi λj ) i N withN fixed powers occur. Using simple combinatorics one X − − N 2 Q λq+N−1 q! = i can see that there are N−1 N−1 terms N (Ni)! q− Ni ! − ( i=1 )( i=1 ) i=1 j=1,j=6 i(λi λj ) N−1 − N N NN−1 q− Ni X Q 1 2 P i=1 N N−2 q+1 q+1 containing the expression λ1 λ2 ...λN−1 λN . Q λ λ λ P q+1 i i N Thus finally we get λN N − . (A22) − (λ λ ) λi λN i=1 j=1,j=6 i i j − N−2 X − 2 q q−ν1 q− i=1 Ni N!Γ(q + 1) Q m = ... P Now one just has to make use of µ−1,N = 0 and one im- q Γ(q + N) N N N mediately gets that the assumption (A17) holds true for X1=0 X2=0 NX−1=0 N−1 N. This result together with (A16) completes the proof N1 N2 NN−1 q− i=1 Ni λ1 λ2 ...λN−1 λN . (A16) of Eq. (8) for integer positive integers q. Since integers P 10 are dense at , we conclude that there is a unique gen- where we were using the assumtion for h − (a + ∞ q,N 1 eralisation to real q. Thus eq (8) is also valid for real xn−1,b + xn−1λn−1. Performing the integration one gets q. Eq (18) for the moments of the Husimi function of a monopartite system differs by a proportionality constant N−2 q+n−1 Γ(q + N) (1 a)λi + b only and its proof is analogous. h = − + q,N Γ(q + 1) (λ λ ) i=1 j=6 i i j X − N−2 q+n−1 APPENDIX B: INTEGRAL REPRESENTATION (1 a)QλN−1 + b OF THE MOMENTS − (B5) (λ − λ ) (λ λ ) i=1 N 1 i j=6 i i j ! X − − In order to prove that Q We are done, if we manage to show that Γ(q + N) q µq,N = dx(~λ~x) , (B1) Γ(q + 1) N−2 Z∆ 1 = where the integration is performed over the propability N−2 i=1 (λN−1 λi) j=6 i (λi λj ) simplex ∆, we consider the function hq,N (a,b) of two real X − − Q 1 variables a and b defined as N−1 , (B6) j=1 (λN−1 λj ) 1−a 1−a− xi − Γ(q + N) i>1 hq,N = dxn−1 ... dx1 Q Γ(q + 1) P Z0 Z0 or eqivalently n−1 n−1 q ( xiλi + 1 a xi)λn + b . (B2) − − N−2 N−2 i=1 i=1 λ − λ X X p = N 1 − j =1 . (B7) λi λj Now we are going to show by induction that h can be i=1 j=6 i X Y − expressed as This is a polynomial of (N 2)-nd order in λN−1. 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