Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states

We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with $d$ levels each. It can be described by the R\'enyi-Ingarden-Urbanik entropy $S_q$ of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case $q=0$ this quantity becomes a function of the rank of the tensor representing the state, while in the limit $q \to \infty$ the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system the entropy $S_1$ coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three and four-qubit systems. In the former case the distributions of $3$-tangle is studied and some of its moments are evaluated, while in the latter case we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.


Introduction
After more than twenty years of intensive research entanglement of the pure states of bipartite quantum systems is rather well understood [1,2] for two subsystems of an arbitrary dimension d. In this case any pure state can be represented in a product basis by a matrix of coefficients of order d, and its standard singular values decomposition allows one to reveal entanglement.
On the other hand, in the case of systems composed of n ≥ 3 subsystems the problem becomes much more complicated, as the state is represented by a tensor of size d and n dimensions. Even though several important results were obtained, especially in the case of three [3][4][5] and four qubits [6][7][8][9][10], and several measures of entanglement in such systems were proposed [11][12][13][14], it is fair to say that the complete understanding of the phenomenon of entanglement in multipartite systems is still missing.
To characterize entanglement of a quantum state of a bi-partite system it is natural to analyze the degree of mixing of the reduced density matrix. For instance, making use of von Neumann entropy, one arrives at one of the most often used measures: the entropy of entanglement [15]. It is often convenient to apply for this purpose also the generalized entropy of Rényi [2] or some other kinds of entropy.
The aim of the present work is to propose a possible generalization of this quantity for the case of multipartite systems, for which a pure state is represented by a tensor. Furthermore, we would like to make a connection with the geometric measure of entanglement, which depends on the distance of the state considered to the closest separable state [16][17][18].
Following the papers of Parker and Rijmen [19] and Bravi [20] we suggest to analyze the entropy of decomposition of a quantum state in a product basis, sometimes called Ingarden -Urbanik entropy [21], minimized over all local unitaries. This quantity can be generalized in the sense of Rényi. Interestingly, to establish a direct link with the geometric measure of entanglement [16,18] it is sufficient to consider the Rényi -Ingarden -Urbanik (RIU) entropy and send the Renyi parameter q to infinity.
Even though the approach advocated here is applicable for arbitrary composite quantum systems, for concreteness we concentrate the majority of this work for the case of three and four qubits. It is demonstrated that there is no a single pure state for which the minimal RIU entropy is the largest for all values of the Renyi parameter q. Investigating the problem for selected values of q we identify certain pure states, which are conjectured to maximize this particular measure of entanglement.
Furthermore, we make also use of a statistical approach to analyze the distribution of the hyperdeterminant and minimal RIU entropy for random quantum pure states. They are distributed with respect to the unique unitarily invariant Haar measure on the space of pure quantum states, the complex projective space, CP N −1 , where the total dimension of the complex Hilbert space is N = d n . Analyzing systems composed of three subsystems of an arbitrary dimension d we obtain a bounds for the geometric measure of entanglement for generic states of such a system. This paper is organized as follows. In Section 2 we introduce the RIU entropy for a pure state of a multipartite system, while in Section 3 different techniques of tensor decomposition are reviewed. In Section 4 we present results obtained for three-qubit system, while analogous results for 4 qubits are presented in Section 5. A more general case of three subsystems consisting of an arbitrary number of levels is discussed in Section 6. Computations of the moments of the distribution of 3-tangle and a derivation of the bound for the geometric measure of entnaglement are relegated to the Appendix.

Minimal Rényi-Ingarden-Urbanik entropy
Consider a quantum state describing a system consisting of n subsystems, with d levels each, |ψ ∈ H N = H ⊗n d . Working in an arbitrary product basis one can represent such a state by a n-index tensor, The standard normalization condition, ψ|ψ = 1, implies that It will be convenient to introduce a multi-index µ = (i 1 , i 2 , . . . , i n ), where µ can be identified with the set {1, . . . , N = n d }, and use a shorter notation p µ = |C µ | 2 = |C i1,i2,...,in | 2 . Hence p(|ψ ) = (p 1 , . . . p N ) represents an N -point probability vector p, which can be characterized by the Rényi entropy S q ( p) = 1 1−q log N µ=1 p q µ . For q → 1 this quantity reduces to the standard Shannon entropy S(p) = − N µ=1 p µ log p µ , which in the context of the decomposition of the state |ψ , is called the Ingarden-Urbanik entropy [21][22][23] and written S IU (|ψ ) = S p(|ψ ) . In will be convenient to use natural logarithms throughout this paper, written log 2 ≈ 0.693.
The product basis |i 1 ⊗ |i 2 ⊗ · · · ⊗ |i n is determined up to a local unitary transformation, U loc = V 1 ⊗ V 2 ⊗ · · · ⊗ V n , where an unitary matrix V j acts on the j-th subsystem. As the Ingarden-Urbanik entropy of the decomposition depends explicitly on the choice of the product basis it is natural to analyze the optimal value, minimized over the set of local unitaries [22,23].
We shall study a more general case of the Rényi entropies S q with Rényi parameter q ≥ 0. For any multipartite state |ψ ∈ H ⊗n d we define the minimal Rényi-Ingarden-Urbanik (RIU) entropy, where the minimum is taken over entire set of local unitary transformations. Proposition 1. For any N qudit state, |ψ ∈ H ⊗n d , its minimal RIU entropy is bounded from above, This statement follows directly from the work of Carteret, Higuchi and Sudbery [22], who showed that performing a local unitary transformation U loc = U 1 ⊗ · · · ⊗ U n , one can always find a product basis such that the decomposition (1) contains no more than R max terms. A suitable choice of n unitary matrices of size d allows one to bring d(d − 1)/2 elements of the tensor C to zero. Therefore out of all its d n entries at least nd(d − 1)/2 can be always set to zero. This fact can be also formulated as the following statement Proposition 2. For any n qudit state, |ψ ∈ H ⊗n d , represented by a tensor C as in Eq. (1), its tensor rank R is bounded by R max = d n − nd(d − 1)/2, where R is the minimal number r of terms in its decomposition involving arbitrary coefficients f ν , Observe that in the case of the Rényi entropy of order zero, q = 0, one has S RIU 0 (ψ) = log R where R is the tensor rank of the tensor C(ψ). The quantity log R is known as the Schmidt measure characterizing entanglement of the multipartite pure state [24,25], while the rank R was used to determine probabilistic conversion of three qubit pure state [26]. Since the Rényi entropy is in general a non-increasing function of the parameter q, one gets the general bound For instance, in the case of a three-qubit system, (n = 3 and d = 2), the bound (4) gives R max = 8 − 3 = 5 in agreement to the five-term standard form of a three-qubit pure state by Acin et al [3].
Besides the case a) q = 0, corresponding to the tensor rank of |ψ , we shall also consider some other particular cases of the definition (3). b) q = 1. The minimal IU entropy S IU 1 (|ψ ) determines the minimal information gained by environment after performing a projective von-Neumann measurement of the pure state |ψ ψ| in an arbitrary product basis [20]. c) q = 2. The minimal decomposition entropy S RIU 2 (|ψ ) characterizes the maximal purity µ p 2 µ = e −S2 of the probability vector p µ associated to the outcomes of a projective measurement in a product basis and is accessible in a coincidence experiment with two copies of the multipartite state |ψ . This quantity was used by Parker and Rijmen [19] to analyze multipartite entanglement in the context of coding theory. d) q = ∞. In the limiting case q → ∞ the minimal RIU entropy gives S RIU ∞ (|ψ ) = − log λ max , where the largest component of the vector p µ reads λ max = max | ψ|χ sep | 2 . The maximum is taken over the set of all separable states, |χ sep = U loc |0 · · · 0 , so λ max is a decreasing function of the Fubini-Study distance to the closest separable state [16,17], D FS = arccos( √ λ max ), and its function called the geometric measure of entanglement [18], In the case of bipartite systems (n = 2) one can use the standard singular value decomposition of the matrix C to show that the probability vector p µ coincides with the vector of Schmidt coefficients λ j determining the Schmidt decomposition [2], |ψ = d j=1

This observation leads to
Proposition 3. For any bipartite state, |ψ ∈ H d ⊗ H d , its minimal RIU entropy coincides with the Rényi entropy of entanglement, so in the case q = 1 one arrives at the standard entropy of entanglement 3 Tensor decompositions Singular value decomposition (SVD) of the matrix containing the expansion coefficients of a biparite pure state plays a crucial role in evaluating the minimal RIU entropy. However, in the case of multipartite systems, and quantum states described by higher order tensors, this decomposition is not directly applicable.
In this section we consider two generalizations of SVD, the higher order singular value decomposition (HO-SVD) [27] and the Parallel Factor model (PARAFAC) [28]. Both decompositions were developed in the framework of Principal Component Analysis [29] and have found several applications in signal processing, numerical linear algebra, graph analysis, and numerical analysis [30]. Throughout this section we will consider such schemes to evaluate some bounds for the minimal RIU entropy and the geometric measure of entanglement.
Let us define a tensor C as a multidimensional or n-way array of numbers, so it can be identified with an element of C d1d2...dn . Accordingly, a matrix and a vector are a 2-way and 1-way tensors, respectively. Note that such a space is linear. Given two tensors A, B ∈ C d1d2...dn their inner product is inherited form the linear space C d1d2...dn and defined as follows where the over-bar denotes the complex conjugation. The corresponding induced norm is the Frobenius norm [27,30], written A := A, A . It implies the Frobenius distance, d(A, B) := A − B . Note that that the coefficients of the state (1) can be arranged in a tensor C ∈ C d n and conversly, a given tensor in such a space defines a certain pure state |ψ in the Hilbert space H ⊗n d .

Higher Order Singular Value Decomposition
Let C ∈ C d1d2...dn be an n-way tensor. We define the k-th unfolding C (k) as the matrix of size The k-mode product of a matrix U (k) ∈ C d k ×d k with C is defined (element-wise) as The higher order singular value decomposition allows one to construct a tensor A, called co-tensor [30], of the same dimension than C such that where each U (k) acts according to (9) and any two sub-tensors A i k =p and A i =q , with p and q fixed, are orthonormal The numbers σ (k) p are called the k-mode singular values of C, they are non-negative and fulfill σ for all p < q. Such a decomposition is accomplished by taking each U (k) in (10) as the matrix of left singular vectors of C (n) . Note that finding the SVD of the k-th unfolding of the coefficients tensor of (1) is equivalent to diagonalize the reduced density matrix of the k-th party of the system. Moreover, according to equation (10) the co-tensor A of C defines a state |ψ HO−SVD that is LU equivalent to (1). In this manner, Liu et al have recently proposed an entanglement classification based on the study of HO-SVD and the local symmetries of the multipartite states [31].

Parallel Factor decomposition
The idea of expressing a tensor as a sum of rank-one tensors was applied in several contexts. For instance, in psychometrics it is known as canonical decomposition (CANDECOMP), while in the brain imaging analysis it is referred to as parallel factor decomposition (PARAFAC) -see [28] and references therein. For other applications of this decomposition see also [30]. The PARAFAC is a decomposition of n-way tensor C ∈ C d1d2...dn into a sum of rank-one tensors, where U ( ) i2=k denotes the k-th column of a matrix of size d × R, with = 1, . . . , n and λ is a vector of size R. The symbol • represents the dyadic product of vectors [32]. In terms of the components, the PARAFAC decomposition reads In practice, we are interested in an approximation given by sum of r rank-one tensors Usually a least squares approximation (12) is concern, in which case one has to minimize the quantity d P = C − C PARAFAC . Since the problem is not linear, usually we obtain an approximate solution only, accomplished through the alternating least squares (ALS) algorithm. In this work we use the algorithm provided by Nion and De Lathauwer [33] to compute the PARAFAC decomposition of a 3-way complex tensor. In the case of r = 1 the state is fully separable. Hence, the quantities d P and λ p = | ψ|ψ P | 2 are bounds for the geometric measure of entanglement E G and the separable state maximum overlap λ max , respectively.

Three qubits
In this section we analyze pure states of a system consisting of three qubits: n = 3 and d = 2. The exact values of the minimal RIU entropy are found for the states W and GHZ for particular values of the Rényi parameter q. Generic properties of the minimal RIU entropy are also discussed. Furthermore, some moments of the 3-tangle τ are computed and the corresponding distribution is analyzed.

Minimal decomposition entropy
In the case of a three-qubit system any pure state |ψ ∈ H ⊗3 2 = H 8 can be represented in the five-terms decomposition of Acin et al. [3], where 5 i=1 |a i | 2 = 1 and four coefficients can be chosen to be real. Observe that selecting a 1 = a 5 = 1/ √ 2 and neglecting others, one obtains the state |GHZ = (|000 +|111 )/ √ 2, while setting a 2 = a 3 = a 4 = 1/ √ 3 one has |W = (|001 + |010 + |100 )/ √ 3. As the number of terms in these states cannot be reduced by any local unitary transformation their tensor ranks are equal to two and three, respectively, so that S RIU 0 (GHZ) = log 2 and S RIU 0 (W ) = log 3. It is possible to write a state |A 5 with all coefficients equal, a i = 1/ √ 5 for which S RIU 0 (A 5 ) = log 5. In the same form, we define |A 4 = (|000 + |010 For any general value of the Rényi parameter q, we are not aware of any constructive procedure which gives the exact value of minimal RIU entropy (3). However, for permutation invariant states we can follow the general scheme of calculating the maximum overlap with the closest separable state. For such states it was first conjectured and later proven that in order to obtain the maximal overlap it is enough to take the product state to be a tensor product of the same single-party real state [34][35][36]. In this spirit, to minimize the minimal RIU entropy for permutation invariant states the product of local unitary matrices U loc in (3) will be taken as This task can be done easily in some special cases, i.e. S RIU q (GHZ) = log 2 for any q and S RIU 1 (W) = log 3. To evaluate the expression (3) for an arbitrary state, we perform a random walk over the space of unitary matrices. Fig. 1 shows the minimal RIU entropy obtained numerically for some particular states and different values of q. In the case of the state |W and |A 5 one can compare the value obtained by this procedure with the analytical results. In the following we are concerned with typical properties of the minimum value of RIU entropy. First, we compute the value of the Rényi entropy of a random state, then the same quantity is evaluated in the corresponding co-tensor and finally we compute the S RIU q by performing the random walk procedure described above. As we pointed before, we are mainly interested in the cases q = 1, q = 2 and q = 100. The latter value serves as an approximation for the limining case S RIU ∞ . The probability distributions of estimations of the minimal RIU entropy for random states are presented in Fig. 2. The mean, second moment and standard deviation for these distributions are reported in Table 1. In the case of q = 1 some analytical results are available [37,38] where N is the size of the system and Ψ(x) denotes the digamma function and Ψ (x) its derivative. In our case we set N = 8 to obtain S 1 = 1.718 and ∆S 1 = 0.160, so our numeric calculations are in good agreement with these analytical predictions. We compute the maximum overlap of the state |ψ with the closest separable state by performing a random walk in the space of unitary matrices, this quantity is denoted as λ LU max . The PARAFAC decomposition of |ψ yields a bound λ PARAFAC max for this overlap. A comparison between these three distributions is presented in Fig. 3.  Table 1: Mean value, second moment and standard deviation of the Renyi entropy of the probability vector corresponding to a three-qubit random pure state (left), its corresponding co-tensor (center) and the minimal RIU entropy for q = 1, 2, 100.
For any value of the Rényi parameter q ≥ 0 one can ask for a state |Φ max q , for which its minimal entropy S RIU q achieves its maximum value. It is known that in the case of three qubits the maximal entangled state with respect the geometric measure is the state |W [39,40], and hence the largest value of the RIU entropy reads S RIU ∞ (W ) = − log λ max (W ) = − log 4/9 ≈ 0.811. We found numerically states for which S RIU

Distribution of 3-tangle
The residual entanglement or 3-tangle τ was introduced by Coffman et al. [42]. It quantifies the genuine entanglement of a system of three qubits A, B and C in the following sense. Let C A,B and C A,C denote the concurrences [41] of the density matrices of the pairs of qubits A, B and A, C, respectively. Since the qubits BC can be regarded as a single subsystem, we can ask for the entanglement between A and BC. Such a quantity will be denoted as C A,BC and it is equal to 2 √ det ρ A [42], where ρ A = tr BC ρ ABC is the reduced density matrix of the qubit A when the partial trace respect to B and C has been performed. The following inequality holds and using it we define the 3-tangle as τ = C 2 A,BC − C 2 A,B − C 2 A,C . This quantity measures the amount of entanglement between the qubit A and the subsystem BC that is not related to the entanglement in the pairs A, B and A, C. The 3-tangle τ is invariant under the permutation of sub-systems and vanishes on all states that are separable under any cut. The 3-tangle for the state (1) with n = 3 and d = 2 is given by where is the Cayley hyperdeterminant of the tensor C. Here i1,i2 stands for the Levi-Civita tensor of rank 2 and the sum is performed over all indexes. The 3-tangle is bounded by 0 and 1. For the |GHZ state the residual entanglement attains its maximum value. Indeed, the pairwise concurrences C A,B and C A,C vanish and τ = C A,BC = 1, so this state is referred as a genuinely three-partite entangled state. The hyperdeterminant Det 3 (C) is a homogeneous polynomial function of degree 4, first introduced by Caley [43]. It is invariant under the action of the group SL(2, C) ⊗3 [44]. This notion of invariance plays an important role in the construction of entanglement monotones for pure states of multiqubit systems [45]. Moreover, a classification of multipartite entangled states has been accomplished by analyzing singularities of the hyperdeterminant [46]. Now we will discuss properties of typical quantum states of a three-qubit system. Kendon et al. [47] studied the distribution of 3-tangle for an ensemble of random pure states drawn according to the Haar measure. By direct integration of equation (21) with respect the unitary invariant measure they obtained the average value, τ = 1 3 . In Fig. 4 we show the probability densities of τ and τ 2 evaluated over a sample of 10 5 random pure states. Moreover, the first six even moments of τ can be calculated by symbolic integration using the Beta integral (see Appendix A). The distribution of 3-tangle can be approximated by the Beta distribution Beta(α, β; τ ), using the moment method to estimate the parameters α = τ By virtue of the chain rule we get an approximated distribution function for τ 2 The above distributions are presented in Fig. 4 together with the estimation obtained by numerical simulations. In order to show the accuracy of our approximation, we show in Table 2 a comparison between the first 6 even moments of the distribution (23) and the moments of P (τ ) obtained by symbolic integration.
k k-th moment of P B (τ ) τ k , k-th moment of P (τ )

Four qubits
In this section we compute both the minimal RIU entropy and the hyperdeterminant for several exemplary 4-qubit states, we also analyze some typical properties of these quantities.

Minimal decomposition entropy
According to Carteret et al [22], a given state |Φ ∈ H ⊗4 2 = H 16 can be written using twelve terms Note that |W = |D(n, 1) , | W = |D(n, n − 1) and |GHZ n = (|D(n, 0) + |D(n, n) )/ √ 2. In computations below we will consider the case of n = 4. The calculation of the minimal RIU entropy for permutation invariant states can be turned into a one-variable optimization by taking U loc = U (p) ⊗4 . We found S RIU q (GHZ 4 ) = log 2, for all q and S RIU 1 (D(4, 1)) = log 4; for the other values of q minimization will be accomplished numerically. On the other hand, the optimal decomposition of the state |D(4, 2) is obtained by taking U loc = U (1/2) ⊗4 . Accordingly, it is possible to get a compact formula for its minimal RIU entropy hence we get S RIU 1 (D(4, 3)) = log(8/ √ 3) by taking the limit q → 1. On the other hand, when q → ∞ one arrives to S RIU ∞ (D(4, 2)) = − log(3/8), which is consistent with the well known result of the overlap of |D(4, 2) with the closest separable state [34]. Note that by symmetry S RIU q (D(4, 3)) = S RIU q (D (4, 1)). In Fig. 5 we compare the results obtained in this way with those acquired by performing a random walk over the space of unitary matrices for several values of q.
Consider also some other exemplary 4-qubit states. The hyperdeterminant state maximizes the 4-qubit hyperdeterminant [48]. The minimal RIU entropy for the former state can be computed in the same way as for the Dicke state (27), as it is shown [34][35][36] that the state (28) presents the optimal decomposition minimizing the RIU entropy for any q. Then we obtain S RIU 1 (HD) = 4 3 log 3, S RIU q (HD) = 1 1 − q log 6 q 4 + 4 q , for q = 1. (29) In the limit q → ∞ we get S RIU ∞ (HS) = − log(2/3) which is consistent with the maximum overlap with the closest separable states computed according to the Ref. [34]. On the other hand, the cluster states identified by Gour and Wallach [8] |C 1 = 1 2 (|0000 + |0011 + |1100 − |1111 ) , are the 4-qubit states that maximize the Rényi α-entropy of partial trace for α ≥ 2. They also found that the state |L = 1 maximizes the average Tsallis α-entropy of the partial trace for all α > 2. The minimal RIU entropy of the former states is shown in Fig. 5 for several values of q. Based on our numerical calculations for the cluster states we conjecture that S RIU q (C k ) = log 4 with k = 1, 2, 3 for any value of the Rényi parameter. We also consider the 4-qubit state found by Higuchi and Sudbery [49] where w = exp(2iπ/3), which has maximum average von Neumann entropy of partial traces averaged over all possible 3 splittings of 4 qubit system into two bipartite systems. Numerical values of S RIU q (HS) are shown in Fig. 5. We are also interested in the typical properties of the minimum of S RIU q for four qubits. For an ensemble of 10 5 random pure states we compute S q (|ψ ), S q (|ψ HOSVD ) and S RIU q (|ψ ) taking q = 1, 2, 100. The corresponding distributions are shown in Fig. 5 Table 3. In the first case and with q = 1 we can we analyzed the entropy with the Renyi parameter q = 100. In Fig. 7 we show the distributions of the minimal RIU entropy S RIU 100 and S max = − log λ LU , where λ LU max is the maximum overlap with the closest separable state computed through a random walk in the space of unitary matrices.  Table 3: Mean value, second moment and standard deviation of the Rényi entropy of a four-qubit random pure state (left), its corresponding co-tensor (center) and the minimal of the RIU entropy for q = 1, 2, 100.

and in
As in the three qubit case, we are looking for the maximal states of the minimal RIU entropy. For the maximal symmetric state with respect to the geometric measure of entanglement |Φ 4 = 1/3|D(4, 0) + 2/3|D(4, 3) the RIU entropy yields S RIU ∞ (Φ 4 ) = − log(1/3) ≈ 1.099 [40]. Moreover, our numerical results allow us to conjecture that S RIU ∞ (HS) = − log(2/9) ≈ 1.504, is the largest value of the minimal RIU entropy with q = ∞ among the four qubit states.
For q = 1 we found numerically a state |Ψ max Further numerical tests support the conjecture that the state |HS , for which S RIU 2 = log 6, maximizes the minimal RIU entropy for q ≥ 2.

Distribution of the hyperdeterminant |Det 4 |
In the case of 4-qubit states the hyperdeterminant Det 4 is a polynomial of degree 24. It can be constructed following the Schläfli's procedure [46]. In analogy to the three-tangle we consider the following function of the coefficients tensor C of the state |ψ ∈ H 16 , T (|ψ ) = 2 6 3 9 |Det 4 (C)|, For a separable four-qubit state one has T (|ψ sep ) = 0 but it vanishes also for the states |D(1, 4) , |D(2, 4) , |C k , |HS and |GHZ . Alsina and Latorre found recently [48] that both |HD and |L maximize the hyperdeterminant and they also discussed relation between these states for which T (|HD ) = T (|L ) = 1. Furthermore, numerical simulations indicate that they have the same minimal RIU entropy S RIU q . We evaluate the quantity (34) over an ensemble of 10 7 random pure states. The mean and the standard deviation read T = 9.74 × 10 −4 and ∆T = 2.39 × 10 −3 , respectively, while the corresponding distribution is shown in Fig. 8. 6 Three qudits: Asymptotic case.
Statistical properties of random tensors and their asymptotic limit became a subject of an intensive research [50,51]. In this section we study random tenors of dimension three and arbitrary size, which describe generic states of a system composed of three qudits. We analyze the bounds for the geometric measure of entanglement of such states provided by the tensor decompositions HOSVD and PARAFAC. Consider a typical state |ψ ∈ H ⊗3 d drawn from the Haar measure. Let λ max and λ H denote the largest component of the probability vectors p(|ψ ) and p(|ψ HOSVD ), respectively, and consider the overlap λ P = | ψ|ψ P | 2 , where |ψ P is the state (15). In the cases d = 2, 3, we also evaluate the maximum overlap of |ψ with the closest separable state by performing a random walk optimization procedure in the space of unitary matrices. Fig. 9 (a) shows the average value of four quantities as function of the size of the quit d computed over an ensemble of 10 5 random states. The mean value of λ max can be expressed [52] in terms of the harmonic numbers H N , where N = d 3 , in our case. Therefore, for a random pure state of three qudits, the average largest overlap to a pure state scales as d −3 . Performing decompositions for an ensemble of such random tensors we find that for large d the largest overlaps optimized by HOSVD and PARAFAC behave as In both cases, the corresponding bound for the geometric measure of entanglement reads E G ∼ 1 − λ k where k = H, P . Fig. 9 (b) shows the geometric measure of entanglement for a three-qudit system as function of the qudit size. Moreover, based on the scaling of λ P we conjecture that the maximum overlap with respect to the closest separable state for a three-qudit system scales as λ max ∼ d −2 for large d.

Conclusions
We have analyzed the Rényi-Ingarden-Urbanik entropy of pure states of quantum multipartite systems minimized over all local unitary operations. For separable states such a quantity is zero irrespectively of the value of the Rényi parameter q > 0. In general it is not easy to get analytical results for a given state and arbitrary value of the Rényi parameter q. In the special case of permutation invariant states the problem becomes easier, as minimization in the space of local unitary operators can be turned into an optimization of a one-variable function.
We computed the minimal RIU entropy of several representative three and four qubit states for various values of the Rényi parameter. Some particular states, which maximize the minimal RIU entropy for q = 1 and q = 2, were identified. Note that this quantity can be considered as a measure of pure states entanglement in multipartite systems, and for q → ∞ it becomes a function of the geometric measure of entanglement. In the case of three qubits the latter quantity is maximal for the state |W , while for four qubits, it achieves maximum for the state (32) of Higuchi and Sudbery [49]. Furthermore, we analyzed the distribution of the minimal RIU entropy for an ensemble of random for some selected values of the Renyi parameter, q = 1, 2, 100. Our numerical simulations demonstrate usefullness of the PARAFAC algorithm which allows one the find the principal component of a tensor representing a multipartite pure state and estimate its geometric measure of entanglement.
We studied also the distribution of 3-tangle for random pure states of three qubit system and derived a few even moments of this distribution. Numerical results show that this distribution may be approximated by a Beta distribution. In the case of four qubits we analyzed the distribution of the absolute value of the hyperdeterminant of a random pure state.
Finally, the behavior of the maximum overlap of a random state of a system consisting of three subsystems of size d and the closest separable states was investigated in the asymptotic limit. Although the size of the largest component of a random state scales as d −3 , numerical results obtained by the PARAFAC decomposition of a tensor allow us to conjecture that the minimal overlap scales in this case as d −2 . This is consistent with an analytic result based on the Marchenko-Pastur asymptotic distribution of singular values of random matrices, which gives an upper bound d −1 .

Acknowledgments
We are thankful to D. Alsina and J. I. Latorre for fruitful discussions and for sharing their results prior to publication. ME is thankful to the Jagiellonian University for kind hospitality and support during his stay in Cracow. We acknowledge support of the Mexican National Council for Science and Technology (Conacyt) (M.E.) and by the grants number DEC-2011/02/A/ST1/00119 (K.Ż) and DEC-2012/04/S/ST6/00400 (ZP) financed by the Polish National Science Center.

A Moments of 3-tangle τ
In the case of even moments of order 2k of 3-tangle τ for a three qubit tensor it is easy to note, that it is a linear combination of moments of rank 8 k for a random normalized vector. The moments of normalized vectors distributed uniformly can be calculated with a use of Beta integral.
One can also note, that all moments of random vector |ψ , which are not in a form presented above, are equal to zero. This follows, for example, form Collins-Śniady formula [53] for integrals of monomials over unitary matrices distributed with Haar measure. Calculation of the second moment, τ 2 = 8/55, is a simple task, but for higher moments the number of term grows so rapidly, that we have used a package for symbolic computations IntU, which yield moments presented in Table 2. This package allows for exact calculation of polynomial integrals over the unitary group with respect to the Haar measure [54].

B Bound for geometric measure of entanglement for tripartite states
The law of Marchenko-Pastur describes asymptotic behavior of singular values of non-hermitian, rectangular random matrices. Let X be a random matrix X of size N × K with entries given by complex random i.i.d. normal variables with zero mean and variance 1. We define Y = XX † /trXX † and for c > 1 consider a random counting measure on a real line, which counts the number of rescaled eigenvalues of Y which belongs to a given set, i.e.
For a measure defined above, if N, K → ∞ with additional assumption K/N → c, there exist a limiting distribution µ M → µ given by with a ± = (1 ± √ c) 2 .
The above theorem gives us a behaviour of the largest eigenvalue of a matrix Y , which is λ 1 (Y ) ∼ 1 cN (1+ √ c) 2 . In the case, when N, K → ∞ but K/N → ∞, the theorem does not give us the limiting distribution, but form the theorem we will extract, the rate of convergence of the largest eigenvalue.
Consider, the case when K = N 2 , then K/N = N , the direct usage of the Marchenko-Pastur law would give us a behaviour of the largest eigenvalue of Y , i.e. λ 1 (Y ) ∼ 1 N 2 (1 + √ N ) 2 ∼ 1 N . Now we will try to use the above asymptotics to bound the geometric measure of entanglement for a tripartite random states.
Consider a random tensor |ψ ∈ C d ⊗ C d ⊗ C d = H 1 ⊗ H 2 ⊗ H 3 . The geometric measure of entanglement is related to the overlap with the nearest product state, i.e. max |φ ∈sep In the above equation a set sep consist of vectors in a form φ 1 ⊗ φ 2 ⊗ φ 3 for φ i ∈ H i . For distinct i, j, k ∈ {1, 2, 3}, we denote sep i|jk vectors of the form φ 1 ⊗ φ 2 for φ 1 ∈ H i and φ 2 ∈ H j ⊗ H k . We have, that sep ⊂ sep i|jk which gives us max The last maximum above is a square of the largest Schmidt coefficient for vector |ψ .
If we consider, the behaviour of a large random tripartite states the above inequality combined with the relations obtained form the Marchenko-Pastur law we obtain max |φ ∈sep