Separability of symmetric states and vandermonde decomposition

Symmetry is one of the central mysteries of quantum mechanics and plays an essential role in multipartite entanglement. In this paper, we consider the separability problem of quantum states in the symmetric space. We establish the relation between the separability of multiqubit symmetric states and the decomposability of Hermitian positive semidefinite matrices. This relation allows us to exchange concepts and ideas between quantum entanglement and Vandermonde decomposition. As an application, we build a suite of tools to investigate the decomposability and show the power of this relation both in theoretical and numerical aspects. For theoretical results, we establish the witness for the decomposability similar to the entanglement witness and characterize the decomposability of some subclasses of matrices. Furthermore, we provide the necessary conditions for the decomposability. Besides, we suggest a numerical algorithm to check whether a given matrix is decomposable. The numerical examples are tested to show the effectiveness.


Introduction
Entanglement is a fundamental phenomenon in the field of quantum mechanics which was first recognized by Einsteinet al [1] and Schrödinger [2]. It has wide applications in the field of quantum computation and quantum information such as cryptography [3], teleportation [4], and key distribution [5,6]. Without the aid of entanglement, impossible tasks can be achieved.
Therefore, it is of great importance to ask whether a given state is entangled or not, which is called the separability problem [7]. For a d-qudit state acting on the Hilbert space  Ä Ä Ä    d 1 2 , it is said to be separable (or non-entangled) if it can be written as a convex combination of pure product quantum states, i.e., , and each | ( ) ñ x i j is the pure state in the subspace  i . Otherwise, ρ is said to be entangled.
Although it is NP-hard [8,9] to find an operational criterion to test the separability in general, several partial works have been done over the past decades [7] including the necessary conditions (not sufficient) for separability and the characterization of some sub-classes of states. For example, the well-known Peres-Horodecki criterion (Positive Partial Transposed (PPT) criterion) tells that a separable quantum state is necessarily PPT. Furthermore, the PPT criterion is proved to be sufficient as well when ( ) · ( )    dim dim 6 1 2 [10,11]. Similarly, the Strong PPT (SPPT) states are considered [12,12,13]. Moreover, the entanglement of some low-rank quantum states have been analyzed. See [14][15][16][17][18] for more details. Another useful tool for investigating the entanglement is the entanglement witness [10,19]. Some numerical algorithms for identifying the entanglement or separability were proposed as well. Doherty et alintroduced a family of separability criteria based on the existence of the extension of a bipartite quantum state to a multipartite symmetric PPT state [20]. That is, if a state ρ is separable on Ä   where P is the swap operator that exchange the first and third parties redand Γ is the partial transpose. In fact, the last condition indicates thatr remains PPT. Similarly, we can find the symmetric extension of ρ in the + s 1 partite system Ä Ä   s 2 1 for any  s 2. The author proved that for every entangled state rho, there exists a number > s 2 such that rho does not have such extension. However, it is impractical since we need to check all the possible numbers s to guarantee the separability. Besides, Ioannou [21] proposed another numerical method, i.e. analytic cutting plane method, to detect the entanglement. It tests the separability by reducing the set of all entanglement witnesses step by step. The algorithm stops until we can find an entanglement witness for ρ or the reduced set is too small to contain any entanglement witness. Dahl et al [22] proposed a method to find the closest separable state for any given quantum state ρ. The key step per iteration in this algorithm is to find a decent direction which is equivalent to finding the largest Z-eigenvalue of a forth-order tensor [23]. However, it is an NP-hard problem as well. Recently, Han [24] introduced a heuristic cross-hill method to solve the largest Z-eigenvalue problem, which, however, can only increase the possibility of obtaining the global solution.
With the development of quantum mechanics in the 1920s, symmetries came to play an essential role in the field of quantum entanglement. In the latter half of the 20th century, symmetry has been one of the most dominant concepts in the exploration and formulation of the fundamental laws of physics [25]. Therefore, the research of the symmetric states, which describes the quantum system of indistinguishable particles [26], is a central topic among the quantum information theory. It is then of great importance to explore the properties of the symmetric states. A quantum state is called symmetric (or bosonic) if it is invariant under the swap of particles. Such system is called the symmetric system (or bosonic system). The separability problem in the symmetric system has attracted more and more attention over recent years [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. Eckert [41] proved that three-qubit PPT symmetric states are all separable. Later, Tura et al [42] characterized the four-qubit entangled PPT symmetric states. Furthermore, Tóth et al [43,44] proved the existence of entangled PPT symmetric states in five-qubit and six-qubit symmetric system. Augusiak et al [45] characterized the PPT entangled states in the symmetric system consisting of an arbitrary number of qubits. In particular, the authors provide criteria for the separability of such states formulated in terms of their ranks. One of which is that these states are separable if they are PPT and not of the maximal rank. Within the symmetric states, a well-known subclass, diagonal symmetric states, are proved to be separable if and only if they are PPT [46]. Another subclass of symmetric states, completely symmetric states, are considered in [47,48] both theoretically and numerically. As an extension of diagonal symmetric states, Turaet al [49] showed that the separability of the multi-qudit diagonal symmetric states can be reformulated as a quadratic conic optimization problem. Recently, Rutkowskiet al [50] investigated the so-called D-invariant diagonal state in the multi-qudit symmetric system. It was proved that d-qudit PPT D-symmetric states are separable when d is even.
However, a successful criterion for the separability of symmetric states is still lacking even for the simplest case, multiqubit system. The existing results do not show the capability to handle the separability problem in general. It is then of great importance to introduce new techniques into this field and investigate this problem from another aspect, with which we are then hopeful to promote the development of the theory of quantum entanglement.
In this paper, we introduce a correspondence between the separability of multiqubit symmetric states and the decomposability of Hermitian positive semidefinite matrices, where a matrix is said to be decomposable if it admits a positive generalized Vandermonde decomposition. We show that any d-qubit symmetric state ρ can be represented by a ( ) + d 1 -dimensional Hermitian matrix r M by the Dicke basis as shown in figure 1. We will prove that ρ is separable if and only if the corresponding Hermitian matrix r M is decomposable. Based on it, we can convert the separability problem into the decomposability problem which does not only reduce the dimension but also provide new directions for studying the entanglement since this hard problem has remained open for too many years. In other words, we will study the decomposability property of r M systematically for some subclasses of matrices, such as the diagonal symmetric state, Toeplitz symmetric state, and Hankel symmetric state. Besides, we will develop a suite of tools for the decomposability similar to that in the separability problem. For example, we propose a sequence of necessary conditions of the decomposability, which come from the properties of the subpart of the matrices and we also characterize the 'decomposability' witnesses. Finally, a numerical algorithm is proposed to detect the decomposability based on the idea of the feasible direction method [51]. Besides, the final result of this algorithm can be used to construct a witness. Numerical examples are tested to show the efficiency and effectiveness.
The rest of the paper is organized as follows. In section 2, we fix some mathematical and physical notations and present the necessary backgrounds of quantum entanglement and Vandermonde decomposition. In section 3, we establish the relation between the separability of multiqubit symmetric states and the Vandermonde decomposition. Next, in section 4, we build a suite of tools for investigating the decomposability as the theoretical applications of our results. In section 5, we propose a numerical algorithm to detect the decomposability as a application of the relation from the numerical aspect. Finally, some concluding remarks are given in section 6.

Preliminaries
The main topic of this paper is the construction of the relation between the separability of multiqubit symmetric states and the Vandermonde decomposition and its applications. It is then necessary to introduce these concepts, including the mathematical notations, entanglement problem, symmetric states, and the Vandermonde decomposition, before handling the problem in detail.

Mathematical notations
Let us first introduce some mathematical notations. For simplicity, we refer to´  , N N N N , and N N as the N × N complex, real, and Hermitian matrices, respectively.
Mathematically, any quantum state can be represented by a Hermitian positive semidefinite matrix with trace one, which is called the density matrix 4 . Here, for a Hermitian matrix Î Given a matrix A, we use ( )  A to denote its range. The space of density matrices is equipped with the standard inner product where ρ and σ are two density matrices acting on the Hilbert space . The associated matrix norm is the Frobenius norm In this paper, for the vector Moreover, we use the notation 'Proj' to denote the projection operator. For any subset , we use ( )  H Proj to denote the projection of H to , i.e.,  is the point in  that is closest to X. Throughout this paper,  denotes the identity operator, whose order will match the operations in context. Especially, i denotes the imaginary unit, i.e., which should be distinguished from the index notation i. The composite quantum system is described by the tensor product of subsystems. We now show the related concepts formally. For any N×N matrix and arbitrary matrix B, the tensor product For any two Hilbert spaces  1 and  2 , the tensor space

Quantum states and separability problem
In this paper, we consider the separability problem of quantum states in the d-partite system . Before considering in depth the separability problem, we will briefly introduce the separability mathematically.
Given a quantum state ρ, which is a Hermitian positive semidefinite matrix, it is said to be a pure state if it is of rank one, i.e., . For pure states, we can also use the vector |fñ to represent the quantum state ρ. In particular, ρ is said to be a pure product state if , respectively. Then a state ρ is said to be fully separable if it can be written as a convex combination of pure product states, i.e., The state ρ is said to be k-separable if it is fully separable with respect to a k-partition of the systems . For example, a bi-partition as: , where the composite system of the lastd 1 ones is regarded as a single system. Moreover, a state ρ is called genuine entangled when it is not k-separable for any  k 2.

Symmetric states
Symmetry plays an essential role in modern physics and the study of symmetric quantum systems have become one of the most popular topics in quantum information theory. The symmetric space (also called bosonic system) is invariant under the exchange of any parties. A quantum state is called symmetric or bosonic if its support space is contained in the symmetric space. Note that for the symmetric system, all the subsystems are identical, that is, Hereafter in this paper, we denote by Ä  d the d-partite symmetric system. In particular, we mainly consider the multiqubit symmetric system, i.e.
Proof. Since r is a pure product state, The condition of trace one is required for explaining quantum states by the hypothesis of quantum physics. For conveniently treating mathematical problems in quantum information such as the spearability problem, distillability problem, which may be omitted the condition unless stated otherwise.

1
Our proof completes. + The above lemma shows that the separable pure state in the symmetric space possess a highly symmetric structure. And we believe that the mixed separable state would have the same property, which is shown in the following lemmas 2 and 17. For symmetric states, we have the following result: Lemma 2 (Augusiak et al [45]). Suppose ρ is a multi-qudit symmetric state, then it is either genuine entangled or fully separable.
Hence, for simplicity, we will just say ρ is separable if it is fully separable throughout this paper. Moreover, we have is contained in the symmetric space, then according to lemma 1, we have Therefore, equation (25) holds.
On the other hand, if equation (25) holds, by the definition of separability, ρ is separable. + For d-qubit symmetric systems, the following Dicke states formulate an orthogonal basis 6 : where the sum runs over all the different permutation operators U π of the N-qubit system and S d is the group of all permutations on d tuples. For example,

Vandermonde decomposition
Vandermonde decomposition is a powerful tool which will be used to study the separability problem in this paper. In linear algebra, the Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with terms of a geometric progression in each column, i.e. anŃ N 1 2 matrix with the following shape: The Vandermonde decomposition is a classical result developed by Caratheodory and Fejér dating back to 1911 [52]. A Hermitian matrix M is said to admit a Vandermonde decomposition if it can be written as where V is a Vandermonde matrix and D is a diagonal matrix. The Vandermonde decomposition has been studied for some subclasses of matrices, especially the Hankel and Toeplitz matrices. However, not all matrices admits the Vandermonde decomposition. Although the so-called Confluent Vandermonde matrices were introduced for studying the Vandermonde decomposition [53], the range of this class is too much wider than necessary to handle the topics in this paper. Therefore, we introduce the following generalized Vandermonde (GV) matrices, which constitute a smaller set but satisfy our requirement. The matrix V is called a Generalized-Vandermonde matrix (or abbreviated as GV matrix) if it is a traditional Vandermonde matrix or of the following form: The GV matrix in type of equation (35) has an extra kind of column, i.e. only the last entry is not zero compared with the original one. Here we state formally the definition of GV-decomposable matrices.
For the sake of convenience, each column of the GV matrix is called a Vandermonde vector, which will be used extensively in the following sections. In particular, for any a Î , let Besides, let Here the dimension N of the Vandermonde vector will match the operations in context. Usually, its dimension would be + d 1 when d-qubit symmetric system are involved in the discussion.
Note that the decomposability of M is invariant if replaced by ( ) > pM p 0 . Moreover, we would just say M is decomposable instead of GV-decomposable if no other confusion caused, regardless of some other types of decomposable matrices defined in linear algebra.

Relation between separability of symmetric states and vandermonde decomposition
In this section, we study the relationship between the separability of symmetric states and the generalized Vandermonde decomposition. We will find that the separability problem of quantum state ρ is equivalent to the decomposability problem of the Dicke matrix r M . This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. The following theorem states our main result in this paper which converts the separability problem to the decomposability problem. This not only reduce the dimension of the problem from2 2 1but also get rid off the tensor structure.
Theorem 4 (Main theorem). Suppose r is a multiqubit symmetric state. Then it is separable if and only if r M is decomposable.
Proof. We first prove this result when ρ is a pure product state. Since the mixed state are just the linear combination of the pure state, we can see later that the associated dicker matrix of mixed state is also the linear combination of that of the pure state. Suppose Next, we represent ρ by the Dicke basis and obtaining r M explicitly. We consider this in two different cases.
Hence, according to the correspondence(31), we have Hence, , and the Vandermonde vector | ñ u is defined as in equation (37). Hence, we proved that ρ is pure product states, then r M is decomposable. Note that the rank of ρ should equal that of r M since they are different representations of a same state. Then if ρ is of rank-1 (pure state), r M is also of rank-1. Suppose r M is decomposable and is rank-1, then it can be either For the former case, according to equation (31), we have , , which is separable. For the latter case, according to equation (31), we have which is also separable. Therefore, for the pure state ρ, it is separable if and only if r M is decomposable. For arbitrary mixed separable state ρ, suppose we have Since each term in the separability decomposition (51) can be represented by Note that V is a GV-matrix and D is positive diagonal. Hence, r M is decomposable. On the other hand, if r M is decomposable, then For each item, we can recover r i by the above discussion for pure states. Sum all the corresponding items, we have The following corollary states clearly about the correspondence of ρ and r M which comes from equations (57)-(60).
Corollary 5. Suppose r is a multiqubit symmetric state and r M is the associated Dicke matrix. Then To find the product vector in the range of the quantum state is a useful method to detect the entanglement. By the relation we constructed above, we also show that to find the product vector in the range of ρ is equivalent to find the Vandermonde vector in the range of r M .
Corollary 6. Suppose r is a multiqubit symmetric state, then the range of r contains a product vector if and only if We want to find more similarity between these two concepts. The following property about the reduced states and principal submatrix is an example about the similarity. However, they are not totally identical. This may suggest that our result is meaningful for the studying of entanglement since the new concept and its leading results cannot be recovered one-to-one by the standard language of density matrix.
Lemma 7. Suppose ρ is a multiqubit symmetric state and r M is the associated Dicke matrix. Then Here the partial trace is a linear positive operator on Ä  d . It seems that the principal submatrix of r M takes the same role as that of the reduced state. They all reduce the dimensions of the system. However, they are not directly connected. For example, consider the state where the orders of these submatrices should be smaller than or equal to d. This does not coincide with the Dicke matrix of r 1 : The above example shows that they are not one-to-one correlated. But it would be of interest to consider the following problem. Unfortunately, this result is still not easy to be verified. We will leave it as one of our future studies. For the quantum states, the separability is invariant under local operations and classical communication. We can also show that the decomposability is also invariant under some operations. This is another similarity between these two concepts. Moreover, let , then A is an invertible matrix. The decomposability of r M is invariant under these operators. After construction of this relation, we will show some applications of this result in the following section both in theoretical and numerical results, which not easy and convenient to establish based on the original quantum states.

Theoretic applications: witnesses and conditions for decomposability
In this section, we will suggest some theoretical applications to show the power of the relation. As for the separability, we will also build a suite of tools that can be used to investigate the decomposability problem.
Like the entanglement witness for the entangled state, we can also define the 'witness' for non-decomposable matrices. Here we first show the concept of the entanglement witnesses and then introduce our decomposability witness. These two concepts are highly connected, one may refer the entanglement witnesses for better understanding the usage of decomposability witness where we also proved a stronger result proposition 13.

Entanglement witnesses
In [44], the entanglement witness for symmetric state has been introduced as a useful tool. For the d-qudit symmetric system Ä  d , a Hermitian operator W acting on Ä  d is called a general entanglement witness of the symmetric system if These entanglement witnesses can be used to detect the entanglement. for all the decomposability witness W .
Note that, for detecting the non-decomposability, the decomposability witness W that we are interested in is the one that has at lest one negative eigenvalue. Otherwise, holds for any positive semidefinite matrix M, which is useless. To sum up, the decomposability witness that are useful should have the following properties: In fact, we can still improve the above result if some subclasses of symmetric states are considered.
Proposition 13. Suppose M is a positive semidefinite non-decomposable matrix which is contained in the convex subspace C. And C  Î , where  is the identity operator. In addition, where D is the set of normalized decomposable matrices. Then there exists a decomposability witness Note that we move the proofs of above results, especially lemma 11 and proposition 13, to appendix A for being focused.
Next, we will consider some subclasses of symmetric states. Proposition 13 provide a useful tool for testing the entanglement of the state ρ if r M possesses a nice structure. And it enables us to parameterize the decomposability witness for some subspaces.

Diagonal symmetric states
If the Dicke matrix r M corresponding to the multiqubit symmetric state ρ is diagonal, then it is called the diagonal symmetric state, i.e., It has been proved that all the diagonal symmetric states are separable if they are PPT [46]. More precisely, ρ is separable if and only the two Hankel matrices According to proposition 13, we know that the decomposability witness used for the diagonal matrix can be chosen to be diagonal as well.

Toeplitz symmetric states
The multiqubit symmetric state ρ is called a Toeplitz symmetric state if r M is a Toeplitz matrix. For these states, we can show that they are separable, which answers the question in [47].
Proposition 15. Suppose r is a symmetric state and the corresponding r M is Toeplitz. Then r M is decomposable, i.e. r is separable.
For the sake of concise, the proof of the above proposition is moved to appendix B.

Hankel symmetric states
The multiqubit state ρ is said to be a completely symmetric state (or Hankel symmetric state) if r M is a Hankel matrix: Equivalently, It is shown in [47] that these states are invariant under any index permutation. In that paper, the authors proved that the multiqubit completely symmetric states are separable.
Proposition 16. Suppose r is a symmetric state and the corresponding r M is Hankel. Then r M is decomposable, i.e. r is separable.
In fact, the completely symmetric states correspond to the symmetric states over real, that is, all the components in the separability decomposition are real. For example, if α is real, then ð87Þ which is Hankel.
Interestingly, if T ( ) ¼ 0, 0, ,0, 1 is not contained in the decomposition, M will has a real positive Vandermonde decomposition. let where Î  t i . Then

t t t t t t t
where Λ is a positive diagonal matrix. This extends the results discovered by Vandermonde [54], where M need to be of full rank or V chosen to be the confluent Vandermonde matrix.

rankρd
Note that the pure Dicke states | ñ D d k , , = ¼ k d 0, 1, , form an orthogonal basis in the d-partite symmetric system, thus the dimension of the space is + d 1. It implies that ρ has rank at most + d 1.
Here we highlight that, for the symmetric states of rank at most d, the entanglement can be distinguished by solving a system of polynomial equations. On the other hand, we can also find the separability decomposition of the separable states via finding the product vectors in their ranges.
Suppose ρ is an d-qubit symmetric state of rank r with  r d. r M is defined as in equation (31)  Therefore, the fact that | ( ) r ñ Î  v is equivalent to the existence of the solution of the following polynomial equation system: nd Therefore, we can find all the product vectors in the range of ρ. By the method in [55], we completely check whether ρ is entangled.
Moreover, we have the following result about the decomposition length and the decomposition uniqueness.
Lemma 17. Suppose r is a d-qubit symmetric state of rank  r r d , . If r is separable, then it is a sum of r pure product states and the decomposition is unique.
Proof. Suppose ρ is separable, then First, we show that L=r. Note that the GV-matrices are always of full rank. Hence, the spanned space is of dimension L provided that L is less than or equal to + d 1, the dimension of the Vandermonde vector. Since ( ) r = < = r d rank 1, we must have L=r. Next, we prove that this decomposition is unique, suppose | ñ + v r 1 is another different Vandermonde vector in the range of r M . Then the matrix are linearly independent, which is a contraction to the assumption that r M has rank r. Therefore, r M has only r different Vandermonde vectors in its range. By corollary 5, ρ has exactly r product vectors in its range and this decomposition is unique. + If we apply the PPT criterion on ρ, according to the result in [45], we have the following result about the decomposition of multiqubit separable symmetric states.
Corollary 18. Suppose r is a d-qubit symmetric state of rank  r r d , . If r is PPT, then it is a sum of r pure product states and the decomposition is unique.

Subpart conditions
In the following, we will also show the relation of matrix r M and its sub-parts, that is, if r M is decomposable, then some sub-part of it would be also decomposable. This may lead to useful necessary conditions. For example, we can show that the diagonal part of r M is always decomposable if r M is decomposable while the decomposability of diagonal one is easy to be detected.
Suppose r M is the Dicke matrix associated with the given multiqubit symmetric state ρ. Here, we introduce some families of matrices that come from parts of r M . Denote by ( ) = ¼ r M k 0, 1, 2, k the matrix whose some diagonal lines are removed:  Then, ⎡ Based on it, we propose a sequence of necessary conditions of the separability of the symmetric states. Moreover, let For general r M k , consider the (i,j)th entry. We claim that the every entry ofM k and r M k is the same, that is, Consequently, r M k is separable for any Î  k . For the mixed state, if r M is separable, then it can be written as a convex combination of the pure product states. For each components, the equation (108) holds, hence it also holds for r M , which completes our proof. + As for the r M 0 , we already have a necessary and sufficient condition for decomposability. This leads to a necessary condition for the general multiqubit symmetric states.

Corollary 20. Suppose r
M is the associated matrix of the given symmetric state r. If r is separable, then ( ) According to proposition 14, we have .
According to proposition 13, we have: 0for any decomposability witness. In other words, each decomposability witness provides a necessary condition for ρ being separable or a sufficient condition for ρ being entangled. The following proposition is an application of this. For focusing on the main topic, we move the proof to appendix C.
This criterion is also sufficient for the diagonal symmetric states. Byproposition 13, the diagonal symmetric state ρ is separable if and only if for any diagonal decomposability witness W. Note the set of diagonal decomposability witnesses is a subset of  T (the set of all the real tridiagonal witnesses, see appendix C). Hence, if equation (117) holds for Î  W T , then it also holds for the diagonal decomposability witness W. Hence, the conclusion of proposition 23 is also sufficient for the multiqubit diagonal symmetric states.

Application for multi-qudit symmetric states
are two linear independent vectors. The state r P P is an d-partite multiqubit symmetric state. The following result provides a necessary condition of the separability of the multi-qudit state ρ.
Lemma 24. If r is separable then so is r P P.
Since we can determine the separability of symmetric multiqubit states by the approach of the Vandermonde matrix, we can investigate the separability of ρ by using the same approach. By choosing various | | ñ ñ e x , i i i = 1,2 and determining the separability of corresponding r P P, we can estimate the separability of ρ with a relatively high probability.

Numerical applications: closest decomposable matrix
In this section, we propose a numerical method to detect the entanglement as an application of our main result in this paper. Instead of considering the original density matrix ρ, we develop the method based on the decomposability of r M which not only save the computations but also get rid off the tensor structure of the data compared with the original one.
By theorem 4, to check whether ρ is separable, it is equivalent to check whether r M is decomposable, where r M is defined by equation (31).
To get rid off the heavy notation r M , in this section, we just consider the decomposability of an arbitrary Hermitian positive semidefinite matrix H.
We propose a numerical method to find its closest decomposable matrix H * . It can detect the decomposability in this way.
• If H is decomposable, then H * will be H itself.
• If H is non-decomposable, then H * is different from H.

Feasible direction method for finding the closest decomposable matrix
We are now ready to show the details of the numerical method to find the closest decomposable matrix of any given Hermitian positive semidefinite matrix H which is based on the Feasible direction method [51]. First of all, let us introduce some frequently used notations. Let The coefficient g a is multiplied to make the extreme points in D normalized (trace-one) and the set D bounded. Hence, D is a convex compact set and all the normalized decomposable matrices are contained in it.
In the extended complex plane, we can assume that where  is the extended complex number system by adding two elements: +¥ and -¥.
In order to check whether H is decomposable, we need to solve the following optimization problem: . 124 if and only if the optimal value of the above optimization problem equals zero. Otherwise, we can find its closest decomposable state which is the optimal solution.
Since D is a compact convex set and the objective function of (124) is a strictly convex quadratic function of X, the optimization problem has a unique minimizer H * , which is essentially the projection of H to D, denoted by . We have the following properties about the unique solution [58]: . Then the following statements are equivalent: See appendix D for the proof.
Here we state the idea of this iterative method in details. Let H 0 be a starting point based on which we want to find a closer point H 1 for approximating in the next step. If equation (126) is satisfied, i.e., then H 0 must be the closest decomposable matrix to H, i.e. the solution of optimization problem(124). Otherwise, we can find an D Î X such that equation (128) is violated. In fact, -X H 0 forms a feasible descent direction. As illustrated in figure 2, we can find a point along the direction -X H 0 that is closer to H. Therefore, we will choose the closest point H 1 along this direction as our candidate for the next step in the algorithm. Suppose However, it is not easy to find a feasible point X which violates equation (126). In order to solve this problem, we need to consider the following optimization problem: Since X can be written as the convex sum of the extreme points ( ) a V , the above optimization problem is equivalent to the following one: In appendix F, we will propose an algorithm to solve this sub-optimization problem(132), where the objective function is a real rational function of a single complex variable. Therefore, our algorithm can be stated as: Given an initial candidate H 0 for approximating H * At kth iteration, solving the optimization problem equation (132), where H 0 is replaced with H k . Denote by a k the solution of the optimization problem.
is less than the given tolerance, then we stop the algorithm and choose H k as the approximation of H * . Otherwise, go to step 2.
Suppose in the kth iteration, we have the decomposition: The coefficients ( ) p i 0 , i=1, K, L, obtained in the above algorithm may not be the best choice such that 2 has the minimal value. In fact, we can accelerate the convergence by solving the following quadratic We first consider the following optimization problem: . 141 Note that we have, (all the elements are nonnegative), then it is the unique solution of the problem(137). Otherwise, we have the following multiplicative iteration method to find the global optimal solution: , , , 144 n n 1 1 2 2 Note that the initial candidate ( ) p 0 must be positive. It can be chosen to be the current coefficients of H k at kiteration as in equation (135). According to the results in [59], this method converges globally.
See appendix E for the pseudocode of and the convergence theorem of the multiplicative method. To satisfy the condition that H K is of trace one, we can project the final result obtained by this method to the normalized one, i.e. ¬ å p p p i i i i . There exist many other fast algorithms for quadratic optimization problem with linear constraints. In this paper, we just adopt one of the simplest methods to illustrate the idea to improve the convergence speed.
Hence, W is a decomposability witness. Further, where the last inequality comes from that H * is different from H.

Numerical examples
In this subsection, we test some examples to show the efficiency of the proposed method (algorithm 1).
where the a i are randomly generated different numbers.   decomposability for different dimension d. We mainly tested = d 5, 10, 15, 20, 25, which should be enough for the results being trusted.
Theoretically, for decomposable H, the closest decomposable matrix should be H itself. For the nondecomposable H, it would be different from H. Our numerical results in the following graphs coincide with this conclusion. For example, for the decomposable case, as shown in figures 3-6, the matrices we find for approximating H * would converge to H themselves. For the nondecomposable ones, as shown in figures 7-10, the distance || | | -H H k would converge to a constant. It shows the capability of our algorithm to detect the nondecomposability and thus the entanglement.
Here the y-axis in these graphs denotes || | | -H H k F 2 and the x-axis denotes the iteration number k. However, one need to point out that if the result || | | -H H k F of our algorithm tends to a positive constant, it may not guarantee completely that H is non-decomposable. This may be caused by the instability of our algorithm which is not considered in our paper. To make the results of our algorithm more convinced, the further convergence analysis should be discussed.

Conclusion
In this paper, we established a relation between the Vandermonde decomposition and the separability of multiqubit symmetric states which is the basic result for the rest paper. Utilizing it, we converted the separability problem to the decomposability problem, i.e. check whether a Hermitian positive semidefinite matrix is decomposable.
Based on the relationship discovered, we built a suite of tools that can be used to investigate the decomposability of Hermitian matrix, i.e. the separability of quantum states.
First, we introduced the decomposability witness for detecting the non-decomposability, where each decomposability witness provides a necessary condition for decomposability. For instance, the real tridiagonal witness possesses a nice structure to be used as a necessary condition.
Second, we studied some subclasses of multiqubit symmetric states to show the power of the Vandermonde decomposition used in the separability problem, including the diagonal symmetric states, Toeplitz symmetric states, completely symmetric states, and rank deficient symmetric states.
Third, we found that the matrices that come from the part of the decomposable matrix inherit the decomposability, for example, the diagonal part of the decomposable matrix will be necessarily decomposable as well. And for such matrices, there exists a decomposability witness that has the same shape to detect the nondecomposability if they are non-decomposable. An application to the multi-qudit symmetric is to project these states into the multiqubit symmetric subspace, where the tools we established can be then used to detect the separability with a high probability. The connection enables us to utilize the tools in both fields, quantum entanglement and Vandermonde decomposition, where the latter one is especially useful in signal processing. We hope our results are helpful to generate new directions on solving the entanglement problem of symmetric states.
Finally, we also utilized the relationship numerically. We proposed a numerical method to detect the decomposability of any given Hermitian positive semidefinite matrix. The idea is based on the well-known feasible direction method. At each step, we need to find a direction along which there exists a better iteration point. Some numerical examples were tested to show the efficiency and effectiveness. Our algorithm is believed as a useful tool for detecting the entanglement and promote the development in the field of entanglement of the symmetric states.
For the future work, there are still lots of researches can be done. For example, many other classes of symmetric states can be studied such as the matrices which have only three diagonal lines. And it is of interest to construct some decomposability witnesses which can be served as the necessary condition for decomposability. Moreover, in [21], the author proposed the numerical method to find the entanglement witness for any given quantum state. Note that decomposability is very similar to the entanglement witness, it is then hopeful to develop a similar numerical algorithm.
Theorem 27 (Hyperplane separation theorem [62] Consider a given non-decomposable matrix M (Hermitian positive semidefinite). We can thus assume M is nonzero since M=0 is a trivial case. Let D be the set of all the decomposable matrices. We need to find a witness such that it can separate M and D.
As described in theorem 27, we need to define the convex sets  and .
In fact, all the Hermitian matrices form a real space. The following matrices form an orthonormal basis of  N N : where E ij is the matrix whose (i, j)th entry is one and all the rests are zeros. Hence, N N is a real vector space of dimension N 2 and the inner product is defined as In fact, this above lemma is used to prove theorem 27. Hoawever, proposition 13 has a extra condition which makes the separating porint special. Unfortunately, theorem 27, which is more general, can not be applied directly. We thus need to find the seprarating point by proving theorem 27 from the begining. Now, we are ready to prove proposition 13.
Proof of proposition 13. Suppose M is a positive semidefinite non-decomposable Hermitian matrix in C. Let By the definition of the decomposable matrices, we know that D is a nonempty closed convex subset and so is .
By lemma 28, there exists a unit A 0 such that : . 168 By the uniqueness of A 0 and equation (81), A 0 is also the unique point in  which is nearest to M.
Let Proof. This lemma is equivalent to prove that every positive semidefinite Toeplitz matrix is decomposable. Every positive Toeplitz matrix admits a positive Vandermonde decomposition. This has been proved by Caratheodory and Fejér in [52]. For completeness, we show a concise proof here.
Since the Hermitian positive semidefinite possesses a Cholesky decomposition, we will utilize it together with the structure of Teoplitz to prove proposition 29. Suppose M is a positive semidefinite Toeplitz matrix, What we want to do next is to represent the equation ( ) r WM Tr with a simple expression which can be tested efficiently. To achieve this, we first need to analyze that which kind of a i and b i ʼs are satisfied to be a witness. If they have a simple expression, then ( ) r WM Tr could be probably easily converted to explicite form. For any entanglement witness, it satisfies the condition . 190 Let = q t re i , we have  .
In our case, n=1, the above lemma implies that any nonnegative polynomial P(r) can be written as   In fact, we claim that each coefficient a i is real. SinceH is Hermitian, then Therefore, ( ) q p r is a real rational function. In order to find the maximal value, we can firstly find the critical points of the function ( ) q p r since the maximal value of the function ( ) q p r always lies at these critical points. Note that, ( ) ¢ = q p r 0 is equivalent to Figure F2. Graph of ( ) q f r, .