A combinatorial criterion for k-separability of multipartite Dicke states
aa r X i v : . [ qu a n t - ph ] O c t A combinatorial criterion for k -separability of multipartite Dicke states Zhihua Chen , Zhihao Ma , Ting Gao , and Simone Severini Department of Mathematics, College of Science,Zhejiang University of Technology, Hangzhou 310024, China Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China College of Mathematics and Information Science,Hebei Normal University, Shijiazhuang 050024, China and Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
We derive a combinatorial criterion for detecting k -separability of N -partite Dicke states. The cri-terion is efficiently computable and implementable without full state tomography. We give examplesin which the criterion succeeds, where known criteria fail. I. INTRODUCTION
While the structure of bipartite entangled states is fairly well-understood, the study of multipartite entanglementstill presents a number of partial successes and difficult open problems (see the recent reviews [1, 2]). General criteriafor multipartite states of any dimension were recently proposed in [3–11, 13].Here, we derive a criterion for detecting k -nonseparability in Dicke states based on various ideas developed in [3–6, 8, 9, 12]. The criterion can be seen as a generalization of a method for detecting genuine multipartite entanglementin Dicke states detailed in [7]. The criterion have the advantages of being computationally efficient and implementablewithout the need of state tomography. We give examples in which the criterion is stronger than the ones proposed in[13].Let us recall some standard terminology and the definition of a Dicke state. An N -partite pure state | ψ i ∈H ⊗ H ⊗ · · · ⊗ H N (dim H i = d i ≥ , ≤ i ≤ N ) is said to be k -separable if there is a k -partition [8, 9] j · · · j m | j · · · j m | · · · | j k · · · j km k such that | ψ i = | ψ i j ··· j m | ψ i j ··· j m · · · | ψ k i j k ··· j kmk , where | ψ i i j i ··· j imi is the state of the subsystems j i , j i , ..., j im i , and S ki =1 { j i , j i , · · · , j im i } = { , , · · · , N } . More gener-ally, an N -partite mixed state ρ is said to be k -separable if it can be written as a convex combination of k -separablepure states ρ = P i p i | ψ i ih ψ i | , where | ψ i i is possibly k -separable under different partitions. An n -partite state is saidto be fully separable when it is N -separable and N -partite entangled if it is not 2-separable. A k -separable mixedstate might not be separable with regard to any specific k -partition, which makes k -separability difficult to deal with.We shall consider pure states as a special case.The N -qubits Dicke state with m excitations (see [14]) is defined as | D Nm i = 1 p C mN X ≤ i j = i l ≤ N | φ i ,...,i m i , where | φ i ,...,i m i = O i i ,...,i m } | i i O i ∈{ i ,...,i m } | i i , where C mN := (cid:0) Nm (cid:1) is the binomial coefficient. For instance, | D i = 6 − / ( | i + | i + | i + | i + | i + | i ) , when N = 4 and m = 2.Section II contains the statements and proofs of the results. Examples are in Section III. II. RESULTS
We construct a set of inequalities which are optimally suited for testing whether a given Dicke state is N -partiteentangled: Theorem 1
Suppose that ρ is an N -partite density matrix acting on a Hilbert space H = H ⊗ H ⊗ · · · ⊗ H N . Let F ( ρ, φ ) := A ( ρ, φ ) − B ( ρ, φ ) , where A ( ρ, φ ) := X ≤ i = j = j ′ ≤ N (cid:18) |h φ i,j | ρ | φ i,j ′ i| − q h φ i,j | ⊗ h φ i,j ′ | Π j ρ ⊗ Π j | φ i,j i ⊗ | φ i,j ′ i (cid:19) , and B ( ρ, φ ) := N k X ≤ i = j ≤ N h φ i,j | ρ | φ i,j i . Here, | φ i,j i := | ... ... ... i ∈ H , with the s in the subspaces H i and H j , N k := max { N − k − , N − k } and Π j , is the operator swapping the two copies of H j in H ⊗ H , for ≤ i = j ≤ N . If the density matrix ρ is k -separablethen F ( ρ, φ ) ≤ . Proof.
We start with a 4-qubit state to get an intuition. Note that for a four-qubit pure state ρ = | ψ ih ψ | , we have F ( ρ, φ ) = 2( | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , + | ρ , | − √ ρ , ρ , (1)+ | ρ , | − √ ρ , ρ , ) − N k ( ρ , + ρ , + ρ , + ρ , + ρ , + ρ , ) . If a 4-qubit pure state ρ = | ψ ih ψ | is biseparable then F ( ρ, φ ) ≤ ρ = | ψ ih ψ | with | ψ i = P i i i i ψ i i i i | i i i i i ( i , i , i , i = 0 ,
1) is k -separable,where k = 3 ,
4. Then, F ( | ψ i , φ ) = 2( | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | ) − N k ( | ψ | + | ψ | + | ψ | + | ψ | + | ψ | + | ψ | ) . Note that there are six 3-partitions 1 | |
34, 1 | |
24, 1 | |
23, 2 | |
14, 2 | |
13, and 3 | |
12. WLOG we prove that F ( | ψ i , φ ) ≤ ρ = | ψ ih ψ | which is 3-separable under the partition 1 | |
34. Suppose that | ψ i = ( a | i + a | i ) ⊗ ( b | i + b | i ) ⊗ ( c | i + c | i + c | i + c | i ) = a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i + a b c | i , then A = 2( | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ |− | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | )= 0;and A = 2( | ψ ψ | − | ψ ψ | + ψ ψ | − | ψ ψ | ) − ( | ψ | + | ψ | + | ψ | + | ψ | + | ψ | + | ψ | ) ≤ . It follows that F ( | ψ i , φ ) = A + A ≤
0. if | ψ i is 3-separable.If | ψ i is fully separable, then | ψ i = ( a | i + a | i ) ⊗ ( b | i + b | i ) ⊗ ( c | i + c | i ) ⊗ ( d | i + d | i ) = a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i + a b c d | i , and F ( | ψ i , φ ) = 2( | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | + | ψ ψ | − | ψ ψ | )= 0 . The equalities above confirms the statement in Eq.(1), when restricted to 4-qubit pure states.For the general case, we use the notation and proof method given in [5, 9].Suppose that ρ = | ψ ih ψ | is a k -separable pure state under the partition of { , , · · · , N } into k pairwise disjointsubsets: { , , · · · , N } = S kl =1 A l , with A l = { j l , j l , · · · , j lm l } and | ψ i = | ψ i j ··· j m · · · | ψ k i j k ··· j kmk = X i , ··· ,i m a i ··· i m | i · · · i m i j ··· j m · · · X i k , ··· ,i kmk a i k ··· i kmk | i k · · · i km k i j k ··· j kmk X i , ··· ,i m , ··· ,i k , ··· ,i kmk a i ··· i m · · · a i k ··· i kmk | i · · · i m · · · i k · · · i km k i j ··· j m ··· j k ··· j kmk . Hence, ρ P s,t i st d jst +1 d jst +2 ··· d N d N +1 +1 , P s,t e i st d jst +1 d jst +2 ··· d N d N +1 +1 = a i ··· i m · · · a i k ··· i kmk a ∗ e i ··· g i m · · · a ∗ e i k ··· g i kmk . The sum is over all possible values of { i st | s ∈ { , , · · · , k } , t ∈ { , , · · · , m s }} , d i = 2, when i = N + 1 and d N +1 = 1.We shall distinguish between the cases in which both indices j and j ′ correspond to different parts A l and A ′ l , orthe same parts A l , 1 ≤ l = l ′ ≤ k , with respect to | ψ i . By direct calculation, one has the following: |h φ i,j | ρ | φ i,j ′ i| = q h φ i,j | ρ | φ i,j ih φ i,j ′ | ρ | φ i,j ′ i ≤ h φ i,j | ρ | φ i,j i + h φ i,j ′ | ρ | φ i,j ′ i , (2)when j and j ′ are in the same part; |h φ i,j | ρ | φ i,j ′ i| = q h φ i | ρ | φ i ih φ i,j,j ′ | ρ | φ i,j,j ′ i = q h φ i,j | ⊗ h φ i,j | Π + j ′ ρ ⊗ Π j | φ i,j i ⊗ | φ i,j ′ i , (3)when j and j ′ are in the different parts ( j ∈ A l , j ∈ A l ′ with l = l ′ ). Here, | φ i i = | · · · · · · i , with | i in the i -thsubspace H i , and | φ i,j,j ′ i = | · · · · · · · · · · · · i , such that all subspaces are in the state | i , except for thesubspaces H i , H j and H ′ j , which are in the state | i .For a given | φ i,j i , the number of | φ i,j ′ i ’s, with j and j ′ in same part, is at most max { N − k − , N − k } . Noticethat the maximal number of subsystems contained in a part of a k -partition is N − k + 1. Suppose that A | A | · · · | A k is a k -partition of { , , · · · , n } , where A l = { j l } , for l = 1 , , · · · , k −
1, and A k = { j k , j k , · · · , j kN − k +1 } . When i , j and j ′ are in the same part A k , the number of | φ i,j ′ i ’s is 2( N − k − i belongs to A ∪ A ∪ · · · ∪ A k − , while j and j ′ belong to A k , the number of | φ i,j ′ i ’s is N − k . Therefore, the number of | φ i,j ′ i ’s satisfying j and j ′ in thesame part is at most max { N − k − , N − k } . This number is denoted as N k .By using the inequalities in (2) and (3), we have X ≤ i,j,j ′ ≤ N |h φ i,j | ρ | φ i,j ′ i| = X i X j ∈ A l ,j ′ ∈ A l ′ ,l = l ′ l,l ′ ∈{ , , ··· ,k } |h φ i,j | ρ | φ i,j i| + X j,j ′ ∈ A l ,j = j ′ l ∈{ , , ··· ,k } |h φ i,j | ρ | φ i,j i|≤ X i X j ∈ A l ,j ∈ A l ′ ,l = l ′ l,l ′ ∈{ , , ··· ,k } q h φ i,j | ⊗ h φ i,j ′ | Π + j ρ ⊗ Π j | φ i,j i ⊗ | φ i,j ′ i + X i X j,j ′ ∈ A l ,j = j ′ l ∈{ , , ··· ,k } (cid:18) h φ i,j | ρ | φ i,j i + h φ i,j ′ | ρ | φ i,j ′ i (cid:19) ≤ X i X j = j ′ q h φ i,j | ⊗ h φ i,j ′ | Π + j ρ ⊗ Π j | φ i,j i ⊗ | φ i,j ′ i + N k X i,j h φ i,j | ρ | φ i,j i . Thus, the inequality in the statement of the theorem is satisfied by all k -separable N -partite pure states.It remains to show that the inequality holds if ρ is a k -separable N -partite mixed state. Indeed, the generalization ofthe inequality to mixed states is a direct consequence of the convexity of the first summation in A ( ρ, φ ), the concavityof B ( ρ, φ ), and the second summation in A ( ρ, φ ), which we can see as follows.Suppose that ρ = X m p m ρ m = X m p m | ψ m ih ψ m | is a k -separable N -partite mixed state, where ρ m = | ψ m ih ψ m | is k -separable. Then, by the Cauchy-Schwarz inequality,( P mk =1 x k y k ) ≤ ( P mk =1 x k )( P mk =1 y k ), we get X i X j = j ′ |h φ i,j | ρ | φ i,j ′ i| ≤ X i X m p m X j = j ′ |h φ i,j | ρ m | φ i,j i|≤ X m p m X i X j = j ′ q h φ i,j | ⊗ h φ i,j | Π + j ρ ⊗ m Π j | φ i,j ′ i ⊗ | φ i,j ′ i + N k X i,j q h φ i,j | ⊗ h φ i,j | Π + j ρ ⊗ m Π j | φ i,j i ⊗ | φ i,j i = X i X j = j ′ X m p h φ i | p m ρ m | φ i i q h φ i,j,j ′ | p m ρ m | φ i,j,j ′ i + N k X i,j X m p m h φ i,j | ρ m | φ i,j i≤ X i X j = j ′ sX m h φ i | p m ρ m | φ i i X m h φ i,j,j ′ | p m ρ m | φ i,j,j ′ i + N k X i,j h φ i,j | ρ | φ i,j i = X i X j = j ′ q h φ i,j | ⊗ h φ i,j | Π + j ρ ⊗ Π j | φ i,j ′ i ⊗ | φ i,j ′ i + N k X i,j q h φ i,j | ⊗ h φ i,j | Π + j ρ ⊗ Π j | φ i,j i ⊗ | φ i,j i , as desired. This completes the proof.We can choose | φ i ad hoc to get different inequalities for detecting k -separability of different classes. For Theorem2, we have chosen | φ i to be an N -qubit product states with m excitations ( i.e. m entries of | φ i are | i , while theremaining N − m entries are | i ). The criterion performs well to detect k -separability for N qubit Dicke states with m excitations mixed with white noises. Theorem 2
Suppose that ρ is an N -partite density matrix acting on Hilbert space H = H ⊗ H ⊗ · · · ⊗ H N , and | φ i i , ··· ,i m i = | · · · · · · · · · · · · i is a state of H , where the local state in H l is | i , for l = i , i , · · · , i m ,and | i , for l = i , i , · · · , i m . Let F ( ρ, φ ) := A ( ρ, φ ) − B ( ρ, φ ) , with A ( ρ, φ ) := X i , ··· ,i j , ··· ,i m ,i ′ j (cid:16) |h φ i ··· ,i j , ··· ,i m | ρ | φ i , ··· ,i ′ j ··· ,i m i|− q h φ i , ··· ,i j , ··· ,i m | ⊗ h φ i , ··· ,i ′ j , ··· ,i m | Π i j ρ ⊗ Π i j | φ i , ··· ,i j , ··· ,i m i ⊗ | φ i , ··· ,i ′ j , ··· ,i m i (cid:19) , and B ( ρ, φ ) := N k X i ,i , ··· ,i m h φ i , ··· ,i j , ··· ,i m | ρ | φ i , ··· ,i j , ··· ,i m i . Here, Π i j is the operator swapping the two copies of H i j in the twofold copy Hilbert space H ⊗ := H ⊗ H , and N k := max { m ( N − k + 1 − m ) , ( m − N − k − m + 2) , · · · , ( N − k ) } . If the density matrix ρ is k -separable then F ( ρ, φ ) ≤ . In the following statement, we consider a criterion which is suitable for any general quantum states. The states tobe chosen are | χ i , | χ α i and | χ β i . Theorem 3
Let V = {| χ i , ..., | χ m i} be a set of product states in H = H ⊗ H ⊗ · · · ⊗ H N . If ρ is k -separable then T ( ρ, χ ) = X | χ α i∈ V X | χ β i∈ K α (cid:18) |h χ α | ρ | χ β i| − q h χ α | ⊗ h χ β | Π αβ ρ ⊗ Π αβ | χ α i ⊗ | χ β i (cid:19) − N k X α h χ α | ρ | χ α i≤ , (4) where K α := {| χ β i : || χ α i ∩ | χ β i| = N − with | χ α i , | χ β i ∈ V } , and || χ α i ∩ | χ β i| is the number of coordinates that are equal in both vectors ( i.e ., | χ α i and | χ β i have only two differentlocal states, say the i αβ -th and i ′ αβ -th local states), while Π αβ is the operator swapping the two copies of H i αβ in H ⊗ .Additionally, N k := max α,i ,i , ··· ,i N − k +1 s α,i ,i , ··· ,i N − k +1 , where s α,i ,i , ··· ,i N − k +1 is the number of states | χ β i in K α such that two of the states for the N − k + 1 particles i , i , · · · , i N − k +1 in | χ β i are different from that of | χ α i , when K α = ∅ . Proof.
By using the same proof method as in [5, 9], we prove that (4) holds for any k -separable pure states ρ = | ψ ih ψ | .Let T ( ρ, χ ) = A + A , where A is the sum of terms |h χ α | ρ | χ β i| − p h χ α | ⊗ h χ β | Π αβ ρ ⊗ Π αβ | χ α i ⊗ | χ β i in the firstsummation in (4). In this expression, the two different bits of | χ α i and | χ β i are in two different parts of a k -partition,while A is the sum containing the summands in (4), such that the different bits of | χ α i and | χ β i are in the samepart of a k -partition.We first prove that T ( ρ, χ ) ≤ V = {| χ i , ..., | χ i} be a set of product states in H ,where | χ i = | i , | χ i = | i , | χ i = | i , and | χ i = | i . Then K = {| χ i , | χ i , | χ i} , K = {| χ i , | χ i} , K = {| χ i , | χ i , | χ i} , and K = {| χ i , | χ i} . Thus, T ( ρ, χ ) = 2 X i =2 (cid:16) |h χ | ρ | χ i i| − p h χ | ⊗ h χ i | Π ρ ⊗ Π | χ i ⊗ | χ i i (cid:17) + |h χ | ρ | χ i| − p h χ | ⊗ h χ | Π ρ ⊗ Π | χ i ⊗ | χ i + |h χ | ρ | χ i| − p h χ | ⊗ h χ | Π ρ ⊗ Π | χ i ⊗ | χ i (cid:17) − N k X i =1 h χ i | ρ | χ i i = 2( | φ φ | − | φ φ | + | φ φ | − | φ φ | + | φ φ | − | φ φ | + | φ φ | − | φ φ | + | φ φ | − | φ φ | ) − N k ( | φ | + | φ | + | φ | + | φ | )= A + A ;When k = 3, there are six 3-partitions, i.e. , 1 | |
34, 1 | |
24, 1 | |
23, 2 | |
14, 2 | |
13, and 3 | |
12. For χ , we have s , = 0, s , = 1, s , = 1, s , = 1, s , = 0, and s , = 0; for χ , we have s , = 1, s , = 0, s , = 1, s , = 0, s , = 0, and s , = 0; for χ , we have s , = 1, s , = 1, s , = 0, s , = 0, s , = 0, and s , = 1;for χ , we have s , = 0, s , = 0, s , = 0, s , = 1, s , = 0, and s , = 1. So, we get N = 1.For the case 1 | | A = 2 " X i =2 (cid:16) |h χ | ρ | χ i i| − p h χ | ⊗ h χ i | Π ρ ⊗ Π | χ i ⊗ | χ i i (cid:17) + |h χ | ρ | χ i| − p h χ | ⊗ h χ | Π ρ ⊗ Π | χ i ⊗ | χ i = 2( | φ φ | − | φ φ | + | φ φ | − | φ φ | + | φ φ | − | φ φ | + | φ φ | − | φ φ | )= 0; A = 2 (cid:16) |h χ | ρ | χ i| − p h χ | ⊗ h χ | Π ρ ⊗ Π | χ i ⊗ | χ i (cid:17) − N ( | φ | + | φ | + | φ | + | φ | )= 2( | φ φ | − | φ φ | ) − ( | φ | + | φ | + | φ | + | φ | ) ≤ . This implies that T ( ρ, χ ) = A + A ≤
0. For the other 3-partitions, we can get the same result T ( ρ, χ ) = A + A ≤ | | k = 4, there is a single 4-partition, 1 | | |
4. Then, it is not possible for any two different bits to be in thesame partition. It follows that N = 0 and T ( ρ, χ ) = A = 0.For a k -separable 4-partite mixed state ρ = P m p m ρ m , where ρ m = | ψ m ih ψ m | is k -separable, we have T ( ρ, χ ) = 2 X i =2 |h χ | X m p m ρ m | χ i i| − s h χ | ⊗ h χ i | Π ( X m p m ρ m ) ⊗ Π | χ i ⊗ | χ i i + |h χ | X m p m ρ m | χ i| − s h χ | ⊗ h χ | Π ( X m p m ρ m ) ⊗ Π | χ i ⊗ | χ i + |h χ | X m p m ρ m | χ i| − s h χ | ⊗ h χ | Π ( X m p m ρ m ) ⊗ Π | χ i ⊗ | χ i − N k X i =1 h χ i | X m p m ρ m | χ i i≤ X m p m " X i =2 (cid:16) |h χ | ρ m | χ i i| − p h χ | ⊗ h χ i | Π ( ρ m ) ⊗ Π | χ i ⊗ | χ i i (cid:17) + |h χ | ρ m | χ i| − p h χ | ⊗ h χ | Π ( ρ m ) ⊗ Π | χ i ⊗ | χ i i = X m p m T ( ρ m , χ ) ≤ k -separable 4-partite states.Notice that for any k -separable pure states ρ = | ψ ih ψ | , if the two different bits of | χ α i and | χ β i are in two differentparts, then |h χ α | ρ | χ β i| − p h χ α | ⊗ h χ β | Π αβ ρ ⊗ Π αβ | χ α i ⊗ | χ β i = 0, otherwise |h χ α | ρ | χ β i| − h χ α | ρ | χ α i + h χ β | ρ | χ β i ≤ k -separable pure N -partite states ρ .Suppose that ρ = P m p m ρ m is a k -separable mixed N -partite state, where ρ m = | ψ m ih ψ m | is k -separable. It followthat T ( ρ, χ ) = X | χ α i∈ V X | χ β i∈ K α |h χ α | X m p m ρ m | χ β i| − s h χ α | ⊗ h χ β | Π αβ ( X m p m ρ m ) ⊗ Π αβ | χ i ⊗ | χ i i − N k X α h χ α | X m p m ρ m | χ α i≤ X m p m X | χ α i∈ V X | χ β i∈ K α (cid:18) |h χ α | ρ m | χ β i| − q h χ α | ⊗ h χ β | Π αβ ( ρ m ) ⊗ Π αβ | χ i ⊗ | χ i i (cid:19) − N k X α h χ α | ρ m | χ α i = X m p m T ( ρ m , χ ) , ≤ III. EXAMPLES
Consider the family of N -qubit mixed states ρ ( D N ) = a | D N ih D N | + (1 − a ) I N N , where | D N i = √ C N P ≤ i = j ≤ N | φ i,j i . By Theorem 1, if a > C N ( N −
2) + N k C N C N ( N −
2) + N k C N − N N k + 2 N +1 ( N − ρ ( D N ) is k -nonseparable. Thus, if a > C N (2 N − C N (2 N −
5) + 2 N then ρ ( D N ) are genuine entangled, which is exactly the same as in [15]; if a > then ρ ( D ) is genuine entangled; if a > then ρ ( D ) is 3-nonseparable; if a > then ρ ( D ) is not fully-separable; if a > = 0 .
23 then ρ ( D ) is notfully-separable. However, the inequality in [13] detects that, if a > then ρ ( D ) is genuine entangled; if a > then ρ ( D ) is 3-nonseparable; if a > then ρ ( D ) is not fully-separable; if a > .
27 then ρ ( D ) is not a fully-separable5-partite state.Consider the N -qubit state ρ ( D Nm ) = (1 − a ) I N N + a | D Nm ih D Nm | , where | D Nm i = √ C mN P ≤ i ≤ i ≤···≤ i m ≤ N | φ i ,i , ··· ,i m i , By Theorem 2, if a > mC mN ( N − m ) + N k C mN mC mN ( N − m ) + N k C mN − N N k + 2 N m ( N − m ) , then ρ ( D Nm ) is k -nonseparable. For N = 5 and m = 3, we get that if a > then ρ ( D Nm ) is 3-nonseparable, while themethod in [13] fails.Consider the one-parameter four-qubit state ρ = − a I + a | φ ih φ | , where | φ i = ( | i + | i + | i + | i ) . By Theorem 3, if a > and a > then ρ is 3-nonseparable and not fully-separable, respectively, while in [13], if a > and a > , then ρ is 3-nonseparable and not fully-separable. Acknowledgments . This work is supported by the National Natural Science Foundation of China under Grant11371005,11371247, 11201427 and 11571313. SS would like to thank the Institute of Natural Sciences (INS) atShanghai Jiao Tong University for the kind hospitality during completion of this work. The support of INS isgratefully acknowledged. [1] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod.Phys. , 865 (2009).[2] O. G¨uhne and G. Toth, Phys. Reports , 1(2009).[3] M. Huber, F. Mintert, A. Gabriel, B. Hiesmayr, Phys. Rev. Lett. , 210501(2010).[4] O. G¨uhne and M. Seevinck, New J. Phys. , 053002(2010).[5] T. Gao and Y. Hong, Phys. Rev. A
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