Detecting entanglement of states by entries of their density matrices
aa r X i v : . [ qu a n t - ph ] O c t DETECTING ENTANGLEMENT OF STATES BY ENTRIES OF THEIRDENSITY MATRICES
XIAOFEI QI AND JINCHUAN HOU
Abstract.
For any bipartite systems, a universal entanglement witness of rank-4 for purestates is obtained and a class of finite rank entanglement witnesses is constructed. In addition,a method of detecting entanglement of a state only by entries of its density matrix with respectto some product basis is obtained. Introduction
Let H and K be separable complex Hilbert spaces. Recall that a quantum state is anoperator ρ ∈ B ( H ⊗ K ) which is positive and has trace 1. Denote by S ( H ) the set of all stateson H . If H and K are finite dimensional, ρ ∈ S ( H ⊗ K ) is said to be separable if ρ can bewritten as ρ = k X i =1 p i ρ i ⊗ σ i , where ρ i and σ i are states on H and K respectively, and p i are positive numbers with P ki =1 p i =1. Otherwise, ρ is said to be inseparable or entangled (ref. [1, 16]). For the case that at leastone of H and K is of infinite dimension, by Werner [21], a state ρ acting on H ⊗ K is calledseparable if it can be approximated in the trace norm by the states of the form σ = n X i =1 p i ρ i ⊗ σ i , where ρ i and σ i are states on H and K respectively, and p i are positive numbers with P ni =1 p i =1. Otherwise, ρ is called an entangled state.Entanglement is a basic physical resource to realize various quantum information and quan-tum communication tasks such as quantum cryptography, teleportation, dense coding and keydistribution [16]. It is very important but also difficult to determine whether or not a statein a composite system is separable or entangled. It is obvious that every separable state hasa positive partial transpose (the PPT criterion). For 2 × × PACS.
Key words and phrases.
Quantum states, separability, entanglement witnesses, positive linear maps.This work is partially supported by National Natural Science Foundation of China (No. 10771157), ResearchGrant to Returned Scholars of Shanxi (2007-38) and the Foundation of Shanxi University. case dim H = dim K = 2 or dim H = 2 , dim K = 3, a state is separable if and only if it isa PPT state, that is, has positive partial transpose (see [7, 17]), but the PPT criterion hasno efficiency for PPT entangled states appearing in the higher dimensional systems. In [3],the realignment criterion for separability in finite-dimensional systems was found, which saysthat if ρ ∈ S ( H ⊗ K ) is separable, then the trace norm of its realignment matrix ρ R is notgreater than 1. The realignment criterion was generalized to infinite dimensional system byGuo and Hou in [6]. A most general approach to characterize quantum entanglement is basedon the notion of entanglement witnesses (see [7]). A self-adjoint operator W acting on H ⊗ K is said to be an entanglement witness (briefly, EW), if W is not positive and Tr( W ρ ) ≥ ρ . It was shown in [7] that, a state ρ is entangled if and only ifit is detected by some entanglement witness W , that is, Tr( W ρ ) <
0. However, constructingentanglement witnesses is a hard task. There was a considerable effort in constructing andanalyzing the structure of entanglement witnesses for finite and infinite dimensional systems[2, 4, 14, 15, 20] (see also [10] for a review). Recently, Hou and Qi in [14] showed that everyentangled state can be recognized by an entanglement witness W of the form W = cI + T with I the identity operator, c a nonnegative number and T a finite rank self-adjoint operatorand provided a way how to construct them.Another important criterion for separability of states is the positive map criterion [7, The-orem 2], which claims that a state ρ ∈ S ( H ⊗ K ) with dim H ⊗ K < ∞ is separable if and onlyif (Φ ⊗ I ) ρ ≥ B ( H ) → B ( K ). Hou [13] generalized thepositive map criterion to the infinite dimensional systems and obtained the following result. Finite rank elementary operator criterion. ([13, Theorem 4.5])
Let H , K be complexHilbert spaces and ρ be a state acting on H ⊗ K . Then the following statements are equivalent. (1) ρ is separable; (2) (Φ ⊗ I ) ρ ≥ holds for every finite-rank positive elementary operator Φ : B ( H ) → B ( K ) . Recall that a linear map Φ : B ( H ) → B ( K ) is an elementary operator if there are operators A , A , · · · , A r ∈ B ( H, K ) and B , B , · · · , B r ∈ B ( K, H ) such that Φ( X ) = P ri =1 A i XB i forall X ∈ B ( H ). It is known that an elementary operator Φ is finite rank positive if and onlyif there exist finite rank operators C , . . . , C k , D , · · · , D l ∈ B ( H, K ) such that ( D , · · · , D l )is a contractive local combination of ( C , · · · , C k ) and Φ( X ) = P ki =1 C i XC † i − P lj =1 D j XD † j for all X ∈ B ( H ) (ref. [13] and the references therein).Therefore, by the finite rank elementary operator criterion, a state ρ on H ⊗ K is entangledif and only if there exists a finite rank positive elementary operator Φ : B ( H ) → B ( K ) suchthat (Φ ⊗ I ) ρ is not positive. Here Φ must be not completely positive (briefly, NCP). Thus it ETECTING ENTANGLEMENT BY ENTRIES 3 is also important and interesting to find as many as possible finite rank positive elementaryoperators that are NCP, and then, to apply them to detect the entanglement of states. In [18],some new finite rank positive elementary operators were constructed and then applied to getsome new entangled states that can not be detected by the PPT criterion and the realignmentcriterion.Due to the Choi-Jamio lkowski isomorphism, any EW on finite dimensional system H ⊗ K corresponds to a linear positive map Φ : B ( H ) → B ( H ). In fact, for system H ⊗ K of anydimension, if Φ : B ( H ) → B ( H ) is a normal positive completely bounded linear map, and if ρ is an entangled state on H ⊗ K , then W = (Φ ⊗ I ) ρ is an entanglement witness whenever W is not positive (see lemma 2.1). Recall that a linear map ∆ : B ( H ) → B ( K ) is said tobe completely bounded if ∆ ⊗ I is bounded; is said to be normal if it is weakly continuouson bounded sets, or equivalently, if it is ultra-weakly continuous (i.e., if { A α } is a boundednet and there is A ∈ B ( H ) such that h x | A α | y i converges to h x | A | y i for any | x i , | y i ∈ H , then h φ | ∆( A α ) | ψ i converges to h φ | ∆( A ) | ψ i for any | φ i , | ψ i ∈ K . ref. [5, pp.59]).The finite rank elementary operator criterion, together with lemma 2.1, gives a way ofconstructing finite rank entanglement witnesses from finite rank positive elementary operatorsfor both finite and infinite dimensional bipartite systems. In the present paper, we constructa rank-4 entanglement witness W that has some what “universal” property for pure statesin any bipartite systems H ⊗ K . We show that, for such a rank-4 entanglement witness W ,a pure state ρ is entangled if and only if there exist unitary operators U on H and V on K such that Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) <
0. In addition, if ρ is a mixed state such thatTr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) <
0, then ρ is 1-distillable (see theorem 2.2). We also construct aclass of entanglement witnesses from the finite rank positive elementary operators obtainedin [18] (see theorem 3.1).So far, by our knowledge, there is no methods of recognizing the entanglement of a state bymerely the entries of its density matrix. Another interesting result of this paper gives a wayof detecting the entanglement of a state in a bipartite system by only a part of entries of itsdensity matrix (see theorems 3.2, 3.3). This method is simple, computable and practicablebecause it provide a way to recognize the entanglement of a state by some suitably chosenentries of its matrix representation with respect to some given product basis. As an illustra-tion, some new examples of entangled states that can be recognized by this way are proposed,which also provides some new entangled states that can not be detected by the PPT criterionand the realignment criterion (see examples 3.4, 3.5). XIAOFEI QI AND JINCHUAN HOU
Recall that a bipartite state ρ is called n -distillable, if and only if maximally entangledbipartite pure states, e.g. | ψ i = ( | ′ i + | ′ i ), can be created from n identical copies ofthe state ρ by means of local operations and classical communication; is called distillable ifit is n -distillable for some n . It has been shown that all entangled pure states are distillable.However it is a challenge to give an operational criterion of distillability for general mixedstates [8]. In [9], it was shown that a density matrix ρ is distillable if and only if there aresome projectors P , Q that map high dimensional spaces to two-dimensional ones such thatthe state ( P ⊗ Q ) ρ ⊗ n ( P ⊗ Q ) is entangled for some n copies.2. Universal entanglement witnesses for pure states
In this section we will give a simple necessary and sufficient condition for separability ofpure states in bipartite composite systems of any dimension.Before stating the main result in this section, we give a basic lemma.
Lemma 2.1.
Let H , K be complex Hilbert spaces of any dimension and let Φ : B ( H ) →B ( H ) be a positive normal completely bounded linear map. Then, for any entangled state ρ on H ⊗ K , W = (Φ ⊗ I ) ρ is an entanglement witness whenever W on H ⊗ K is not positive. Proof.
Because Φ is completely bounded, W = (Φ ⊗ I ) ρ is a bounded self-adjoint operatoron H ⊗ K . Note that B ( H ) = T ( H ) ∗ , where T ( H ) denotes the Banach space of all trace classoperators on H endowed with the trace norm. Then the normality of Φ implies that thereexists a bounded linear map ∆ : T ( H ) → T ( H ) such that Φ = ∆ ∗ . We claim that ∆ is alsopositive. In fact, for any unit vector | φ i ∈ H and any positive operator A ∈ B ( H ), we haveTr( A ∆( | φ ih φ | )) = Tr(Φ( A )( | φ ih φ | )) = h φ | Φ( A ) | φ i ≥ . This implies that ∆( | φ ih φ | ) is positive for any | φ i . So, ∆ is a positive linear map.Now, for any separable state ρ ∈ S ( H ⊗ K ), we haveTr( W ρ ) = Tr((Φ ⊗ I ) ρ · ρ ) = Tr( ρ · (∆ ⊗ I ) ρ ) ≥ ⊗ I ) ρ ≥
0. So, if W is not positive, then it is an entanglement witness. (cid:3) Since every elementary operator is normal and completely bounded, by Lemma 2.1, if Φis a positive elementary operator and if ρ is an entangled state, then W = (Φ ⊗ I ) ρ is anentanglement witness whenever W is not positive. Also note that, if W is an entanglementwitness, then for any positive number b , bW is an entanglement witness, too.Let W be an entanglement witness on H ⊗ K . We say that W is universal (for all states) if,for any entangled state ρ on H ⊗ K , there exist unitary operators U on H and V on K such thatTr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) < W is universal for pure states if, for any entangled pure state ρ on ETECTING ENTANGLEMENT BY ENTRIES 5 H ⊗ K , there exist unitary operators U on H and V on K such that Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) <
0. The following is the main result of this section, which gives a universal entanglement witnessof rank-4 for pure states. Particularly, we conclude that the separability of pure states can bedetermined by a special class of rank-4 witnesses, and every 1-distillable state can be detectedby one of such rank-4 entanglement witnesses. However, we do not know whether or not thereexists a universal entanglement witness for all states.Let U ( H ) (resp. U ( K )) be the group of all unitary operators on H (resp. on K ). Theorem 2.2.
Let H and K be Hilbert spaces and let {| i i} dim H ≤∞ i =1 and {| j ′ i} dim K ≤∞ j =1 beany orthonormal bases of H and K , respectively. Let W = | i| ′ ih |h ′ | − | i| ′ ih |h ′ | − | i| ′ ih |h ′ | + | i| ′ ih |h ′ | . (2 . Then W is an entanglement witness of rank-4. Moreover, the following statements are true. (1) If ρ is a pure state, then ρ is separable if and only if Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) ≥ . hold for all U ∈ U ( H ) and V ∈ U ( K ) . So W is a universal entanglement witness for purestates. (2) Let ρ be a state. If there exist U ∈ U ( H ) and V ∈ U ( K ) such that Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) < , then ρ is entangled and 1-distillable. Proof.
We first prove that W is an entanglement witness. It is obvious that W is notpositive. Define a map Φ : B ( H ) → B ( H ) byΦ( A ) = E AE † + E AE † + E AE † + E AE † − ( E + E ) A ( E + E ) † (2 . A ∈ B ( H ), where E ij = | i ih j | ∈ B ( H ). It is obvious that Φ is a positive map becausethe map a a a a a − a − a a on M ( C ) is positive. Note that W = 2(Φ ⊗ I ) ρ + , where ρ + = | ψ + ih ψ + | with | ψ + i = √ ( | ′ i + | ′ i ). Thus, by Lemma 2.1, W is an entanglement witness.If ρ is separable, then Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) ≥ U † ⊗ V † ) ρ ( U ⊗ V ) are separable.Conversely, assume that ρ = | ψ ih ψ | is inseparable. Consider its Schmidt decomposition | ψ i = P N ψ k =1 δ k | k, k ′ i , where δ ≥ δ ≥ · · · > P N ψ k =1 δ k = 1, {| k i} N ψ k =1 and {| k ′ i} N ψ k =1 are orthonormal in H and K , respectively. As | ψ i is inseparable, we must have its Schmidtnumber N ψ ≥
2. Thus ρ = P N ψ k,l =1 δ k δ l | k, k ′ ih l, l ′ | . Up to unitary equivalence, we may assume XIAOFEI QI AND JINCHUAN HOU that {| k i} k =1 = {| i i} i =1 and {| k ′ i} k ′ =1 = {| j ′ i} j =1 . Then Tr( W ρ ) = Tr( − δ δ | ′ ih ′ | − δ δ | ′ ih ′ | ) = − δ δ <
0. Hence the statement (1) is true.For the statement (2), assume that there exist U ∈ U ( H ) and V ∈ U ( K ) such that Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) <
0. Then ρ is entangled. Moreover, ρ has a matrix representation ρ = X i,j,k,l α ijkl | U i i| V j ′ ih U k |h V l ′ | . Thus, one gets0 > Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) = Tr( W ( U † ⊗ V † ) ρ ( U ⊗ V ))= Tr( P i,j,k,l α ijkl ( | i| ′ ih |h ′ | − | i| ′ ih |h ′ | − | i| ′ ih |h ′ | + | i| ′ ih |h ′ | ) · ( U † ⊗ V † ) | U i i| V j ′ ih U k |h V l ′ | ( U ⊗ V ))= Tr( P i,j,k,l α ijkl ( | i| ′ ih |h ′ | − | i| ′ ih |h ′ | − | i| ′ ih |h ′ | + | i| ′ ih |h ′ | ) ·| i i| j ′ ih k |h l ′ | )= − α − α . Now let P and Q be the projectors from H and K onto the two dimensional subspaces spannedby {| i , | i} and {| ′ i , | ′ i} , respectively. ThenTr( P ⊗ Q )( U ⊗ V ) W ( U † ⊗ V † )( P ⊗ Q ) ρ ( P ⊗ Q )) = − α − α < , which implies that ( P ⊗ Q ) ρ ( P ⊗ Q ) is entangled. It follows from [9] that ρ is 1-distillable.The proof is complete. (cid:3) Detecting entanglement of states by their entries
In this section, we give a method of detecting entanglement of a state in any bipartitesystem only by some entries of its matrix representation.Let H and K be complex Hilbert spaces of any dimension with {| i i} dim Hi =1 and {| j ′ i} dim Kj =1 be orthonormal bases of them respectively. Denote E ij = E i,j = | i ih j | , which is an operatorfrom H into H . Let n ≤ min { dim H, dim K } be a positive integer. By [18, Remark 5.2], forany permutation κ of (1 , , · · · , n ), the linear map Φ κ : B ( H ) → B ( H ) defined byΦ κ ( A ) = ( n − n X i =1 E ii AE † ii + n X i =1 E i,κ ( i ) AE † i,κ ( i ) − ( n X i =1 E ii ) A ( n X i =1 E ii ) † (3 . A ∈ B ( H ) is a positive elementary operator that is not completely positive if κ = id.Then, for any unitary operators U and V on H , the map Φ U,Vκ defined byΦ
U,Vκ ( A ) = ( n − P ni =1 ( V E ii U ) A ( V E ii U ) † + P ni =1 ( V E i,κ ( i ) U ) A ( V E i,κ ( i ) U ) † − ( P ni =1 V E ii U ) A ( P ni =1 V E ii U ) † (3 . ETECTING ENTANGLEMENT BY ENTRIES 7 for every A ∈ B ( H ) is positive, too. Let ρ + = | ψ + ih ψ + | , where | ψ + i = 1 √ n ( | i| ′ i + | i| ′ i + · · · + | n i| n ′ i ) . Then, by Lemma 2.1, we get a class of entanglement witnesses of the form W U,Vκ = n (Φ U,Vκ ⊗ I ) ρ + = (Φ U,Vκ ( E ij )) . (3 . W U,Vκ is of finite rank because ρ + is.Particularly, for permutations π, σ of (1 , , · · · , n ), if U and V are the unitary operatorsdefined by U † | i i = | π ( i ) i , V | i i = | σ ( i ) i for i = 1 , , · · · n and U † | i i = | i i , V | i i = | i i for i > n ,then we haveΦ π,σκ ( A ) = Φ U,Vκ ( A ) = ( n − P ni =1 E σ ( i ) ,π ( i ) AE † σ ( i ) ,π ( i ) + P ni =1 E σ ( i ) ,π ( κ ( i )) AE † σ ( i ) ,π ( κ ( i )) − ( P ni =1 E σ ( i ) ,π ( i ) ) A ( P ni =1 E σ ( i ) ,π ( i ) ) † (3 . A . And correspondingly, we get entanglement witnesses of the concrete form W π,σκ = (Φ π,σκ ( E ij )) , (3 . π,σκ ( E ij ) = − E σ ( π − ( i )) ,σ ( π − ( j )) (3 . ≤ i = j ≤ n ,Φ π,σκ ( E ii ) = ( n − E σ ( π − ( i )) ,σ ( π − ( i )) + E σ ( κ − π − ( i )) ,σ ( κ − π − ( i )) (3 . ≤ i ≤ n , and Φ π,σκ ( E ij ) = 0 (3 . i > n or j > n .Thus we have proved the following result. Theorem 3.1.
Let H and K be complex Hilbert spaces of any dimension with {| i i} dim H ≤∞ i =1 and {| j ′ i} dim K ≤∞ j =1 be orthonormal bases of them respectively. For any positive integer ≤ n ≤ min { dim H, dim K } and any permutations κ, π, σ of (1 , , · · · , n ) with κ = id , the finite rankoperator W π,σκ defined by W π,σκ = ( n − P ni =1 | σπ − ( i ) , i ′ ih σπ − ( i ) , i ′ | + P ni =1 | σκ − π − ( i ) , i ′ ih σκ − π − ( i ) , i ′ |− P ≤ i = j ≤ n | σπ − ( i ) , i ′ ih σπ − ( j ) , j ′ | is an entanglement witness. XIAOFEI QI AND JINCHUAN HOU
Assume that dim H = dim K = n . By applying the witnesses W π,σκ in Theorem 3.1, weget a method of detecting the entanglement of states by the entries of their density matrix.Write the product basis of H ⊗ K in the order {| e i = | i| ′ i , | e i = | i| ′ i , · · · , | e n i = | n i| ′ i , | e n +1 i = | i| ′ i , · · · , | e n − i = | ( n − i| n ′ i , | e n i = | n i| n ′ i} . (3 . ρ ∈ S ( H ⊗ K ) has a matrix representation ρ = ( α kl ) n × n . Theorem 3.2.
Let ρ ∈ B ( H ⊗ K ) with dim H = dim K = n < ∞ be a state with thematrix representation ρ = ( α kl ) n × n with respect to the product basis in Eq.(3.9). If thereexist distinguished positive integers ( i − n < k i , h i ≤ in , i = 1 , , · · · , n such that n X i =1 k i = n X i =1 h i = 12 n ( n + 1) , (3 . and ( n − n X i =1 α k i k i + n X i =1 α h i h i − X ≤ i = j ≤ n α k i k j < , (3 . then ρ is entangled. Proof.
Eq.(3.10) implies that, there exist permutations π and σ such that ( k , k − n, · · · , k n − ( n − n ) = π (1 , , · · · , n ) and ( h , h − n, · · · , h n − ( n − n ) = σ (1 , , · · · , n ).It is clear that π ( i ) = σ ( i ) as k i = h i for every i = 1 , , · · · , n .For any permutations κ , π and σ , by Theorem 3.1, we haveTr( W π,σκ ρ ) = ( n − P ni =1 α σ ( π − ( i ))+( i − n,σ ( π − ( i ))+( i − n + P ni =1 α σ ( κ − π − ( i ))+( i − n,σ ( κ − π − ( i ))+( i − n − P ni = j α σ ( π − ( i ))+( i − n,σ ( π − ( j ))+( j − n . (3 . κ , π and σ so that π ( i ) = σ ( π − ( i )) and σ ( i ) = σ ( κ − π − ( i )) (3 . i , that is, π = σπ − and σ = σκ − π − . Take π = id. Then we get σ = π and σ = σκ − = π κ − . Thus, κ = σ − π , π = id and σ = π satisfy Eq.(3.13). With such κ, π and σ , by Eqs.(3.11) and (3.12), we haveTr( W π,σκ ρ ) = ( n − n X i =1 α k i k i + n X i =1 α h i h i − X ≤ i = j ≤ n α k i k j < . Hence, ρ is entangled with W π,σκ an entanglement witness for it. (cid:3) The general version of Theorem 3.2 is the following result, which is applicable for bipartitesystems of any dimension.
ETECTING ENTANGLEMENT BY ENTRIES 9
Theorem 3.3.
Let H and K be complex Hilbert spaces with {| i i} dim H ≤∞ i =1 and {| j ′ i} dim K ≤∞ j =1 be orthonormal bases of them respectively. Assume that ρ is a state on H ⊗ K and n ≤ min { dim H, dim K } is a positive integer. If there exist permutations π and σ of (1 , , · · · , n ) with π ( i ) = σ ( i ) for any i = 1 , , · · · , n such that ( n − n X i =1 h π ( i ) , i ′ | ρ | π ( i ) , i ′ i + n X i =1 h σ ( i ) , i ′ | ρ | σ ( i ) , i ′ i − X ≤ i = j ≤ n h π ( i ) , i ′ | ρ | π ( j ) , j ′ i < , (3 . then ρ is entangled. The idea of the proof of Theorem 3.3 is the same as that of Theorem 3.2 and we omit ithere.Theorems 3.2 and 3.3 tell us, some times we can detect the entanglement of a state bysuitably chosen n + n entries of its matrix representation with respect to some product basis,where n ≤ min { dim H, dim K } .To illustrate how to use Theorem 3.2 and Theorem 3.3 to detect entanglement of a state,we give some examples. Example 3.4.
Let q , q , q be nonnegative numbers with q + q + q = 1 and let a, b, c ∈ C with | a | ≤ q q , | b | ≤ q q , | c | ≤ q q . Let ρ be a state of 3 × ρ = 13 q q q q a a q q b q q q b q q c
00 0 0 0 0 0 ¯ c q q q q . (3 . ρ in Eq.(3.15) is a new kind of states, and ρ degenerates to the state as that in[18, Example 3.3] when a = b = c = 0.We claim that, if q < q or q < q , then ρ is entangled.In fact, choosing ( k , k , k ) = (1 , , h , h , h ) = (3 , ,
8) or (2 , , X i =1 α k i k i + X i =1 α h i h i − X ≤ i = j ≤ α k i k j = 13 (3 q + 3 q − q ) = q − q or X i =1 α k i k i + X i =1 α h i h i − X ≤ i = j ≤ α k i k j = 13 (3 q + 3 q − q ) = q − q . By Theorem 3.2, we see that ρ is entangled if q < q or q < q .It is clear that the partial transpose of ρ in Eq.(3.15) with respect to the first subsystem is ρ T = 13 q q ¯ a q a q q q q b q b q q
00 0 q q ¯ c
00 0 0 0 0 q c q
00 0 0 0 0 0 0 0 q . Particularly, if we take q = , q = , q = and a = b = c = , then, by what provedabove, we see that ρ is PPT entangled because its partial transpose has eigenvalues {
160 (8 ± √ , , , , , , , } that are all positive. Example 3.5.
Let ρ be a state in 4 × ρ = 14 q q q q q a a q q q q q
00 0 0 q q q q q q q q q b b q q c q q q q q q q q c q q d d q q q q q q q q q , (3 . ETECTING ENTANGLEMENT BY ENTRIES 11 where q i ≥ P i =1 q i = 1, | a | , | b | , | c | and | d | are all ≤ q q . ρ defined by Eq.(3.16)is also a new example, and when a = b = c = d = 0 we get states in [18, Example 4.4].We claim that, if q i < q for some i ∈ { , , } ; or if q i < q for some i ∈ { , , } , then ρ isentangled.In fact, we can take( k , k , k , k ) = (1 , , ,
16) and ( h , h , h , h ) = (2 , , , , or ( k , k , k , k ) = (1 , , ,
16) and ( h , h , h , h ) = (3 , , , , or ( k , k , k , k ) = (1 , , ,
16) and ( h , h , h , h ) = (4 , , , , or ( k , k , k , k ) = (4 , , ,
15) and ( h , h , h , h ) = (1 , , , , or ( k , k , k , k ) = (4 , , ,
15) and ( h , h , h , h ) = (2 , , , , or ( k , k , k , k ) = (4 , , ,
15) and ( h , h , h , h ) = (3 , , , . Then, it follows from the first three choices that2 X i =1 α k i k i + X i =1 α h i h i − X ≤ i = j ≤ α k i k j = q i − q with i = 2 , ,
4. Hence, by Theorem 3.2 we see that ρ is entangled if there exists some i ∈ { , , } such that q i < q . Similarly, by the last three choices one sees that ρ is entangledif there exists some i ∈ { , , } such that q i < q .The kind of states in Eq.(3.16) allow us give some new examples of entangled states thatcan not be recognized by PPT criterion and the realignment criterion. It is obvious that the partial transpose of ρ in Eq.(3.16) with respect to the first subsystem is ρ T = 14 q q q ¯ a q q a q q q q q q q q q q ¯ b q q q b q q q q q c q q q q q c q q
00 0 0 q q q ¯ d q q d q q q
00 0 q q and that the realignment of ρ is ρ R = 14 q q ¯ a a q q q q q q q q
00 0 0 q q q q q ¯ b b q q q q q q q q q q c q q c q q q q q q q q q q ¯ d d q q q . ETECTING ENTANGLEMENT BY ENTRIES 13
If we take q = , q = , q = q = and a = b = c = d = , ρ is PPT entangled because q < q and its partial transpose ρ T has eigenvalues { . , . , . , . , . , . , . , . , . , . , . , . , . , . , . , . } that are all positive. Moreover, the trace norm of the realignment ρ R of ρ is k ρ R k . = 0 . <
1. Hence, we get another example of entangled states that is PPT and cannot be detected bythe realignment criterion.It is not difficult to give some examples of applying Theorem 3.3 to infinite dimensionalsystems based on examples 3.4 and 3.5.4.
Conclusions
Let H and K be Hilbert spaces and let {| i i} dim H ≤∞ i =1 and {| j ′ i} dim K ≤∞ j =1 be any orthonormalbases of H and K , respectively. By the finite rank elementary operator criterion [13], a state ρ on H ⊗ K is entangled if and only if there exists a finite rank positive elementary operatorΦ : B ( H ) → B ( K ) that is not completely positive such that (Φ ⊗ I ) ρ is not positive. By thiscriterion and the finite rank positive elementary operators constructed in [18], we construct acollection of finite rank entanglement witnesses.By using these witnesses we obtain a rank-4 entanglement witness W = | i| ′ ih |h ′ | −| i| ′ ih |h ′ | − | i| ′ ih |h ′ | + | i| ′ ih |h ′ | which is universal for pure states, that is, for a purestate ρ , ρ is separable if and only if Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) ≥ U on H and V on K . In addition, for a mixed state ρ , if there exist unitary operators U on H and V on K such that Tr(( U ⊗ V ) W ( U † ⊗ V † ) ρ ) <
0, then ρ is entangled and 1-distillable.Another interesting result, maybe for the first time, gives a way of detecting the entangle-ment of a state in H ⊗ K by only a part entries of its density matrix. This method is simple,computable and practicable. Assume that ρ is a state on H ⊗ K and n ≤ min { dim H, dim K } is a positive integer. If there exist permutations π and σ of (1 , , · · · , n ) with π ( i ) = σ ( i ) forany i = 1 , , · · · , n such that( n − n X i =1 h π ( i ) , i ′ | ρ | π ( i ) , i ′ i + n X i =1 h σ ( i ) , i ′ | ρ | σ ( i ) , i ′ i − X ≤ i = j ≤ n h π ( i ) , i ′ | ρ | π ( j ) , j ′ i < , then ρ is entangled. Thus we provide a way of detecting the entanglement of a state by finitesuitably chosen entries of its matrix representation with respect to some product basis. Asan illustration how to use this method, some new examples of entangled states that can berecognized by this way are proposed, which also provides some new entangled states that cannot be detected by the PPT criterion and the realignment criterion. References [1] I. Bengtsson, K. Zyczkowski, Cambridge University Press, Cambridge, 2006.[2] D. Bruß, J. Math. Phys. 43 (2002) 4237.[3] K. Chen, L. Wu, Quant. Inf. Comput 3 (2003) 193.[4] D. Chru´ s ci´ n ski and A. Kossakowski, Open Systems and Inf. Dynamics 14 (2007) 275; D. Chru´ s ci´ n ski andA. Kossakowski, J. Phys. A: Math. Theor. 41 (2008) 145301.[5] J. Dixmier, Von Neumann Algebras, North-Holland Publishing Com., Amsterdan, New York, Oxford,1981.[6] Y. Guo, J. Hou, arXiv:1009.0116v1.[7] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223 (1996) 1.[8] R. Horodecki, M. Horodecki, P. Horodecki, Phys. Lett. A 222 (1996) 21.[9] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 80 (1998) 5239.[10] R. Horodecki, P. Horodecki, M. Horodecki, Rev. Mod. Phys. 81 (2009) 865.[11] J. Hou, Sci. in China (ser.A), 36(9) (1993), 1025-1035.[12] J. Hou, J. Operator Theory, 39 (1998), 43-58.[13] J. Hou, J. Phys. A: Math. Theor. 43 (2010) 385201; arXiv[quant-ph]: 1007.0560v1.[14] J. Hou, X. Qi, Phys. Rev. A 81 (2010) 062351.[15] M. A. Jafarizadeh, N. Behzadi, Y. Akbari, Eur. Phys. J. D 55 (2009) 197.[16] M. A. Nielsen, I. L. Chuang, Cambridge University Press, Cambridge, 2000.[17] A. Peres, Phys. Lett. A 202 (1996) 16.[18] X. Qi, J. Hou, Positive finite rank elementary operators and characterizing entanglement of states,arXiv:1008.3682v2[19] S. Simon, S. P. Rajagopalan, R. Simon, Pramana-Journal of Physics, 73(3) (2009) 471-483.[20] G. T´ o th, O. G¨ u hne, Phys. Rev. Lett. 94 (2005) 060501.[21] R. F. Werner, Phys. Rev. A 40 (1989) 4277.(Xiaofei Qi) Department of Mathematics, Shanxi University , Taiyuan 030006, P. R. of China;
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