Detection of genuine multipartite entanglement based on local sum uncertainty relations
aa r X i v : . [ qu a n t - ph ] F e b Detection of genuine multipartite entanglement based on localsum uncertainty relations
Jun Li ∗ and Lin Chen
1, 2, † School of Mathematical Sciences, Beihang University, Beijing 100191, China International Research Institute for Multidisciplinary Science,Beihang University, Beijing 100191, China
Abstract
Genuine multipartite entanglement (GME) offers more significant advantages in quantum infor-mation compared with entanglement. We propose a sufficient criterion for the detection of GMEbased on local sum uncertainty relations for chosen observables of subsystems. We apply the crite-rion to detect the GME properties of noisy n -partite W state when n = 3 , , n ranges from 4 to 6. Moreover, the criterionis also used to detect the genuine entanglement of 3-qutrit state. The result is stronger than thatbased on GME concurrence and fisher information. PACS numbers: ∗ Corresponding author: [email protected] † Corresponding author: [email protected] . INTRODUCTION Quantum entanglement [1–4] is a remarkable resource in the theory of quantum infor-mation, which is one of the most distinctive features of quantum theory as compared toclassical theory. Entangled states play the essential roles in quantum cryptography [5], tele-portation [6] and dense coding [7]. Genuine multipartite entanglement has more significantadvantages compared with entanglement. It is beneficial in various quantum communicationprotocols, such as secret sharing [8], extreme spin squeezing [9], quantum computing usingcluster states [10], high sensitivity in general metrology tasks [11], and multiparty quantumnetwork [12, 13]. To certify GME, Bell-like inequalities [14], various entanglement witness[15–20], and generalized concurrence for multi genuine entanglement [21–25] were derived.Some entanglement criteria for bipartite entangled state and multipartite non fully separablestates have been also proposed [26–28]. In particular, the entanglement criteria based onlocal sum uncertainty relations (LUR) have been proposed for bipartite systems [29] and tri-partite systems [30]. Although non-fully separable states contain genuinely entangled states,the criterion of GME based on LUR has not been studied.In this paper, we study the criterion of GME based on LUR and obtain the sufficientconditions in Theorems 3 and 5. First, for any quantum states, we show that we canalways find the lower bound of LUR. Second, we apply the sum of local observables to themultipartite biseparable state, and obtain the lower bound of LUR by using the method in[29, 30]. The converse negative process is the criterion of detecting GME. When we choosespin observables, the criterion is better than that in [31]. Third, we use the criterion detect3-qutrit state and noisy W state for n -qubit system ( n = 3 , , ,
6) and find that the criterionis strong than the exiting ones [24, 32]. Moreover, it can detect more noisy W states when n changes from 4 to 6, which is consistent with the fully-separability of noisy W states [33].In the rest of this paper, we will introduce the criterion of bipartite separability andtripartite fully separability based on LUR in Sec. II. We investigate the GME criterionbased on LUR in Sec. III, that is, Theorem 3 and Theorem 5. In Sec. IV, we apply thecriterion to noisy W state and 3-qutrit state to verify its effectiveness. We conclude in Sec.V. 2 I. PRELIMINARIES
A multipartite state that is not the convex sum of bipartite product states is said to begenuine multipartite entangled [34]. Take the tripartite system as an example. Let H dA , H dB , H dC denote d − dimensional Hilbert spaces of system A , B , C , respectively. A tripartite state ρ ∈ B ( H dA ⊗ H dB ⊗ H dC ) is biseparable if it can be expressed ρ BS = P X R η (1) R ρ R ⊗ ρ R + P X R ′ η (2) R ′ ρ R ′ ⊗ ρ R ′ + P X R ′′ η (3) R ′′ ρ R ′′ ⊗ ρ R ′′ , (1)where P k =1 P k = 1 , P k ≥ , and P R η ( k ) R = 1 . Here ρ Rmn is an arbitrary density operatorfor the subsystems m and n . Otherwise, ρ is called genuinely tripartite entangled. Thedefinition can be extended to genuine multipartite entangled states. Next, we introducethe criterion of bipartite separability and tripartite fully separability based on local sumuncertainty relations. They can also be detected for the criterion of GME.Consider the set of local observables { A k } and { B k } for systems A and B respectively.The sum uncertainty relations for arbitrary state ρ are as follows X k ∆ A k ≥ U A , X k ∆ B k ≥ U B , (2)where the non-negative constants U A and U B are independent of ρ and ∆ O k = h O k i−h O k i =Tr( O k ρ ) − Tr ( O k ρ ) with O ∈ { A, B } . An entanglement criterion based on local sumuncertainty relation was introduced for bipartite system AB . Lemma 1 [29] For bipartite separable state ρ AB , the following inequality holds, F ABρ AB := X k ∆( A k ⊗ I + I ⊗ B k ) − ( U A + U B + M AB ) ≥ , (3) where M AB = pP k ∆ A k − U A − pP k ∆ B k − U B . The violation of inequality implies en-tanglement of ρ AB . For tripartite system, we consider the set of local observables { A k } , { B k } and { C k } forsubsystem A , B and C , respectively. Suppose that the sum uncertainty relations for theseobservables have non-negative constants bounds U A , U B and U C independent of states, i. e. X k ∆ A k ≥ U A , X k ∆ B k ≥ U B , X k ∆ C k ≥ U C . (4)3ecently, the criterion (3) has been extended to a non-fully separability criterion for thetripartite system based on local sum uncertainty relations as follows. Lemma 2 [30] For any tripartite fully separable state ρ ABC , ρ ABC = X i p i ρ Ai ⊗ ρ Bi ⊗ ρ Ci , (5) the reduced states ρ AB , ρ AC and ρ BC are also separable. Therefore, ρ AB must satisfy theinequality (3) and also similar statements must hold for ρ AC and ρ BC . That is, F ABρ AB ≥ , F ACρ AC ≥ , F BCρ BC ≥ , (6) where F ACρ AC and F BCρ BC have similar definitions with F ABρ AB . So the following inequalities musthold simultaneously, F AB | Cρ ABC ≥ , F AC | Bρ ABC ≥ , F BC | Aρ ABC ≥ , (7) with F AB | Cρ ABC = F ρ ABC − ( U A + U B + U C + M AB + M ABC ) ,F AC | Bρ ABC = F ρ ABC − ( U A + U B + U C + M AC + M ACB ) ,F BC | Aρ ABC = F ρ ABC − ( U A + U B + U C + M BC + M BCA ) , where F ρ ABC = X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ , (8) and M ABC = q F ABρ AB − sX k ∆ C k − U C , and M ACB and M BCA have similar definitions. Violation of any inequality in Eqs. (6) and(7) implies non fully separability of ρ ABC . The method of Lemma 1 and 2 can be used to find the criterion of genuine entanglementin Theorem 3. It may be related to the lower bounds of quantum uncertainty relations forsingle system and bipartite system. In Eqs. (2) and (4), the lower bound of uncertaintyrelations U A , U B and U C are also independent of states. Moreover, some lower bound related4o states have also been studied. We know some well-known formula of uncertainty relationfor two observables [35],(∆ A ) + (∆ B ) ≥ ± i h ψ | [ A, B ] | ψ i + |h ψ | A + iB | ψ ⊥ i| , (∆ A ) + (∆ B ) ≥ |h ψ ⊥ A + B | A + B | ψ i| = 12 [∆( A + B )] , where h ψ | ψ ⊥ i = 0, | ψ ⊥ A + B i ∝ ( A − B − h A + B i ) | ψ i , and the sign on the right hand sideof the inequality takes +( − ) while i [ A, B ] is positive (negative). Let us mark the right sideof the inequality as U ρ . Furthermore, some multiple observables uncertainty relations wereproposed [36–38]. We consider the local observables { A k } and { B k } for systems A and B respectively, the multi-observables sum uncertainty relations are as follows X k ∆ A k ≥ U ρ A , X k ∆ B k ≥ U ρ B . (9)Similarly, for bipartite states, the multi-observables sum uncertainty relations are asfollows X k ∆( A k ⊗ I + I ⊗ B k ) ≥ U ρ AB , (10)where U ρ A , U ρ B , and U ρ AB can be obtained by the right side of multi-observables sum un-certainty relations in [36–38]. We will use the forementioned notions and facts in the nextsection. III. MAIN RESULTS
In this section, we investigate the genuine tripartite and multipartite entanglement basedon local sum uncertainty relations. We apply the observables in Eq. (8) to the tripartitebiseparable state, and obtain the lower bound of inequality by using Eqs. (9) and (10).Thus, we construct a sufficient condition for genuine tripartite entanglement in Theorem 3.Further, we extend this criterion to multipartite system in Theorem 5.5 . CRITERIA FOR GENUINE TRIPARTITE ENTANGLEMENT
Theorem 3
For a tripartite quantum state ρ ABC , let Eqs. (9) and (10) be satisfied. If ρ ABC is biseparable, then F ρ ABC ≥ min { U ρ A + U ρ BC + W ABC ,U ρ B + U ρ AC + W BAC ,U ρ C + U ρ AB + W CAB } (11) where F ρ ABC is defined in (8), and W ABC = sX k ∆( A k ) ρ A − U ρ A − sX k ∆( B k ⊗ I C + I B ⊗ C k ) ρ BC − U ρ BC , and W BAC and W CAB can be similarly defined.
Proof. If ρ ABC is biseparable, it can be written as Eq. (1) [31, 39, 40], ρ BS = P X R η R ρ R ⊗ ρ R + P X R ′ η R ′ ρ R ′ ⊗ ρ R ′ + P X R ′′ η R ′′ ρ R ′′ ⊗ ρ R ′′ with 0 ≤ P k ≤ P k P k = 1 and P R η kR = 1.For any mixture of type ρ mix = P R ≥ P R ρ R , the variance ∆ u satisfies [41]∆ u ≥ X R P R ∆ u R . (12)6ence for the biseparable state, X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ BS ≥ P X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ R + P X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ R ′ + P X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ R ′′ ≥ min { X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ R , X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ R ′ , X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ R ′′ } . (13)We can always choose as the lower bound the smallest value of P k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ ζ in (13). So the second inequality can be obtained using the fact that P k P k = 1.Then we consider P k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) that corresponds to thebipartition P R η R ρ R ⊗ ρ R , X k ∆( A k ⊗ I BC + B k ⊗ I AC + I AB ⊗ C k ) ρ R = X k {h [ A k ⊗ I BC + I A ⊗ ( B k ⊗ I C + I B ⊗ C k )] i−h A k ⊗ I BC + I A ⊗ ( B k ⊗ I C + I B ⊗ C k ) i } = X k ∆( A k ) ρ A + X k ∆( B k ⊗ I C + I B ⊗ C k ) ρ BC +2 X k [ h A k ⊗ ( B k ⊗ I C + I B ⊗ C k ) i −h A k ⊗ I BC ih I A ⊗ ( B k ⊗ I C + I B ⊗ C k ) i ] ≥ X k ∆( A k ) ρ A + X k ∆( B k ⊗ I C + I B ⊗ C k ) ρ BC − s [ X k ∆( A k ) ρ A − U ρ A ] · s [ X k ∆( B k ⊗ I C + I B ⊗ C k ) ρ BC − U ρ BC ]= U ρ A + U ρ BC + W ABC , (14)where W ABC = qP k ∆( A k ) ρ A − U ρ A − qP k ∆( B k ⊗ I C + I B ⊗ C k ) ρ BC − U ρ BC . The in-equality is due to Lemma 1 in [29]. 7ombining Eq. (13) and Eq. (14), we can obtain Eq. (11). In Eq. (11), the first term inthe bracket {} , namely, U ρ A + U ρ BC + W ABC is implied by the biseparable state P R η R ρ R ⊗ ρ R .Similarly, the second term is implied by the biseparable state P R η R ′ ρ R ′ ⊗ ρ R ′ , and the thirdterm is implied by the biseparable state P R η R ′′ ρ R ′′ ⊗ ρ R ′′ . Violation of the inequality (11)is sufficient to confirm genuine tripartite entanglement of ρ ABC . (cid:3) When we choose spin observables as the observables A , B , and C , the criteria in Theorem3 require only the statistics of a set of observables. In this sense, it is state independent,which is similar to [31]. In order to compare Theorem 3 with criterion 1 in [31], we considerthe sum of ∆ u and ∆ v where u = h J x, + h J x, + h J x, v = g J y, + g J y, + g J y, (15)and h k and g k ( k = 1 , ,
3) are real numbers. Here J x,k , J y,k , J z,k are the spin operators forsubsystem k , satisfying [ J x,k , J y,k ] = J z,k . Then F ρ ABC in Eq. (11) is equal to ∆ u + ∆ v when A = h J x, , B = h J x, , C = h J x, , A = g J y, , B = g J y, , C = g J y, and k = 2. This leads us to the following criterion, Corollary 4
Violation of the inequality ∆ u + ∆ v ≥ min {| g h h J z, i| + | g h h J z, i + g h h J z, i| + W , | g h h J z, i| + | g h h J z, i + g h h J z, i| + W , | g h h J z, i| + | g h h J z, i + g h h J z, i| + W } (16) is sufficient to confirm genuine tripartite entanglement. Here W = q ∆ ( h J x, ) + ∆ ( g J y, ) − | g h h J z, i| − q ∆ ( h J x, + h J x, ) + ∆ ( g J y, + g J y, ) − | g h h J z, i + g h h J z, i| ,W and W can be similarly defined. If W = W = W = 0, Eq. (16) is reduced to the result in [31], so the abovecriterion is better than criterion 1 in [31]. For the specific spin state, we can choose theoptimal values for h k , g k . 8 . CRITERIA FOR GENUINE MULTIPARTITE ENTANGLEMENT Now we extend the method in previous section used to derive criteria for genuine tripartiteentanglement to N -partite system. One can show that the number of possible bipartition is2 N − −
1. In order to investigate the criteria of genuine N -partite entanglement, we shouldconsider every bipartition. Here we generalize the criterion in Eq. (11) and Eq. (16) for N -partite system. We denote every bipartition by S r − S s , where S r and S s are the sets oftwo part in a specific bipartition. Theorem 5
If a N -partite quantum state ρ A A ...A N is biseparable, then F ρ A A ...AN ≥ min { U BS } , (17) where U BS is the set of the quantity U ρ kr + U ρ ks + W ρ kr | ks defined for each partition S r − S s , ρ k r and ρ k s are the states in set S r and S s respectively. Violation of the inequality (17) issufficient to confirm genuine N -partite entanglement. The proof of the inequality followsfrom the proof in Eq. (11). When the observables in Eq. (17) are spin observables, the following criterion can beobtained.
Corollary 6
Violation of the inequality ∆ u + ∆ v ≥ min { S B } (18) implies genuine N -partite entanglement. Where S B is the set of | Σ mk r =1 h k r g k r h J z,k r i| + | Σ nk s =1 h k s g k s h J z,k s i| + W ρ kr | ks defined for each partition S r − S s . When every W ρ kr | ks = 0 , theinequality is reduced to criterion 6 in [31]. For N = 4, there will be 2 − − − − − − −
34, 13 −
24, 14 −
23. Using them we obtain the criterion for genuine four-partiteentanglement. 9 orollary 7
If a four-partite quantum state is biseparable, then ∆ u + ∆ v ≥ min {| g h h J z, i| + | g h h J z, i + g h h J z, i + g h h J z, i| + W | , | g h h J z, i| + | g h h J z, i + g h h J z, i + g h h J z, i| + W | , | g h h J z, i| + | g h h J z, i + g h h J z, i + g h h J z, i| + W | , | g h h J z, i| + | g h h J z, i + g h h J z, i + g h h J z, i| + W | , | g h h J z, i + g h h J z, i| + | g h h J z, i + g h h J z, i| + W | , | g h h J z, i + g h h J z, i| + | g h h J z, i + g h h J z, i| + W | , | g h h J z, i + g h h J z, i| + | g h h J z, i + g h h J z, i| + W | } (19) where W | = q ∆ ( h J x, ) + ∆ ( g J y, ) − | g h h J z, i| − q ∆ ( h J x, + h J x, + h J x, ) + ∆ ( g J y, + g J y, + g J y, ) − | g h h J z, i + g h h J z, i + g h h J z, i| ,W | , W | , W | , W | , W | , and W | can be similarly defined. The violation of the inequality in Eq. (19) implies genuine four-partite entanglement. If W | = W | = W | = W | = W | = W | = W | = 0, Eq. (19) is reduced tothe criterion 8 in [31], so the above inequality is better than the result in [31]. IV. EXAMPLE
In this section, we illustrate the utility of the criteria by a few examples.
Example 8
Consider the n -qubit W state mixed with the white noise, ρ W n ( q ) = 1 − q n I + q | W n ih W n | , where ≤ q ≤ , | W n i = √ ( | . . . i + | . . . i + . . . + | . . . i ) and I is the n × n identity matrix. Set A = B = − C = σ x , A = B = − C = σ y , and A = B = C = σ z . The criterionin Theorem 3 is computed to be f ( q ) = − q − q + − ( q − q − q − q − q + ) ,which means the left side of (11) minus the right of that, as shown in FIG 1. Comparing10 .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q -2-1.5-1-0.500.511.5 f (q) FIG. 1: The abscissa and ordinate represent q and f ( q ), respectively. Below the abscissaaxis means that Theorem 3 can detect genuinely entangled state for 0 . ≤ q ≤ ρ W n ( q ) for three cases.1. When n = 4, we set A = B = C = − D = σ x , A = B = C = − D = σ y ,and A = B = C = D = σ z in Theorem 5. By calculation, we can obtain f ( q ) = − q + 3 − ( p − q + 2 q − p − q − q + 3) , which means the left side of (17) minusthe right side.2. when n = 5, we set A = B = C = − D = − E = σ x , A = B = C = − D = − E = σ y , and A = B = C = D = E = σ z . By calculation, we can obtain f ( q ) = − q + q + − ( q − q − q + − q − q + 2 q ) .3. When n = 6, we set A = B = C = − D = − E = − F = σ x , A = B = C = − D = − E = − F = σ y , and A = B = C = D = E = F = σ z . By calculation,we can obtain f ( q ) = − q + 6 q + − ( q − q + 4 q + − q − q ) .We describe the three cases in FIG 2. The same method can be used when n ≥ .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q -25-20-15-10-505 f n (q) FIG. 2: The abscissa and ordinate represent q and f n ( q ), respectively. The blue, red andblack lines represent f ( q ), f ( q ), and f ( q ), respectively. Below the abscissa axis meansthat Theorem 5 can detect genuinely entangled state for 0 . ≤ q ≤ n = 4.Similarly, we have 0 . ≤ q ≤ n = 5, and 0 . ≤ q ≤ n = 6. With theincrease of n , more genuinely entangled states can be detected. n q FIG. 3: The abscissa and ordinate represent critical value q and the number of system n ,respectively. Above the stars are entangled states that can be detected. With the increaseof n , the criterion can detect more states.12t is worth mentioning that [33] the noisy W state ρ W n ( q ) is fully separable if q ≤ n √ n − n if ≤ n ≤ nn +( n − n if n ≥ . (20)The condition is necessary and sufficient when n ≤
5. This is similar to the genuine entan-glement criterion in Theorems 3 and 5, that is, (20) can detect more states with the increaseof n . We describe these results in FIG 3.This criterion can not only detect the GME of qubit states, but also detect that of quritstates. Here is an example of the 3-qutrit state. Example 9
Consider a -qutrit state mixed with the white noise [24], ρ = 1 − x I + x | ϕ ih ϕ | , where ≤ x ≤ , | ϕ i = √ ( | i + | i + | i ) and I is the × identity matrix. Set A = − B = − C = J x , A = − B = − C = J y , and A = B = C = J z .Here J x , J y , J z are spin operators. The criterion in Theorem 3 is computed to be f ( x ) = − x − ( q − x − x + − q − x + x + ) , which means the left side of (11) minus theright of that, as shown in FIG 4. The criterion can detect GME better than the criterion in[24]. V. CONCLUSION
The detection of GME is a basic and important object in quantum theory. In view of thebipartite entanglement and tripartite non-fully separable criteria based on LUR, we havestudied the GME based on LUR. We have obtained an effective criterion to detecting theGME for tripartite system, which be extended to multipartite system. Comparing withsome existing criteria, the criterion can detect more genuinely entangled states by theo-retical analysis and numerical examples. Also, we found the relation of n and genuinelyentanglement for n - qubit noisy W state. The method used in this paper can also be gen-eralized to arbitrary multipartite qudit systems. It would be also worthwhile to investigatethe k -separability of multipartite systems. 13 .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x -4-3-2-10123 f(x) FIG. 4: The abscissa and ordinate represent x and f ( x ), respectively. Below the abscissaaxis means that Theorem 3 can detect genuinely entangled state for 0 . ≤ x ≤ Acknowledgments
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