aa r X i v : . [ qu a n t - ph ] M a y Generalized W-Class State and its MonogamyRelation
Jeong San Kim and Barry C. Sanders
Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4,CanadaE-mail: [email protected] , [email protected] Abstract.
We generalize the W class of states from n qubits to n qudits and provethat their entanglement is fully characterized by their partial entanglements even forthe case of the mixture that consists of a W-class state and a product state | i ⊗ n .PACS numbers: 03.67.-a, 03.65.Ud, eneralized W-Class State and its Monogamy Relation
1. Introduction
Quantum entanglement is one of the most non-classical features in quantum mechanicsand provides us a lot of applications. Due to its variety of usages, much attentionhas been shown, so far, for the quantification of entanglement and, thus, the conceptof entanglement measure has been naturally arisen.
Concurrence [1] is one of the mostwell-known bipartite entanglement measures with an explicit formula for 2-qubit systemwhile there does not exist any analytic way of evaluation yet for the general case ofhigher-dimensional mixed states. Another entanglement measure that can be consideredas a dual to concurrence is the concurrence of assistance (CoA) [2], and this can beinterpreted as the maximal average concurrence that two parties in the bipartite systemcan locally prepare with the help of the third party who has the purification of thebipartite system.In multipartite quantum system, there can be several inequivalent types ofentanglement among the subsystems and the amount of entanglement with differenttypes might not be directly comparable to each other. For 3-qubit pure states, it isknown that there are two inequivalent classes of genuine tripartite entangled states [3];one is the Greenberger-Horne-Zeilinger (GHZ) class [4], and the other one is the W-class [3]. This can be characterized by means of stochastic local operations and classicalcommunication (SLOCC), that is, the conversion of the states in a same class can beachieved through local operation and classical communication with non-zero probability.Another way to characterize the different types of entanglement distribution is byusing monogamy relation of entanglement. Unlike classical correlations, the amount ofentanglement that can be shared between any of two parties and the others is stronglyconstrained by the entanglement between two parties.In 3-qubit systems, Coffman, Kundu and Wootters (CKW) [5] first introduced amonogamy inequality in terms of a bipartite entanglement measure, concurrence, as C A ( BC ) ≥ C AB + C AC , (1)where C A ( BC ) = C ( | ψ i A ( BC ) ) is the concurrence of a 3-qubit state | ψ i A ( BC ) for a bipartitecut of subsystems between A and BC and C AB = C ( ρ AB ). Its generalization into n -qubitcase was also proved [6] and, symmetrically, its dual inequality in terms of the CoA for3-qubit states, C A ( BC ) ≤ ( C aAB ) + ( C aAC ) , (2)and its generalization into n -qubit cases have been shown in [7, 8].In 3-qubit system, two inequivalent classes of genuine tripartite entangled states,GHZ and W classes, show extreme difference in terms of CKW inequality and its dualone. In other words, CKW inequality is saturated by W-class states, while it becomesthe most strict inequality with the states in GHZ class. Here, W-class states are of ourspecial interest, since the saturation of the inequality implies that a genuine tripartiteentanglement can have a complete characterization by means of the bipartite ones insideit. In other words, the entanglement A - BC , measured by concurrence, is completely eneralized W-Class State and its Monogamy Relation A - B and A - C . For the case of n -qubit W-classstates, generalized CKW and its dual inequalities are also saturated, and thus the sameinterpretation can be applied.In this paper, we generalized the concept of W-class states from n qubits to n quditsand show that their entanglement is fully characterized by their partial entanglements.We also prove that the complete characterization of the global entanglement in termsof its partial entanglement is possible even for the case of the mixture consisting of aW-class state and a product state | i ⊗ n .This paper is organized as follows. In Section 2 we recall the the monogamy relationof n -qubit W-class states in terms of CKW and its dual inequalities. In Section 3.1,we provide more general monogamy relations of n -qubit W-class states with respect toarbitrary partitions by investigating the structure of n -qubit W-class states. In Section3.2 we generalize the concept of W-class states to arbitrary n -qudit system as well asits monogamy relations with respect to arbitrary partitions. In Section 4, we considerthe class of multipartite mixed states that is a mixture of a W-class state and a productstate. We also provide its monogamy relations in terms of its partial entanglement bystudying its structural properties. In Section 5 we summarize our results.
2. Monogamy Relation of n -qubit W-class States For any bipartite pure state | φ i AB ∈ C d ⊗ C d ′ , concurrence of | φ i AB is defined as C ( | φ i AB ) = q − tr ρ A ) , (3)where ρ A = tr B | φ i AB h φ | . For any mixed state ρ AB , it is defined as C ( ρ AB ) = min X k p k C ( | φ k i AB ) , (4)where the minimum is taken over its all possible pure state decompositions, ρ AB = P k p k | φ k i AB h φ k | .Another entanglement measure that can be considered as a dual to concurrence isCoA [2], which is defined as C a ( ρ AB ) = max X k p k C ( | φ k i AB ) , (5)where the maximum is taken over all possible decompositions of ρ AB .For 3-qubit W-class states | W i ABC = a | i ABC + b | i ABC + c | i ABC , (6)with | a | + | b | + | c | = 1, CKW and its dual inequalities (1), (2) are saturated, that is, C A ( BC ) = C AB + C AC , C AB = C aAB , C AC = C aAC . (7)In other words, the entanglement of W-class states between one party and the restcan have a complete characterization in terms of the partial entanglement that is thebipartite entanglement between one party and each of the rest parties. eneralized W-Class State and its Monogamy Relation n -qubit systems A ⊗ · · · ⊗ A n where A i ∼ = C for i = 1 , . . . , n , CKW and itsdual inequalities can be generalized as [6, 8] C A ( A ··· A n ) ≥ C A A + · · · + C A A n , C A ( A ··· A n ) ≤ ( C aA A ) + · · · + ( C aA A n ) . (8)For n -qubit W-class states, | W i A ··· A n = a | · · · i A ··· A n + · · · + a n | · · · i A ··· A n , n X i =1 | a i | = 1 , (9)the inequalities (8) are saturated; that is, C A ( A ··· A n ) = C A A + · · · + C A A n , C A A i = C aA A i , i = 2 , . . . , n. (10)In fact, there can be several ways to show Equation (10). Since any two-qubitreduced density matrix ρ A A i of | W i A ··· A n can have analytic formulas for concurrenceand concurrence of assistance [1, 2], one of the ways to check Equalities (10) is usingthe formulas in [1, 2]. However, there is no formula for the general case of a bipartitequantum state with arbitrary dimension, so here we use the method of consideringall possible decompositions of the bipartite mixed state ρ A A i so that the optimizationprocess in (4, 5) can also be used later in this paper for higher-dimensional quantumsystems.Let us first consider C A ( A ··· A n ) = 2(1 − tr ρ A ) where ρ A = tr A ··· A n ( | W i A ··· A n h W | ).Since ρ A = | a | | i A h | + ( P ni =2 | a i | ) | i A h | , we can easily see C A ( A ··· A n ) = 4 | a | ( n X i =2 | a i | ) . (11)For C A A i with i ∈ { , . . . , n } , let us consider ρ A A i = tr A ··· b A i ··· A n ( | W i A ··· A n h W | )= ( a | i + a i | i ) A A i ( a ∗ h | + a ∗ i h | )+ ( | a | + · · · + | b a i | + · · · + | a n | ) | i A A i h | , (12)where A · · · b A i · · · A n = A · · · A i − A i +1 · · · A n and ( | a | + · · · + | b a i | + · · · + | a n | ) =( | a | + · · · + | a i − | + | a i +1 | + · · · + | a n | ). Now, let | ˜ x i A A i = a | i A A i + a i | i A A i and | ˜ y i A A i = p | a | + · · · + | b a i | + · · · | a n | | i A A i , where | ˜ x i A A i and | ˜ y i A A i areunnormalized states of the subsystems A A i . Then, by the Hughston-Jozsa-Wootters(HJW) theorem [9], any other decomposition of ρ A A i = P rh =1 | ˜ φ h i A A i h ˜ φ h | with size r ≥ r × r unitary matrix ( u hl ) where | ˜ φ h i A A i = u h | ˜ x i A A i + u h | ˜ y i A A i . (13)Let ρ A A i = P h p h | φ h i A A i h φ h | where √ p h | φ h i A A i = | ˜ φ h i A A i and p h = |h ˜ φ h | ˜ φ h i| ; then,by straightforward calculation, we can easily see that the average concurrence of the eneralized W-Class State and its Monogamy Relation ρ A A i = P h p h | φ h i A A i h φ h | is, r X h =1 p h C ( | φ h i A A i ) = r X h =1 p h q − tr( ρ hA ) )= 2 | a || a i | , (14)where ρ hA = tr A i ( | φ h i A A i h φ h | ). In other words, the average concurrence remains thesame for any pure state decomposition of ρ A A i , and thus, we can have,min X h p h C ( | φ h i A A i ) = max X h p h C ( | φ h i A A i ) (15)where the maximum and minimum are taken over all possible decomposition of ρ A A .This implies C A A i = C aA A i = 2 | a || a i | , (16)which leads to (10).
3. General Monogamy Relation of Multipartite W-Class States
In Section 2, we have seen the monogamy relations of n -qubit W-class states betweensubsystems in terms of CKW and its dual inequalities. Here, we investigate the structureof W-class states in n -qubit system by considering arbitrary partitions of subsystems andderive more general concept of monogamy relations between the parties. Furthermore,we generalize the concept of W-class states to arbitrary n -qudit system and also considerthe monogamy relations in terms of arbitrary partitions. For n -qubit W-class states, | W i A ··· A n = a | · · · i A ··· A n + · · · + a n − | · · · i A ··· A n , n − X i =0 | a i | = 1 , (17)let us consider a partition P = { P , . . . , P m } , m ≤ n for the set of subsystems S = { A , . . . , A n } where each of P s contains several qubits, that is, P s = { A i j } , j = 1 , . . . , n s , X s n s = n,P s ∩ P t = ∅ for s = t, [ s P s = S. (18)For simplicity, let us first consider the case when P = { P , P , P } with | P s | = n s where s ∈ { , , } . Figure 1 shows an example of partition for a set of subsystems S = { A , . . . A } where P = { A , A } , P = { A , A , A } and P = { A , A } .Without loss of generality, we may assume P = { A , . . . , A n } , P = { A n +1 , . . . , A n + n } and P = { A n + n +1 , . . . , A n } ; otherwise we can have some properreordering of the subsystems. eneralized W-Class State and its Monogamy Relation Figure 1.
A partition for the set of subsystem S = { A , . . . A } where P = { A , A } , P = { A , A , A } and P = { A , A } . Here, we use the representation, that is, | · · · · · · i P s = | i i P s (19)where | · · · · · · i P s is an n s -qubit product state of the party P s whose i th subsystemfrom the right is 1 and 0 elsewhere. Then (9) can be rewritten as, | W i P P P = | ˜ x i P | ~ i P | ~ i P + | ~ i P | ˜ y i P | ~ i P + | ~ i P | ~ i P | ˜ z i P , (20)where | ~ i P s = | · · · i P s and | ˜ x i P , | ˜ y i P and | ˜ z i P are unnormalized states in P , P and P respectively such that | ˜ x i P = P n − j =0 a n + n + j | j i P , | ˜ y i P = P n − k =0 a n + k | k i P and | ˜ z i P = P n − l =0 a l | l i P .Here, we note that | ˜ x i P , | ˜ y i P and | ˜ z i P are unnormalized W-class states of theparties P , P and P respectively. Thus any n -qubit W-class state can have this typeof representation, that is, | W i P P P = √ q | W i P | ~ i P | ~ i P + √ q | ~ i P | W i P | ~ i P + √ q | ~ i P | ~ i P | W i P , (21)where q = P n − j =0 | a n + n + j | , q = P n − k =0 | a n + k | and q = P n − l =0 | a l | with thenomalization condition, q + q + q = 1.If we just rename | W i P s = | i P s and | ~ i P s = | i P s , then | i P s and | i P s areorthogonal to each other, and (21) can be rewritten as, | W i P P P = √ q | i P | i P | i P + √ q | i P | i P | i P + √ q | i P | i P | i P , (22)which is a tripartite W-class state in ( C ) n ⊗ ( C ) n ⊗ ( C ) n quantum systems.Similarly, for an arbitrary partition P = { P , . . . , P m } of size m , we can have | W i P ··· P m = √ q | W i P | i P · · · | i P m + √ q | i P | W i P · · · | i P + · · · + √ q m | i P | i P · · · | W i P m eneralized W-Class State and its Monogamy Relation √ q | i P | i P · · · | i P m + √ q | i P | i P · · · | i P + · · · + √ q m | i P | i P · · · | i P m . (23)For any partition P = { P , . . . , P m } of the set of subsystems S = { A , . . . , A n } , the n -qubit W-class state (17) can be also considered as an m -partite W-class state withdifferent names of the basis, and thus we can have following lemma. Lemma 1.
For any n -qubit W-class states | W i A ··· A n and a partition P = { P , . . . , P m } of the set of subsystems S = { A , . . . , A n } , C P s ( P ··· b P s ··· P m ) = X k = s C P s P k = X k = s ( C aP s P k ) , (24) and C P s P k = ( C aP s P k ) , (25) for all k = s .3.2. n -qudit W-Class States Now, we generalize the concept of W-class states to arbitrary n -qudit systems withsimilar properties of monogamy relations as in Lemma 1.Let us consider a class of n -qudit quantum states, (cid:12)(cid:12) W dn (cid:11) A ··· A n = d − X i =1 ( a i | i · · · i A ··· A n + · · · + a ni | · · · i i A ··· A n ) , (26)with P ns =1 P d − i =1 | a si | = 1. In the case d = 2, Equation (26) is reduced to n -qubitW-class states in (17).Let | ˜ ψ s i A s = P d − i =1 a si | i i A s be an unnormalized state of subsystem A s for s ∈{ , . . . , n } and |h ˜ ψ s | ˜ ψ s i| = P d − i =1 | a si | = α s , then the normalization condition can berephrased as P ns =1 α s = 1.Now, we will see that the class of states (26) has similar properties as in Equation(10); that is, C A ( A ··· A n ) = C A A + · · · + C A A n , C A A t = C aA A t , t = 2 , . . . , n. (27)In other words, CKW and its dual inequalities are saturated by the class ofstates (26). To see this, let us first consider C A ( A ··· A n ) = 2(1 − tr ρ A ) where ρ A = tr A ··· A n ( (cid:12)(cid:12) W dn (cid:11) A ··· A n (cid:10) W dn (cid:12)(cid:12) ). Since ρ A = | ˜ ψ i A h ˜ ψ | + P ns =2 α s | i A h | , C A ( A ··· A n ) = 4 α ( n X s =2 α s ) . (28)For C A A t with t ∈ { , . . . , n } , let us consider ρ A A t = tr A ··· b A t ··· A n ( (cid:12)(cid:12) W dn (cid:11) A ··· A n (cid:10) W dn (cid:12)(cid:12) ) eneralized W-Class State and its Monogamy Relation d − X i,j =1 ( a i | i i + a ti | i i ) A A t ( a ∗ j h j | + a ∗ tj h j | )+ ( α + · · · + b α t + · · · + α n ) | i A A t h | , (29)where A · · · b A i · · · A n = A · · · A i − A i +1 · · · A n and ( α + · · · + b α t + · · · + α n ) =( α + · · · + α t − + α t +1 + · · · + α n ).Now, let us denote | ˜ x i A A t = P d − i =1 ( a i | i i + a ti | i i ) A A t and | ˜ y i A A t = p α + · · · + b α t + · · · + α n | i A A t , where | ˜ x i A A t and | ˜ y i A A t are unnormalized statesof the subsystems A A t . Then, by the HJW theorem, any other decomposition of ρ A A t = P rh =1 | ˜ φ h i A A t h ˜ φ h | with size r ≥ r × r unitary matrix( u hl ) where | ˜ φ h i A A t = u h | ˜ x i A A t + u h | ˜ y i A A t . (30)Let ρ A A t = P h p h | φ h i A A t h φ h | where √ p h | φ h i A A t = | ˜ φ h i A A t and p h = |h ˜ φ h | ˜ φ h i| ; then,after a tedious calculation, we can see that the average concurrence of the pure statedecomposition ρ A A t = P h p h | φ h i A A t h φ h | is r X h =1 p h C ( | φ h i A A t ) = r X h =1 p h q − tr( ρ hA ) )= 2 α α t , (31)where ρ hA = tr A t ( | φ h i A A t h φ h | ). Similar to the n -qubit case, the average concurrenceremains the same for any pure state decomposition of ρ A A t , and thus, we can have,min X h p h C ( | φ h i A A t ) = max X h p h C ( | φ h i A A t ) , (32)where the maximum and minimum are taken over all possible decomposition of ρ A A t .This implies C A A t = C aA A t = 2 α α t , (33)which leads to (27).Now, let us consider a partition P = { P , . . . , P m } , m ≤ n where each of P s with s ∈{ , . . . , m } contains several qudits such that | P s | = n s and n + · · · + n m = n . Withoutloss of generality, we may assume P = { A , . . . , A n } , P = { A n +1 , . . . , A n + n } , . . . , P m = { A n + ··· + n m − +1 , . . . , A n } ; otherwise we can have some proper reordering of thesubsystems. For each party P s of the partition P , let | ˜ x si i P s = a ( n + ··· + n s − +1) i | i · · · i P s + a ( n + ··· + n s − +2) i | i · · · i P s + · · · + a ( n + ··· + n s ) i | · · · i i P s , (34)then | ˜ x si i P s is an unnormalized state of the party P s and (26) can be rewritten as (cid:12)(cid:12) W dn (cid:11) P ··· P m = d − X i =1 (cid:16) | ˜ x i i P ⊗ | ~ i P ⊗ · · · ⊗ | ~ i P m + · · · + | ~ i P ⊗ | ~ i P ⊗ · · · ⊗ | ˜ x mi i P m (cid:17) , (35) eneralized W-Class State and its Monogamy Relation | ~ i P s = | · · · i P s . If we consider the normalized state | x si i P s = √ q si | ˜ x si i P s with |h ˜ x si | ˜ x si i| = q si and rename | x si i P s = | i i P s and | ~ i P s = | i P s , then (35) can be representedas (cid:12)(cid:12) W dn (cid:11) P ··· P m = d − X i =1 (cid:0) √ q i | i i P ⊗ | i P ⊗ · · · ⊗ | i P m + · · · + √ q mi | i P ⊗ | i P ⊗ · · · ⊗ | i i P m (cid:1) , (36)which is an m -partite generalized W-class state and, thus, we can have the second lemmawhich incorporates Lemma 1. Lemma 2.
For any n -qudit generalized W-class states | W i A ··· A n in (26) and a partition P = { P , . . . , P m } for the set of subsystems S = { A , . . . , A n } , C P s ( P ··· ˆ P s ··· P m ) = X k = s C P s P k = X k = s ( C aP s P k ) , (37) and C P s P k = ( C aP s P k ) , (38) for all k = s . Furthermore, if we consider the state | ˜ x s i P s of the partition P s such that | ˜ x s i P s = d − X i =1 ( a ( n + ··· + n s − +1) i | i · · · i P s + a ( n + ··· + n s − +2) i | i · · · i P s + · · · + a ( n + ··· + n s ) i | · · · i i P s ) , (39)then | ˜ x s i P s is an unnormalized W-class state of n s -qudit system. Let | ˜ x s i P s = √ q s (cid:12)(cid:12) W dn s (cid:11) P s with q s = P d − i =1 q si ; then (35) can be also represented as | W i P ··· P m = √ q (cid:12)(cid:12) W dn (cid:11) P ⊗ | i P ⊗ · · · ⊗ | i P m + · · · + √ q m | i P ⊗ | i P ⊗ · · · ⊗ (cid:12)(cid:12) W dn m (cid:11) P m , (40)which is the same type of representation as the n -qubit W-class states in (23). n -qudit Mixed States Concurrence is one of most well-known entanglement measures for bipartite quantumsystem with an explicit formula for 2-qubit system. However, in higher-dimensionalquantum system, there does not exist any explicit way of evaluation yet for mixedstate. The lack of an analytic evaluation technique is mostly due to the difficulty ofoptimization problem which is minimizing over all possible pure state decompositionsof the given mixed state. The difficulty of optimization problem for its dual, CoA, alsoarises in forms of maximization.Recently, the optimal pure state decomposition for the mixture of generalized GHZand W states in 3-qubit system was found [10, 11], and the optimal decomposition, here,was assured by the saturation of CKW inequality. In [12], another monogamy relation eneralized W-Class State and its Monogamy Relation n -qubit W-class stateand a product state | i ⊗ n .Here, we investigate the structure of the mixed states that consists of n -qubit W-class states and the product state | i ⊗ n and provide an analytic proof for its saturationof CKW and its dual in equality. This saturation of the inequalities are also true for anypartition of the set of subsystems. Noting that the average of squared concurrences isalways an upper bound of the square of average concurrences, the result in [12] becomesa special case of the result here.For any n -qudit W-class state in (26), if we consider the reduced density matrix ofthe subsystem { A s , . . . , A s l } for 2 ≤ l ≤ n −
1, we can easily check that it is always amixture of some l -qudit W-class state and a product state | · · · i , that is, ρ A s ··· A sl = p (cid:12)(cid:12) W dl (cid:11) A s ··· A sl (cid:10) W dl (cid:12)(cid:12) + (1 − p ) | · · · i A s ··· A sl h · · · | , (41)for some 0 ≤ p ≤ n -qudit W-class state in (26) and aproduct state | · · · i A ··· A n , ρ A ··· A n = p (cid:12)(cid:12) W dn (cid:11) A ··· A n (cid:10) W dn (cid:12)(cid:12) + (1 − p ) | · · · i A ...A n h · · · | . (42)Since ρ A ··· A n is an operator of rank two, we can always have a purification | ψ i A ··· A n A n +1 ∈ ( C d ) ⊗ n +1 of ρ A ··· A n such that, | ψ i A ··· A n A n +1 = √ p (cid:12)(cid:12) W dn (cid:11) A ··· A n ⊗ | i A n +1 + p − p | · · · i A ··· A n ⊗ | x i A n +1 , (43)where | x i A n +1 = P d − i a n +1 i | i i A n +1 is a 1-qudit quantum state of A n +1 which isorthogonal to | i A n +1 where P d − i | a n +1 i | = 1. Now, we can easily see that (43) can berewritten as | ψ i A ··· A n +1 = d − X i =1 [ √ p ( a i | i · · · i A ··· A n +1 + · · · + a ni | · · · i i ) A ··· A n +1 + p − pa n +1 i | · · · i i A ··· A n +1 ] , (44)and this is an ( n + 1)-qudit W-class state.In other words, the reduced density matrix of a generalized W-class state ontoany subsystem is a mixture of a W-class state and a product state. Furthermore, anymixture of a W-class state and a product state | · · · i can be considered as a reduceddensity matrix of some W-class state in a quantum system with a larger number ofparties.Then, we can have following theorem. Theorem 1.
Let ρ A ··· A n be a n -qudit mixed state in B (( C d ) ⊗ n ) , which is a mixture of ageneralized W-class state | W i A ··· A n and a product state | · · · i A ··· A n with any weightingfactor ≤ p ≤ such that ρ A ··· A n = p (cid:12)(cid:12) W dn (cid:11) A ··· A n (cid:10) W dn (cid:12)(cid:12) + (1 − p ) | · · · i A ··· A n h · · · | . (45) eneralized W-Class State and its Monogamy Relation Then C A ( A ··· A n ) = C A A + · · · + C A A n , C A A t = C aA A t , t = 2 , . . . , n. (46) Furthermore, for any partition P = { P , . . . , P m } of the set of subsystems S = { A , . . . , A n } , C P s ( P ··· ˆ P s ··· P m ) = X k = s C P s P k = X k = s ( C aP s P k ) , (47) and C P s P k = ( C aP s P k ) , (48) for all k = s .Proof. Let (44) be a purification of ρ A ··· A n in ( C d ) ⊗ n +1 , then, by Lemma 2, we can have, C A [( A ··· A n ) A n +1 ] = C A ( A ··· A n ) + C A A n +1 = n X s =2 C A A s + C A A n +1 , (49)and, thus, C ( ρ A ( A ··· A n ) ) = C A ( A ··· A n ) = n X s =2 C A A s . (50)Furthermore, for any partition P = { P , . . . , P m } of the set of subsystems S = { A , . . . , A n } , C P i [( P ··· ˆ P i ··· P m ) A n +1 ] = C P i ( P ··· ˆ P i ··· P m ) + C P i A n +1 = X s = i C P i P s + C P i A n +1 , (51)and we can have C P i ( P ··· ˆ P i ··· P m ) = X s = i C P i P s . (52)Note that Theorem 1 encapsulates the first two lemmas. In other words, if p = 1,Theorem 1 deals with the monogamy relations of n -qudit W-class states that werepresented in Lemma 2, and for the case p = 1 , d = 2, it is about the monogamyrelations of n -qubit W-class states that were presented in Lemma 1. Thus, Theorem 1deals with the most general case of high-dimensional multipartite mixed states, so far,whose entanglement is completely characterized by their partial entanglements. eneralized W-Class State and its Monogamy Relation
5. Conclusions
We have investigated general monogamy relations for W-class states by considering thestructure of W-class states in terms of arbitrary partitions of subsystems. We havegeneralized the concept of W-class states from n -qubit systems to arbitrary n -quditsystems and have shown that their entanglement is completely characterized by theirpartial entanglements in terms of any partition of the set of subsystems. The structuralproperties of a class of mixed states that consist of a n -qubit W-class state and a productstate | i ⊗ n have been shown, and we have proved that the monogamy relations of themixture have the same characterization as the case of W-class states.The structural properties of W-class states by considering an arbitrary partitionof the set of subsystems show that the structure of W-class states is inherent withrespect to any arbitrary partition by the choice of a proper basis. Our novel techniquecan also be helpful to study the structural properties of other genuine multipartiteentangled states, such as n -qudit GHZ-class states and cluster states [13]. Noting theimportance of the study on high-dimensional multipartite entanglement, although therehave not been much so far, our results can provide a rich reference for future work onthe characterization of multipartite entanglement. Acknowledgments
This work was supported by the Korea Research Foundation Grant funded by theKorean Government (MOEHRD) (KRF-2007-357-C00008), Alberta’s informatics Circleof Research Excellence (iCORE) and a CIFAR Associateship.
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