Applie Mathematics & Information Sciences 4(3) (2010), 281 288 An International Journal c 2010 Dixie W Publishing Corporation, U. S. A. Monogamy an Polygamy of Entanglement in Multipartite Quantum Systems Barry C. Saners 1 an Jeong San Kim 2 1 Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canaa Email Aress: sanersb@ucalgary.ca 2 Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canaa Email Aress: jekim@ucalgary.ca Receive November 11, 2009; Revise Jamuary 2, 2010 We summarize our recent results about monogamy an polygamy of entanglement in multipartite quantum systems. We also consier convex-roof extene negativity as a strong caniate for bipartite entanglement measure for general monogamy an polygamy relations for multipartite quantum systems. Keywors: Multipartite quantum entanglement, monogamy, polygamy, convex-roof extene negativity. 1 Introuction One istinct property of quantum entanglement from other classical correlations is that multipartite entanglements cannot be freely share among the parties: If two parties in a multi-party system share a maximally entangle state, then they cannot share any entanglement with the rest. This is known as monogamy of entanglement (MoE) [1], which is a key ingreient for secure quantum cryptography [2, 3], an it also plays an important role in conense-matter physics such as the N-representability problem for fermions [4]. Whereas MoE is the restricte sharability of multipartite entanglement, quantifying entanglement itself is about bipartite entanglement among the parties in multipartite systems. Thus, it is important an necessary to have a proper way of quantifying bipartite entanglement for a goo escription of the multipartite entanglement monogamy. For this reason, certain criteria of bipartite entanglement measure were recently propose for a goo escription of the monogamy nature of entanglement in multipartite quantum systems [5]: 1. Monotonicity: the property that ensures entanglement cannot be increase uner local operations an classical communications.
282 Barry C. Saners an Jeong San Kim 2. Separability: capability of istinguishing entanglement from separability. 3. Monogamy: upper boun on a sum of bipartite entanglement measures thereby showing that bipartite sharing of entanglement is boune. Using concurrence [6] as a bipartite entanglement measure, MoE was first shown to have a mathematical characterization in three-qubit systems as an inequality [1], an it was generalize for arbitrary multi-qubit systems [7]. As a ual concept of MoE, a polygamy inequality of multi-qubit entanglement was also establishe later in terms of Concurrence of Assistance (CoA). However, multi-qubit monogamy inequality using concurrence is known to fail in its generalization for higher-imensional quantum systems [8, 9], an this exposes the importance of having proper entanglement measure for general MoE in higher-imensional quantum systems. 2 Concurrence-Base Monogamy an Polygamy Inequalities For any bipartite pure state φ AB, its concurrence is efine as [6] C( φ AB )= 2(1 trρ 2 A ), (2.1) where ρ A =tr B ( φ AB φ ), an for any mixe state ρ AB, it is efine as C(ρ AB ) = min k p k C( φ k AB ), (2.2) where the minimum is taken over all possible pure state ecompositions, ρ AB = k p k φ k AB φ k. As a ual quantity to concurrence, CoA is efine as C a (ρ AB ) = max k p k C( φ k AB ), (2.3) where the maximum is taken over all possible pure state ecompositions of ρ AB [10], Concurrence an CoA are known to have an analytic formula for two-qubit systems [6, 10]: For any two-qubit mixe state ρ AB in B ( C 2 C 2), its concurrence an CoA are C(ρ AB ) = max{0,λ 1 λ 2 λ 3 λ 4 }, C a (ρ AB )=λ 1 + λ 2 + λ 3 + λ 4, (2.4) where λ i s are the eigenvalues, in ecreasing orer, of ρab ρ AB ρab an ρ AB = σ y σ y ρ AB σ y σ y with the Pauli operator σ y. In three-qubit systems, monogamy inequality in terms of concurrence was first propose as [1], C 2 A(BC) C2 AB + C 2 AC, (2.5)
Monogamy an Polygamy of Entanglement 283 for a pure state ψ ABC, where C A(BC) = C( ψ A(BC) ) is the concurrence of a 3-qubit state ψ A(BC) with respect to the bipartite cut between A an BC, an C AB an C AC are the concurrences of the reuce ensity matrices onto subsystems AB an AC respectively. Later, Eq. (2.5) was generalize into arbitrary multi-qubit systems as C 2 A 1(A 2 A n) C2 A 1A 2 + + C 2 A 1A n, (2.6) for an n-qubit state ρ A1 A n [7]. As a ual inequality to Eq. (2.6), a polygamy inequality for multi-qubit systems in terms of CoA was also introuce as, C 2 A 1(A 2 A n) (Ca A 1A 2 ) 2 + +(C a A 1A n ) 2, (2.7) for an n-qubit pure state ψ A1 A n [11]. Although concurrence is a goo entanglement measure for multi-qubit systems satisfying the criteria in [5], it is also known that there are some counter examples in higher-imensional quantum systems violating concurrence-base monogamy inequality in Eq. (2.6) [8, 9]. Let us first consier a pure state ψ in 3 3 3 quantum systems [8] such that ψ ABC = 1 6 ( 123 132 + 231 213 + 312 321 ). (2.8) Then, we can easily check CA(BC) 2 = 4 3, while C2 AB = C2 AC =1, an therefore we have C 2 AB + C 2 AC =2 4 3 = C2 A(BC), (2.9) which is a violation of the inequality in Eq. (2.6) for higher-imensional quantum systems. Now, let us consier a pure state ψ in 3 2 2 quantum systems such that ψ ABC = 1 6 ( 2 010 + 2 101 + 200 + 211 ). (2.10) Again, it can be easily seen that CA(BC) 2 = 12 9 whereas C2 AB = C2 AC = 8 9, which implies the violation of the inequality in Eq. (2.6). In other wors, the concurrence-base monogamy inequality in Eq. (2.6) only hols for multi-qubit systems, an even a tiny extension in any of the subsystems leas to a violation. 3 Convex-Roof Extene Negativity Besies concurremce, another well-known quantification of bipartite entanglement is the negativity [12, 13], which is base on the positive partial transposition (PPT) criterion [14, 15]. For a bipartite pure state φ AB, its negativity is efine as N ( φ AB )= φ AB φ TB 1 1, (3.1)
284 Barry C. Saners an Jeong San Kim where φ AB φ TB is the partial transpose of φ AB φ, an 1 is the trace norm. However, for a mixe state ρ AB, its negativity, N (ρ AB )= ρab T B 1 1, (3.2) oes not even give us a separability criterion because there exist some entangle states with PPT [15, 16]. To overcome this lack of separability criterion, a moifie version of negativity was introuce, an it is calle Convex-Roof Extene Negativity (CREN) [12]. For a bipartite mixe state mixe state ρ AB, CREN is efine as N c (ρ) min k p k N ( φ k ), (3.3) where the minimum is taken over all possible pure state ecompositions of ρ = k p k φ k φ k. Unlike usual negativity of bipartite mixe states in Eq. (3.2), CREN provies a perfect iscrimination of PPT boun entangle states an separable states in any bipartite quantum system. Now, let us consier the relation between CREN an concurrence. For any bipartite pure state φ AB with Schmit rank 2, φ AB = λ 0 00 AB + λ 1 11 AB, (3.4) we have N ( φ AB )= φ φ TB 1 1=2 λ 0 λ 1 = C( φ AB ), (3.5) where ρ A =tr B ( φ φ ). In other wors, negativity is reuce to concurrence for any pure state with Schmit rank 2, an consequently, for any 2-qubit mixe state ρ AB = i p i φ i φ i,wehave N c (ρ AB )=C(ρ AB ). (3.6) Similar to the uality between concurrence an CoA, we can also efine a ual to CREN by taking the maximum value of average negativity over all possible pure state ecomposition, namely Negativity of Assistance (NoA). Furthermore, for a two-qubit state ρ AB,wehave N a (ρ AB ) = max i p i N ( φ i ) = max i p i C( φ i )=C a (ρ AB ), (3.7) where N a (ρ AB ) is the NoA of ρ AB, an the maxima are taken over all pure state ecompositions of ρ AB. Thus, the monogamy an polygamy inequalities in Eqs. (2.6) an
Monogamy an Polygamy of Entanglement 285 (2.7) base on concurrence can be rephrase in terms of negativity: For any n-qubit state ρ A1 A n,wehave an for any n-qubit pure state ψ A1 A n, N ca1(a 2 A n) 2 N ca1a 2 2 + + N ca1a n 2, (3.8) N ca1(a 2 A n) 2 (N c a A1A 2 ) 2 + +(N c a A1A n ) 2, (3.9) where N ca1(a 2 A n) = N ( ψ A1(A 2 A n) ), N ca 1A i = N c (ρ A1A i ) an N c a A1A i = N c a (ρ A1A i ) for i =2,...,n. In other wors, multi-qubit monogamy an polygamy relation can be well-characterize in terms of CREN an NoA. In fact, the states in Eqs. (2.8) an (2.10) are all known counterexamples showing the violation of the concurrence-base monogamy inequality in higher-imensional quantum systems. However, they still have a monogamy relation in terms of CREN: For the state in Eq. (2.8), it can be irectly checke that N A(BC) =2an N cab = N cac =1, an thus N ca(bc) 2 =4 1+1=N cab 2 + N cac 2. (3.10) Similarly, we can also show N 2 ca(bc) =4an N 2 cab = N 2 cab = 8 9 for the state in Eq. (2.10) [5]. In other wors, the states in Eqs. (2.8) an (2.10) still show the monogamy of entanglement in terms of CREN, although they are counterexamples of the concurrence-base monogamy inequality. Thus, CREN monogamy an polygamy inequalities in Eqs. (3.8) an (3.9) are strong caniates for general monogamy an polygamy inequalities in multipartite higher-imensional quantum systems without any known counterexample. 4 Monogamy an Polygamy of Entanglement in Higher-Dimensional Quantum Systems Multipartite entanglement is known to have many inequivalent classes, which are not convertible to each other uner stochastic local operations an classical communications (SLOCC) [18]. For example, there are two inequivalent classes in three-qubit systems: the Greenberger-Horne-Zeilinger (GHZ) class [17] an the W-class [18]. In terms of monogamy an polygamy relations, W-class states saturate concurrence-base monogamy an polygamy inequalities in Eqs. (2.6) an (2.7), while the ifferences between terms in the inequalities can assume their largest values for GHZ-class states. In other wors, monogamy an polygamy of multipartite entanglement can also be use for an analytical characterization of entanglement in multipartite quantum systems. As the first step towar general CREN MoE stuies in higher-imensional quantum systems, we consier here a class of quantum states in higher-imensional quantum systems,
286 Barry C. Saners an Jeong San Kim which are in partially coherent superposition of a generalize W-class state [9] an the vacuum. We further show that this class of states saturates CREN monogamy an polygamy inequalities in Eqs. (3.8) an (3.9), regarless of the ecoherency in the superposition. A partially coherent superposition of a generalize W-class state an 0 n is given as ρ A1 A n where W n is the n-quit W-class state [9], =p W n W n +(1 p) 0 n 0 n + λ p(1 p)( W n 0 n + 0 n Wn ), (4.1) 1 W n = (a A 1 A n 1i i0 0 + a 2i 0i 0 + + a ni 00 0i ), (4.2) i=1 with n 1 j=1 i=1 a ji 2 =1, an λ is the egree of coherence for 0 λ 1. By using the metho introuce in [5,9], we can irectly evaluate the average negativity of the reuce ensity matrices ρ A1A i for i =2,...,n, an we can also show that the average negativity is invariant for any possible choice of pure state ecomposition of ρ A1A i. Furthermore, after a teious calculation, we obtain the following equalities; n n N 2 ca1ai = N 2 ( ) a 2 ca1(a2 A n) = NcA1A i, (4.3) i=2 which is the saturation of CREN monogamy an polygamy inequalities in Eqs. (3.8) an (3.9). Besies the case of multi-qubit systems an the counterexamples in in Eqs. (2.8) an (2.10), CREN also shows a strong possibility of general monogamy an polygamy relation of entanglement by proviing saturate inequalities for a large class of higher-imensional quantum states. i=2 5 Conclusion We have shown that CREN is a powerful caniate to characterize general monogamy an polygamy relation of multipartite entanglement in higher-imensional quantum systems. We have shown that multi-qubit monogamy an polygamy inequalities can be rephrase in terms of CREN, an CREN monogamy inequality is also true even for the counterexamples of concurrence-base inequalities in higher-imensional quantum systems. We further teste the possibility of CREN monogamy an polygamy inequalities in higher-imensional quantum systems by showing its saturation for a large class of quantum states in n-quit systems that are in a partially coherent superpositions of a generalize W- class state an the vacuum. Thus, CREN is a strong caniate for the general monogamy relation of multipartite entanglement in higher-imensional quantum systems with no obvious counterexamples.
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288 Barry C. Saners an Jeong San Kim [18] W. Dür, G. Vial an J. I. Cirac, Three qubits can be entangle in two inequivalent ways, Phys. Rev. A 62 (2000), 062314. Dr. Barry Saners is icore Chair of Quantum Information Science an Director of the Institute for Quantum Information Science at the University of Calgary. He receive his Bachelor of Science egree from the University of Calgary in 1984 an then complete a Diploma of Imperial College. Subsequently he complete a PhD at Imperial College. Following completion of the PhD in 1987, he was a postoctoral research fellow uner the supervision of Professor Gerar Milburn. He joine the Physics Department of Macquarie University in 1991 an was there for 12 years, incluing 6.5 years as Department Hea, before moving to Calgary in 2003. He is especially well known for seminal contributions to theories of quantum-limite measurement, highly nonclassical light, practical quantum cryptography, an optical implementations of quantum information tasks. Dr. Jeong San Kim is a postoctoral associate of the Institute for Quantum Information Science at the University of Calgary. He receive his Ph.D. in mathematical science from Seoul National University, Korea in 2006, an he was also a postoctoral research fellow at Seoul National University. In December 2007, he move to the Institute for Quantum Information Science at the University of Calgary working as a postoc. uner the supervision of Professor Barry C. Saners. Dr. Kim is currently working on the mathematical aspect of quantum information theory an quantum cryptography, which are mainly about quantum entanglement theory in multipartite quantum systems an the structural property of quantum cryptosystems using boun entanglement.