Multipartite Monogamy of the Entanglement of Formation. Abstract

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1 Multipartite Monogamy of the Entanglement of Formation Xian Shi Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , China University of Chinese Academy of Sciences, Beijing , China arxiv: v1 [quant-ph] Oct 018 UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information Processing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , China Abstract Characterizing the distribution of entanglement in multi-partite systems is one of the most interesting topics on entanglement theory. Here we consider a linear monogamy relation between all bipartite partitions for a three-qubit system in terms of entanglement of formation (EoF). We also present the upper bound of the sum of all bipartite partitions for a three qubit pure state in terms of concurrence. PACS numbers: 0.67.Mn, 0.65.Ud Electronic address: shixian01@gmail.com 1

2 I. INTRODUCTION Monogamy of entanglement is an interesting property that characterizes the distribution of entanglement, it presents that entanglement cannot be shareable arbitrarily among many parties, which is different from classical correlations [1]. If party A has strong correlation with party B such that ψ AB = 1 ( ), then the correlations between A and B cannot be shared by party C, that is, ρ ABC = ψ AB ψ ρ C. This property has been applied on many tasks in quantum information, in particularly, it can be applied on the proof of the security of quantum cryptography []. Mathematically, for a tripartite system A,B and C, the general monogamy in terms of an entanglement measure E implies that the entanglement between A and BC satisfies E A BC E AB +E AC. (1) This relation was first proved for qubit systems with respect to the squared concurrence [, 4]. However, the inequality (1) is not valid with respect to all entanglement measure. It is well known that the EoF E does not satisfy the inequality (1). Bai et al. showed that the squared EoF satisfies the inequality (1) for qubit systems [5]; Zhu et al. showed that the monogamy relation is valid in terms of the α-th power of EoF when α for qubit systems [6]; Oliveira et al. considered a three-qubit system and numerically obtained a bound on E AB +E AC [7]; in 015, Liu et al. proved this bound analytically [8]. Moreover, Cornelio proposed an interesting monogamous representation for three-qubit systems [9]. In this work, we consider the bound of E AB +E BC +E CA for a three-qubit system, then based on the relations between the EoF and the discord, we also get a similar bound for the discord. At last, we talk about the upper bound for a three qubit pure state in terms of concurrence. This article is organized as follows. First we review preliminary knowledge needed. Then we prove our main results, based on the relation between the EoF and the discord, we can get a similar result for the sum of all bipartite quantum discord for a three-qubit pure state. And we also talk about the upper boundfor a three qubit pure state in terms of concurrence. At last, we end with a conclusion.

3 II. PRELIMINARY KNOWLEDGE Apure state ψ AB can always bewrittenas ψ AB = λ i ii withλ i 0 and λ i = 1 due to the Schmidt decomposition. The EoF of ψ AB is given by E( ψ AB ) = S(ρ A ) = λ i logλ i. () Here λ i are the eigenvalues of ρ A = Tr B ψ AB ψ. For a mixed state ρ AB, the EoF is defined by the convex roof method, E(ρ AB ) = min {p i, φ i AB } p i E( φ i AB ), () where the minimum takes over all the decompositions of ρ AB = i p i φ i φ i with p i 0 and p i = 1. Another important entanglement measure is the concurrence C. The concurrence of a pure state ψ AB is defined as C( ψ AB ) = (1 Trρ A ) = (1 i i λ i ). (4) For a mixed state ρ AB, it is defined as C(ρ AB ) = min {p i, φ i } p i C( φ i ), (5) where the minimum takes over all the decompositions of ρ AB = i p i φ i φ i with p i 0 and p i = 1. For a two-qubit mixed state ρ AB, Wootters derived an analytical formula [10]: i E(ρ AB ) = h( 1+ 1 CAB ), (6) where h(x) = xlog x (1 x)log (1 x) is the binary entropy and C AB = max{ λ 1 λ λ λ 4,0}, here the λ i are the eigenvalues of the matrix ρ AB (σ y σ y )ρ AB (σ y σ y ) with nonincreasing order. Note that we denote f(x) = h( 1+ 1 x ) below.

4 III. MAIN RESULTS Forathree-qubitpurestate ψ ABC, thepairwisecorrelationsaredescribedbythereduced density operators ρ AB,ρ BC and ρ CA. Oliveira et al. [7] numerically obtained a bound on E AB +E AC by randomly sampling 10 6 uniformly distributed states which is much less than, they also considered a three-qubit pure state ψ ABC = and obtained E(ρ AB )+E(ρ AC ) = Recently, Liu et al. [8] showed the upper bound E(ρ AB )+E(ρ AC ) for a three-qubit pure state is , there the authors also considered the upper bound of C(ρ AB )+C(ρ AC ) for a three-qubit pure state is Here it is easily to see that both and are less than, which may be seen as another way to consider the problem on the distribution of entanglement. In 01, Cornelio [9] proposed a generalized monogamy relation involving multipartite concurrence, bipartite concurrence, that is, for a three-qubit pure state ψ ABC, C (ρ AB )+ C (ρ AC )+C (ρ BC ) C ( ψ ABC ), here C ( ψ ABC ) was defined by Mintert et al. [11]: C = 1 / ( Trρ i Trρ ) = A Trρ B Trρ C. i There the author presented counterexamples which showed this result cannot be generalized to n-qubit systems generally. Then it may be meaningful to consider the upper bound of E(ρ AB )+E(ρ AC )+E(ρ BC ) for a three-qubit system. Afterwards we first consider the upper bound of E(ρ AB )+E(ρ AC )+E(ρ BC ) for a threequbit pure state ψ ABC. Here we denote that x = C AB, y = C AC +C AB, c = C AC +C AB +C BC, g(x,y) = f(x)+f(y x)+f(c y), (7) here we have x,y (0,c],g(x,y) = E(ρ AB )+E(ρ AC )+E(ρ BC ), then g x = f (x) f (y x), g y = f (y x) f (c y), (8) from the equality (8), we see that when g x = g y = 0, C AB = C AC = C BC. As f (x) < 0 [1], then the above condition is the only case when the two equalities in (8) are 0. As f(x) is a monotonic function [1], we have when C AB = C AC = C BC, E(ρ AB)+E(ρ AC )+E(ρ BC ) achieve the upper bound for a three-qubit pure state. 4

5 From [1], we have that a three-qubit pure state ψ ABC can be written in the generalized Schmidt decomposition: ψ = l l 1 e iθ 100 +l 101 +l 110 +l 4 111, (9) here θ [0,π),l i 0,i = 0,1,,,4, 4 i=0 l i = 1. From simple computation, we have C AB = 4l 0l,C AC = 4l 0l,C BC = 4l l + 4l 1l 4 8l 1 l l l 4 cosθ, as f(x) is an increasing function [1] and C AB = C AC = C BC, then we only need to get the maximum of 4l 0 l by using the Lagrange multiplier techniques, m(l 0,l 1,l,l 4 ) = 4l 0 l +λ(l 0 +l 1 +l +l 4 1) +µ(l 0l l 4 l 1l 4 +l 1 l l 4 cosθ), (10) l 0 = 8l 0 l +λl 0 +µl 0 l l 1 = λl 1 +µl l 4 cosθ µl 1 l 4 l = 8l 0l +4λl +µl 0l 4µl +4µl 1 l l 4 cosθ l 4 = λl 4 µl 1 l 4 +µl 1 l cosθ = 0 θ = µl 1l l 4sinθ λ = l 0 +l 1 +l +l 4 1 µ = l 0 l l4 l 1 l 4 +l 1l l 4cosθ (11) then we have l 0 = l = l = 1,l 1 = l 4 = 0, that is, max (E(ρ AB )+E(ρ AC )+E(ρ BC )) ψ ABC =h( 1+ 5/9 ) (1) We can generalize this result to the mixed state ρ ABC. Assume that {s h, ψ h ABC } is a 5

6 decomposition of ρ ABC, then we have E(ρ AB )+E(ρ AC )+E(ρ BC ) = i p i E( φ i AB )+ j q j E( θ j AC )+ k r k E( ζ k BC ) h s h (E(ρ h AB)+E(ρ h AC)+E(ρ h BC)) h s h 1.65 = (1) Here we assume that in the first equality, {p i, φ i }, {q j, θ j } and {r k, ζ k } are the optimal decompositions of ρ AB, ρ AC and ρ BC in terms of the EoF correspondingly. The first equality is due to the definition of the EoF for the mixed states, the second inequality is due to the equality (1), in the first inequality, assume {s h, φ h } is a decomposition of ρ ABC, Tr C φ h φ h = ρ h AB,Tr B φ h φ h = ρ h AC,Tr A φ h φ h = ρ h BC, Next we consider the case when E(ρ AB )+E(ρ AC )+E(ρ BC ) gets the upper bound, i.e. ψ ABC = 1 ( ), it is easy to see when we take the operation σ x on the first partite, we get the W state 1 ( ). For the pure states in three qubit systems, Dür et al. [14] showed that there are two inequivalent kinds of genuinely entangled states, i.e. the W-class states and the GHZ-class states. However, as the entanglement of formation and the partial trace operation is continuous, the GHZ class pure states in the system is dense [15], this function cannot distinguish between the W class states and the GHZ class states. As the W class states ψ are all LU equavilent to the following states: φ = r r r 010 +r 100, (14) where i=0 r i = 1. From simple computation, we have C (ρ AB ) = 4 r r,c (ρ AC ) = 4 r 1 r,c (ρ BC ) = 4 r 1 r, then we see that the function E(ρ AB )+E(ρ AC )+E(ρ BC ) ranges over(0,1.65]forthewclassstates. When ψ = ,E(ρ AB )+E(ρ AC )+E(ρ BC ) = 0, and as the GHZ class states is dense, the function E(ρ AB )+E(ρ AC )+E(ρ BC ) ranges over (0,1.65]. Here we recall an interesting result that there is a conservation law for distributed EoF and quantum discord [16] for a three-qubit pure state, E AB +E AC = δ AB +δ AC, (15) 6

7 here δab = I AB JAB = I AB max {Π B x }(S(ρ A ) x p xs(ρ x A )), where the maximum takes over all the positive-operator-valued-measurements {Π B x } performed on the subsystem B, p x = TrΠ B xρ AB Π B x, and ρ x A = Tr B(Π B xρ AB Π B x)/p x. This result depends on the Koashi-Winter (KW) relation E AB +J AC = S A [17], then from the equality (15), we have E AB +E AC +E BC +E BA +E CA +E CB =δ AB +δ BC +δ CA +δ BA +δ AC +δ CB 1.65 =., (16) here we propose a simple and interesting result, it tells us an upper bound of the sum of all bipartite quantum discord for a three-qubit pure state. At last, we present the upper bound of C(ρ AB )+C(ρ AC )+C(ρ BC ) for a pure three-qubit state. From the equality (6), we denote E(x) = h( 1+ 1 x ), E (x) = x 1 x E (x) = 1 x ln ln(1+ 1 0, 1 x) 1 ln(1 x ) /[ 1 x +ln( 1+ 1 x 1 > 0 (17) 1 x)] when x (0,1], we have E (x) > 0, the inverse function exists. From the rule of the differential of the inverse function, we see that dx = 1, de E d x = E, then with the similar d E (E ) method as above, we have when ψ = , C(ρ AB )+C(ρ AC )+C(ρ BC ) gets the upper bound. IV. CONCLUSION In this article, we mainly consider the shareablility of the entanglement for a threequbit state in terms of the EoF. We present an upper bound on the sum of the EoF, E AB +E AC +E BC,thistellsusthattheentanglement cannotbesharedfreelyforathree-qubit system. Then we generalize the relation to the sum of all the bipartite quantum discord for a three-qubit pure state. Finally, we present the upper bound of C(ρ AB )+C(ρ AC )+C(ρ BC ) for a pure state ψ in system. At last, we think this method can be generalized to consider the upper bound of the linear monogamy relation in terms of other bipartite entanglement measures for an n-qubit pure state. 7

8 V. ACKNOWLEDGMENTS This work was partially supported by the National Key Research and Development Program of China (Grant No. 016YFB100090), the National Natural Science Foundation of China (Grant Nos , , and ), the Beijing Science and Technology Project (016), Tsinghua-Tencent-AMSS-Joint Project (016), and the Key Laboratory of Mathematics Mechanization Project: Quantum Computing and Quantum Information Processing. [1] B. M. Terhal. IBM J. Res. Dev. 48, 71 (004). [] M. Tomamichel, S. Fehr, J. Kaniewski, S. Wehner. New J. Phys. 15, 1000 (01). [] Coffman V, Kundu J and Wootters W K, Phys. Rev. A 61, 0506 (000). [4] T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 050 (006). [5] Y. K. Bai, Y. F. Xu and Z. D. Wang. Phys. Rev. Lett. 11, (014). [6] X. N. Zhu, S. M. Fei. Phys. Rev. A 90?0404 (014). [7] T. R. de Oliveira, M. F. Cornelio and F. F. Fanchini. Phys. Rev. A 89, 040 (014). [8] F. Liu, F. Gao and Q. Y. Wen. Sci. Rep. 5?16745 (015). [9] M. F. Cornelio. Phys. Rev. A 87, 00 (01). [10] W. K. Wootters, Phys. Rev. Lett. 80, 45 (1998). [11] F. Mintert, M. Kus, and A. Buchleitner, Phys. Rev. Lett. 95, 6050 (005). [1] Y. K. Bai, Y. F. Xu and Z. D. Wang. Phys. Rev. A 90, 064 (014). [1] A. Acin, A. Andrianov, L. Costa, E. Jane, J. I. Latorre and R. Tarrach. Phys. Rev. Lett. 85, 1560 (000). [14] W. Dür, G. Vidal, and J. I. Cirac. Phys. Rev. A 6, 0614 (000). [15] A. Acin, D. Bruβ, M. Lewenstein and A. Sanpera. Phys. Rev. Lett. 87, (001). [16] F. F. Fanchini, M. F. Cornelio, M. C. de Oliveira and A. O. Caldeira. Phys. Rev. A 84, 011 (011). [17] M. Koashi and A. Winter. Phys. Rev. A 69, 009 (004). 8