-
The problem of sharing quantum correlations is an interesting problem in the study of quantum information theory. Silva et al. proposed the study of sharing quantum nonlocality at first. They studied the fundamental limits on nonlocality, asking whether a single pair of entangled qubits could generate a long sequence of nonlocal correlations. At the same time, the sequential scenario was also introduced first, in which Alice and Bob each have half of a pair of entangled qubit states. The first Bob measures his half and then passes his part to a second Bob who measures again and so on. Obviously, even partial preservation of entanglement in a shared state in spite of a few sequences of local operations performed by the sharing parties can be important for information processing schemes in which entanglement is utilized as a resource. Thus, the problem of sharing quantum entanglement has also been extensively investigated. Recently, Srivastava et al. proved that there exist a class of T-states whose entanglement can be shared by arbitrarily many independent observers in [
Phys. Rev. A 2022 105 062413 ]. Here, we want to find whether there are other entangled states that can be shared entanglement arbitrarily many times. In this paper, we consider the problem of sharing quantum entanglement when the initial shared state is a two-qubit entangled Werner state. The goal is to maximize the number of Bobs that can share entanglement with a single Alice. By suitably choosing the entanglement witness operator and the unsharp measurement settings by the Bobs, we prove that there exist two-qubit entangled initial shared Werner states whose entanglement can be detected by arbitrarily many sequential observers Bobs with a single Alice. Then, we also consider the special case of the Werner state, that is, the maximally entangled state as the initial shared state. In this case, its entanglement can also be witnessed arbitrarily many times, and the number of Bobs increases with the decrease of parameter.-
Keywords:
- entanglement witness /
- Werner state /
- sharing quantum entanglement
[1] Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
Google Scholar
[2] Bennett C H, Wiesner S J 1992 Phys. Rev. Lett. 69 2881
Google Scholar
[3] Scarani V, Lblisdir S, Gisin N, Acin A 2005 Rev. Mod. Phys. 77 1225
Google Scholar
[4] Ekert A K 1991 Phys. Rev. Lett. 67 661
Google Scholar
[5] Wang K, Wang X, Zhan X, Bian Z, Li J, Sanders B C, Xue P 2018 Phys. Rev. A 97 042112
Google Scholar
[6] Ladd T D, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O'Brien J L 2010 Nature 464 45
Google Scholar
[7] Silva R, Gisin N, Guryanova Y, Popescu S 2015 Phys. Rev. Lett. 114 250401
Google Scholar
[8] Brown P J, Colbeck R 2020 Phys. Rev. Lett. 125 090401
Google Scholar
[9] Zhang T G, Fei S M 2021 Phys. Rev. A 103 032216
Google Scholar
[10] Bera A, Mal S, Sen (De) A, Sen U 2018 Phys. Rev. A 98 062304
Google Scholar
[11] Curchod F J, Johansson M, Augusiak R, Hoban M J, Wittek P, Acin A 2017 Phys. Rev. A 95 020102(R
Google Scholar
[12] Cheng S, Liu L, Baker T J, Hall M J W 2021 Phys. Rev. A 104 L060201
Google Scholar
[13] Das D, Ghosal A, Sasmal S, Mal S, Majumdar A S 2019 Phys. Rev. A 99 022305
Google Scholar
[14] Kumari A, Pan A K 2019 Phys. Rev. A 100 062130
Google Scholar
[15] Saha S, Das D, Sasmal S, Sarkar D, Mukherjee K, Roy A, Bhattacharya S S 2019 Quantum Inf. Process. 18 42
Google Scholar
[16] Bowles J, Baccari F, Salavrakos A 2020 Quantum 4 344
Google Scholar
[17] Foletto G, Calderaro L, Tavakoli A, Schiavon M, Picciariello F, Cabello A, Villoresi P, Vallone G 2020 Phys. Rev. Appl. 13 044008
Google Scholar
[18] Foletto G, Padovan M, Avesani M, Tebyanian H, Villoresi P, Vallone G 2021 Phys. Rev. A 103 062206
Google Scholar
[19] Feng T, Ren C, Tian Y, Luo M, Shi H, Chen J, Zhou X 2020 Phys. Rev. A 102 032220
Google Scholar
[20] Hu M J, Zhou Z Y, Hu X M, Li C F, Guo G C, Zhang Y S 2018 npj Quantum Inf. 4 63
Google Scholar
[21] Anwer H, Muhammad S, Cherifi W, Miklin N, Tavakoli A, Bourennane M 2020 Phys. Rev. Lett. 125 080403
Google Scholar
[22] Anwer H, Wilson N, Silva R, Muhammad S, Tavakoli A, Bourennane M 2021 Quantum 5 551
Google Scholar
[23] Foletto G, Calderaro L, Vallone G, Villoresi P 2020 Phys. Rev. Res. 2 033205
Google Scholar
[24] Mohan K, Tavakoli A, Brunner N 2019 New J. Phys. 21 083034
Google Scholar
[25] Schrödinger E 1936 Math. Proc. Cambridge Philos. Soc. 32 446
Google Scholar
[26] Einstein A, Podolsky B, Rosen N 1935 Phys. Rev. 47 777
Google Scholar
[27] Sasmal S, Das D, Mal S, Majumdar A S 2018 Phys. Rev. A 98 012305
Google Scholar
[28] Srivastava C, Pandit M, Sen U 2022 Phys. Rev. A 105 062413
Google Scholar
[29] Gühne O, Tóth G 2009 Phys. Rep. 474 1
Google Scholar
[30] Werner R F 1989 Phys. Rev. A 40 4277
Google Scholar
[31] Horodecki M, Horodecki P 1999 Phys. Rev. A 59 4206
Google Scholar
[32] Simmons G F 1963 Introduction to Topology and Modern Analysis (New York: McGraw-Hill)p224-231
[33] Gühne O, Hyllus P, BrußD, Ekert A, Lewenstein M, Macchiavello C, Sanpera A 2002 Phys. Rev. A 66 062305
Google Scholar
[34] Gühne O, Hyllus P, Bruss D, Ekert A, Lewenstein M, Macchiavello C, Sanpera A 2003 J. Mod. Opt. 50 1079
Google Scholar
-
图 1 Alice和第一个Bob, 即
${\rm{B}}_{1}$ , 共享一个两体纠缠态$\rho_{{\rm{AB}}_{1}}$ .${\rm{B}}_{1}$ 对他所拥有的部分共享态进行测量, 然后将其传递给第二个Bob, 即${\rm{B}}_{2}$ . 测量后的状态是$\rho_{{\rm{AB}}_{2}}$ . 之后,${\rm{B}}_{2}$ 测量$\rho_{{\rm{AB}}_{2}}$ , 然后再传递给${\rm{B}}_{3}$ , 以此类推Fig. 1. A bipartite entangled quantum state
$\rho_{{\rm{AB}}_{1}}$ is initially shared by Alice and the first Bob (${\rm{B}}_{1}$ ).${\rm{B}}_{1}$ performs his measurement on his part and then passes it to the second Bob (${\rm{B}}_{2}$ ). The post-measurement state is$\rho_{{\rm{AB}}_{2}}$ . Then,${\rm{B}}_{2}$ measures$\rho_{{\rm{AB}}_{2}}$ on his part and passes it to${\rm{B}}_{3}$ and so on
图 2 当
$t = - 1$ 时初始共享态是最大纠缠态. 这里绘制了相对于参数$\epsilon$ , 可以与单个Alice见证纠缠的Bobs的数量. 横轴表示参数$\epsilon $ 的取值, 图中将(0, 1)按长度0.01均匀分段,$\epsilon $ 取值为每段的右端点. 纵轴表示能成功检测到与单个Alice纠缠的Bobs的个数, 用k表示Fig. 2. When
$t = -1$ , the initial shared state is the maximally entangled state. We plot here the number of Bobs who can witness entanglement with Alice, with respect to$\epsilon$ . The horizontal axis represents the value of parameter$\epsilon$ which belongs to 100 uniformly distributed points in (0, 1). The vertical axis counts the number of Bobs k who can succeed in detecting entanglement with Alice -
[1] Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
Google Scholar
[2] Bennett C H, Wiesner S J 1992 Phys. Rev. Lett. 69 2881
Google Scholar
[3] Scarani V, Lblisdir S, Gisin N, Acin A 2005 Rev. Mod. Phys. 77 1225
Google Scholar
[4] Ekert A K 1991 Phys. Rev. Lett. 67 661
Google Scholar
[5] Wang K, Wang X, Zhan X, Bian Z, Li J, Sanders B C, Xue P 2018 Phys. Rev. A 97 042112
Google Scholar
[6] Ladd T D, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O'Brien J L 2010 Nature 464 45
Google Scholar
[7] Silva R, Gisin N, Guryanova Y, Popescu S 2015 Phys. Rev. Lett. 114 250401
Google Scholar
[8] Brown P J, Colbeck R 2020 Phys. Rev. Lett. 125 090401
Google Scholar
[9] Zhang T G, Fei S M 2021 Phys. Rev. A 103 032216
Google Scholar
[10] Bera A, Mal S, Sen (De) A, Sen U 2018 Phys. Rev. A 98 062304
Google Scholar
[11] Curchod F J, Johansson M, Augusiak R, Hoban M J, Wittek P, Acin A 2017 Phys. Rev. A 95 020102(R
Google Scholar
[12] Cheng S, Liu L, Baker T J, Hall M J W 2021 Phys. Rev. A 104 L060201
Google Scholar
[13] Das D, Ghosal A, Sasmal S, Mal S, Majumdar A S 2019 Phys. Rev. A 99 022305
Google Scholar
[14] Kumari A, Pan A K 2019 Phys. Rev. A 100 062130
Google Scholar
[15] Saha S, Das D, Sasmal S, Sarkar D, Mukherjee K, Roy A, Bhattacharya S S 2019 Quantum Inf. Process. 18 42
Google Scholar
[16] Bowles J, Baccari F, Salavrakos A 2020 Quantum 4 344
Google Scholar
[17] Foletto G, Calderaro L, Tavakoli A, Schiavon M, Picciariello F, Cabello A, Villoresi P, Vallone G 2020 Phys. Rev. Appl. 13 044008
Google Scholar
[18] Foletto G, Padovan M, Avesani M, Tebyanian H, Villoresi P, Vallone G 2021 Phys. Rev. A 103 062206
Google Scholar
[19] Feng T, Ren C, Tian Y, Luo M, Shi H, Chen J, Zhou X 2020 Phys. Rev. A 102 032220
Google Scholar
[20] Hu M J, Zhou Z Y, Hu X M, Li C F, Guo G C, Zhang Y S 2018 npj Quantum Inf. 4 63
Google Scholar
[21] Anwer H, Muhammad S, Cherifi W, Miklin N, Tavakoli A, Bourennane M 2020 Phys. Rev. Lett. 125 080403
Google Scholar
[22] Anwer H, Wilson N, Silva R, Muhammad S, Tavakoli A, Bourennane M 2021 Quantum 5 551
Google Scholar
[23] Foletto G, Calderaro L, Vallone G, Villoresi P 2020 Phys. Rev. Res. 2 033205
Google Scholar
[24] Mohan K, Tavakoli A, Brunner N 2019 New J. Phys. 21 083034
Google Scholar
[25] Schrödinger E 1936 Math. Proc. Cambridge Philos. Soc. 32 446
Google Scholar
[26] Einstein A, Podolsky B, Rosen N 1935 Phys. Rev. 47 777
Google Scholar
[27] Sasmal S, Das D, Mal S, Majumdar A S 2018 Phys. Rev. A 98 012305
Google Scholar
[28] Srivastava C, Pandit M, Sen U 2022 Phys. Rev. A 105 062413
Google Scholar
[29] Gühne O, Tóth G 2009 Phys. Rep. 474 1
Google Scholar
[30] Werner R F 1989 Phys. Rev. A 40 4277
Google Scholar
[31] Horodecki M, Horodecki P 1999 Phys. Rev. A 59 4206
Google Scholar
[32] Simmons G F 1963 Introduction to Topology and Modern Analysis (New York: McGraw-Hill)p224-231
[33] Gühne O, Hyllus P, BrußD, Ekert A, Lewenstein M, Macchiavello C, Sanpera A 2002 Phys. Rev. A 66 062305
Google Scholar
[34] Gühne O, Hyllus P, Bruss D, Ekert A, Lewenstein M, Macchiavello C, Sanpera A 2003 J. Mod. Opt. 50 1079
Google Scholar
下载:
计量
- 文章访问数: 1994
- PDF下载量: 75
- 被引次数: 0
