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In previous paper (
2019 Int. J. Mod. Phys. B 33 1950197 ;2020 Int. J. Mod. Phys. B 34 2050022 ), we presented a method to judge the entanglement of 2-qubit system. The necessary and sufficient conditions for the 2qubit system being separable are that if the relevant coefficients is positive and the principal density matrix is separable, then the system is separable, otherwise it is entangled. Now in this paper, we try to generalize this criterion to a 3-qubit system, and then, we further generalize the criterion of 3-qubit system to an N-qubit system. This is a complicated and interesting issue.-
Keywords:
- Born probability density /
- N qubit system /
- entangle states
[1] Rossi R 2013 Physica A 392 2615
Google Scholar
[2] Horst B, Bartkiewicz K, Miranowicz A 2013 Phys. Rev. A 87 042108
Google Scholar
[3] Bartkiewicz K, Horst B, Lemr K, Miranowicz A 2013 Phys. Rev. A 88 052105
Google Scholar
[4] Ekert A K 1991 Phys. Rev. Lett. 67 661
Google Scholar
[5] Bennett C H, Wiesner S J 1992 Phys. Rev. Lett. 69 2881
Google Scholar
[6] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
Google Scholar
[7] Werner R F 1989 Phys. Rev. A 40 4277
Google Scholar
[8] Horodecki M, Horodecki P, Horodecki R 1996 Phys. Lett. A 223 1
Google Scholar
[9] Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
Google Scholar
[10] Peres A 1996 Phys. Rev. Lett. 77 1413
Google Scholar
[11] Horodecki P 1997 Phys. Lett. A 232 333
Google Scholar
[12] Duan L M, Giedke G, Cirac J I, Zoller P 2000 Phys. Rev. Lett. 84 2722
Google Scholar
[13] Simon R 2000 Phys. Rev. Lett. 84 2726
Google Scholar
[14] Lewenstein M, Kraus B, Cirac J I, Horodecki P 2000 Phys. Rev. A 62 052310
Google Scholar
[15] Samsonowicz J, Kuś M, Lewenstein M, 2007 Phys. Rev. A 76 022314
Google Scholar
[16] Bennett C H, Bernstein H J, Popescu S, Schumacher B 1996 Phys. Rev. A 53 2046
Google Scholar
[17] Zhao C Y, Guo Q Z, Tan W H 2019 Int. J. Mod. Phys. B 33 1950197
Google Scholar
[18] Zhao C Y, Guo Q Z, Tan W H 2020 Int. J. Mod. Phys. B 34 2050022
Google Scholar
[19] Schiff L I 1968 Quantum Mechanics (3rd Ed.) (NewYork: McGraw-Hill Book Company) pp8, 154
[20] Boyd R W 2009 Nonlinear Optics (3rd Ed.) (NewYork: Academic Press) p130
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图 1 纠缠态的“N-AB-tree结构”
Figure 1. 24-AB-tree to four 2 qubit string.
图 2 2量子比特系统主密度矩阵系数
${\kern 1 pt} {p_p}$ 随$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ 变化的三维曲线图,$r = 1$ (a)$\varDelta = 0.002$ ; (b)$\varDelta = 0.00103$ ; (c)$\varDelta = 0.0002$ ; (d)$\varDelta = 0.0001$ Figure 2. Principal entangled state
${p_p}$ of 2 qubit system with the parameters$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ ,$r = 1$ : (a)$\varDelta = 0.002$ ; (b)$\varDelta = $ $ 0.00103$ ; (c)$\varDelta = 0.0002$ ; (d)$\varDelta = 0.0001$ .
图 3 3量子比特系统主密度矩阵系数
${\kern 1 pt} {p_p}$ 随$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ 变化的三维曲线图 (a)${r_a} = 0.867$ ,$\varDelta = 0.0001$ ; (b)${r_b} = 0.547$ ,$\varDelta = 0.001$ ,$\xi + \eta = r_a^2 + r_b^2 = 1$ Figure 3. Principal entangle state
${p_p}$ of 3 qubit system with the parameters$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ : (a)${r_a} = 0.867$ ,$\varDelta = 0.0001$ ; (b)${r_b} = $ $ 0.547$ ,$\varDelta = 0.001$ ,$\xi + \eta = r_a^2 + r_b^2 = 1$ .
图 5 3量子比特系统主密度矩阵系数
${\kern 1 pt} {p_p}$ 随$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ 变化的三维曲线图,${r_a} = {r_b} = 0.707$ (a)$\varDelta = 0.0001$ ; (b)$\varDelta = $ $ 0.0002$ ; (c)$\varDelta = 0.00103$ Figure 5. Principal entangled state
${p_p}$ of 3 qubit system with the parameters$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ ,${r_a} = {r_b} = 0.707$ ; (a)$\varDelta = 0.0001$ ; (b)$\varDelta = 0.0002$ ; (c)$\varDelta = 0.00103$ .
图 4 3量子比特系统主密度矩阵系数
${\kern 1 pt} {p_p}$ 随$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ 变化的三维曲线图 (a)${r_a} = 0.724$ ,$\varDelta = 0.0001$ ; (b)${r_b} = 0.652$ ,$\varDelta = 0.001$ ,$\xi + \eta = r_a^2 + r_b^2 = 1$ Figure 4. Principal entangled state
${p_p}$ of 3 qubit system with the parameters$\{ \theta , 0, 0.2\} $ ,$\{ w, - \pi , \pi \} $ : (a)${r_a} = 0.724$ ,$\varDelta = 0.0001$ ; (b)${r_b} = 0.652$ ,$\varDelta = 0.001$ ,$\xi + \eta = r_a^2 + r_b^2 = 1$ .
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[1] Rossi R 2013 Physica A 392 2615
Google Scholar
[2] Horst B, Bartkiewicz K, Miranowicz A 2013 Phys. Rev. A 87 042108
Google Scholar
[3] Bartkiewicz K, Horst B, Lemr K, Miranowicz A 2013 Phys. Rev. A 88 052105
Google Scholar
[4] Ekert A K 1991 Phys. Rev. Lett. 67 661
Google Scholar
[5] Bennett C H, Wiesner S J 1992 Phys. Rev. Lett. 69 2881
Google Scholar
[6] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
Google Scholar
[7] Werner R F 1989 Phys. Rev. A 40 4277
Google Scholar
[8] Horodecki M, Horodecki P, Horodecki R 1996 Phys. Lett. A 223 1
Google Scholar
[9] Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
Google Scholar
[10] Peres A 1996 Phys. Rev. Lett. 77 1413
Google Scholar
[11] Horodecki P 1997 Phys. Lett. A 232 333
Google Scholar
[12] Duan L M, Giedke G, Cirac J I, Zoller P 2000 Phys. Rev. Lett. 84 2722
Google Scholar
[13] Simon R 2000 Phys. Rev. Lett. 84 2726
Google Scholar
[14] Lewenstein M, Kraus B, Cirac J I, Horodecki P 2000 Phys. Rev. A 62 052310
Google Scholar
[15] Samsonowicz J, Kuś M, Lewenstein M, 2007 Phys. Rev. A 76 022314
Google Scholar
[16] Bennett C H, Bernstein H J, Popescu S, Schumacher B 1996 Phys. Rev. A 53 2046
Google Scholar
[17] Zhao C Y, Guo Q Z, Tan W H 2019 Int. J. Mod. Phys. B 33 1950197
Google Scholar
[18] Zhao C Y, Guo Q Z, Tan W H 2020 Int. J. Mod. Phys. B 34 2050022
Google Scholar
[19] Schiff L I 1968 Quantum Mechanics (3rd Ed.) (NewYork: McGraw-Hill Book Company) pp8, 154
[20] Boyd R W 2009 Nonlinear Optics (3rd Ed.) (NewYork: Academic Press) p130
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