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Parameterized entanglement measures have demonstrated their superiority compared with kinds of unparameterized entanglement measures. Entanglement concurrence has been widely used to describe entanglement in quantum experiments. As an entanglement measure it is related to specific quantum Rényi-α entropy. In the work, we propose a parameterized bipartite entanglement measure based on the general Rényi-α entropy, which is named α-logarithmic concurrence. This measure, different from existing parameterized measures, is defined first for pure states, then extended to the mixed states. Furthermore, we verify three necessary conditions for α-logarithmic concurrence to satisfy the entanglement measures. We show that this measure is easy to calculate for pure states. However, for mixed states, analytical calculations are only suitable for special two-qubit states or special higher-dimensional mixed states. Therefore, we devote our efforts to developing the analytical lower bound of the-logarithmic concurrence for general bipartite states. Surprisingly, this lower bound is a function on positive partial transposition criterion and realignment criterion of this mixed state. This shows the connection among the three entanglement measures. The interesting feature is that the lower bound depends on the entropy parameter associated with the detailed state. This allows us to choose appropriate parameter α such that
$ G_\alpha({\boldsymbol{\rho}})\gg0$ for experimental entanglement detection of specific state ρ. Moreover, we calculate expressions of the α-logarithmic concurrence for isotropic states, and give a the analytic expressions for isotropic states with$ d = 2$ . Finally, the monogamy of the α-logarithmic concurrence is also discussed. We set up a mathematical formulation for the monogamous property in terms of α-logarithmic concurrence. Here we set up the functional relation between concurrence and α-logarithmic concurrence in two qubit systems. Then we obtain some useful properties of this function, and by combining the Coffman–Kundu–Wootters (CKW) inequality, we establish the monogamy inequality about α-logarithmic concurrence. We finally prove that the monogamy inequality holds true for α-logarithmic concurrence.-
Keywords:
- parameterized measure /
- Rényi-α entropy /
- entanglement measure /
- concurrence
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Google Scholar
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Google Scholar
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[1] Bennett C H, Wiesner S J 1992 Phys. Rev. Lett. 69 2881
Google Scholar
[2] Bennett C H, Brassard G, Crepeau G, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
Google Scholar
[3] Hillery M, Buzek V, Berthiaume A 1999 Phys. Rev. A 59 1829
Google Scholar
[4] Gisin N, Ribordy G, Tittel W, Zbinden H 2002 Rev. Mod. Phys. 74 145
Google Scholar
[5] Peres A 1996 Phys. Rev. Lett. 77 1413
Google Scholar
[6] Horodecki M, Horodecki P, Horodecki R 1996 Phys. Lett. A 223 1
Google Scholar
[7] Horodecki M, Horodecki P 1996 Phys. Rev. A 59 4206
[8] Rudolph O 2005 Quantum Inf. Process. 4 219
Google Scholar
[9] Chen K, Wu L A 2003 Quantum Inf. Comput. 3 193
Google Scholar
[10] Hill S, Wootters W K 1997 Phys. Rev. Lett. 78 5022
Google Scholar
[11] Rungta P, Buzek V, Caves C M 2001 Phys. Rev. A 64 042315
Google Scholar
[12] Wootters W K 1998 Phys. Rev. Lett. 80 2245
Google Scholar
[13] Zyczkowski K, Horodecki P, Sanpera A, Lewenstein M 1998 Phys. Rev. A 58 883
Google Scholar
[14] Vidal G, Werner R F 2002 Phys. Rev. A 65 032314
Google Scholar
[15] Bennett C H, DiVincenzo D P, Smolin J A, Wootters W K 1996 Phys. Rev. A 54 3824
Google Scholar
[16] Horodecki M 2001 Quantum Inf. Comput. 1 3
Google Scholar
[17] Gour G, Bandyopadhyay S, Sanders B C 2007 J. Math. Phys. 48 012108
Google Scholar
[18] Kim J S, Sanders B C 2010 J. Phys. A 43 445305
Google Scholar
[19] Kim J S 2010 Phys. Rev. A 81 062328
Google Scholar
[20] Simon C, Kempe J 2002 Phys. Rev. A 65 052327
Google Scholar
[21] Yang X, Luo M X, Yang Y H, Fei S M 2021 Phys. Rev. A 103 052423
Google Scholar
[22] Wei Z W, Luo M X, Yang Y H, Fei S M 2022 Quantum Inf. Process. 21 210
Google Scholar
[23] Wei Z W, Fei S M 2022 J. Phys. A: Math. Theor. 55 275303
Google Scholar
[24] Lee S, Chi D P, Oh S D, Kim J 2003 Phys. Rev. A 68 062304
Google Scholar
[25] Rungta P, Caves C M 2003 Phys. Rev. A 67 012307
Google Scholar
[26] Vollbrecht K G H, Werner R F 2001 Phys. Rev. A 64 062307
Google Scholar
[27] Terhal B M, Vollbrecht K G H 2000 Phys. Rev. Lett. 85 2625
Google Scholar
[28] Buchholz L E, Moroder T, Guhne O 2016 Ann. Phys. 528 278
Google Scholar
[29] Chen K, Albeverio S, Fei S M 2005 Phys. Rev. Lett. 95 210501
Google Scholar
[30] Chen K, Sergio A, Fei S M 2005 Phys. Rev. Lett. 95 040504
Google Scholar
[31] Liu L G, Tian C L, Chen P X, Yuan N C 2009 Chin. Phys. Lett. 26 060306
Google Scholar
[32] Li M, Wang J, Shen S Q, Chen Z H, Fei S M 2018 Sci. Rep. 7 17274
[33] Gour G, Sanders B C 2004 Phys. Rev. Lett. 93 260501
Google Scholar
[34] Nielsen M A, Chuang I L 2010 Quantum Computation and Quantum Information (10th Ed.) (Cambridge: Cambridge University Press) pp109–111
[35] Vedral V, Plenio M B, Rippin M A, Knight P L 1997 Phys. Rev. Lett. 78 2275
Google Scholar
[36] Vedral V, Plenio M B 1998 Phys. Rev. A 57 1619
Google Scholar
[37] Vidal G, Tarrach R 1999 Phys. Rev. A 59 141
Google Scholar
[38] Mintert F, Carvalho A, Kus M, Buchleitner A 2005 Phys. Rep. 415 207
Google Scholar
[39] Ando T 1989 Linear Algebr. Appl. 118 163
Google Scholar
[40] Bhatia R 1997 Matrix Analysis (New York: Springer-Verlag) pp40–47
[41] Manne K K, Caves C M 2008 Quantum Inf. Comput. 8 295
Google Scholar
[42] Wang Y X, Mu L Z, Vedral V, Fan H 2016 Phys. Rev. A 93 022324
Google Scholar
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