Sumiyoshi Abe
College of Information Science and Engineering, Huaqiao University, Xiamen, 361021 China
Mie University, Tsu, 514-8507 Japan
Kazan Federal University, Kazan, 420008 Russia
E-mail: suabe@sf6.so-net.ne.jp
Received November 24, 2017
Abstract
The logarithmic derivative (or quantum score) of a positive definite density matrix appearing in the quantum Fisher information has been discussed, and its exact expression has been presented. The problem of estimating the parameters in a class of the Werner-type N-qudit states has been studied in the context of the quantum Cramer–Rao inequality. The largest value of the lower bound to the error of estimate by the quantum Fisher information has been shown to coincide with the separability point only in the case of two qubits. It has been found, on the other hand, that such largest values give rise to the universal fidelity that is independent of the system size.
Keywords: estimation of entanglement, Werner-type N-qudit states, quantum Fisher information
Acknowledgements. This work was supported in part by the grants from the National Natural Science Foundation of China (project no. 11775084) and the grant-in-aid for scientific research from the Japan Society for the Promotion of Science (projects nos. 26400391 and 16K05484) and performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.
References
1. Helstrom C.W. Quantum Detection and Estimation Theory. Acad. Press, 1976. 308 p.
2. Holevo A.S. Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, 1982. xii+312 p.
3. Barndorff-Nielsen O.E., Gill R.D., Jupp P.E. On quantum statistical inference. J. R. Stat. Soc. B, 2003, vol. 65, no. 4, pp. 775–816.
4. Petz D., Ghinea C. Introduction to quantum Fisher information. Quantum Probability and Related Topics: Proc. 30th Conf. Rebolledo R., Orszag M. (Eds.). Singapore, World Sci. Publ., 2011, pp. 261–281. doi: 10.1142/8059.
5. Sarovar M., Milburn G.J. Optimal estimation of one-parameter quantum channels. J. Phys. A: Math. Gen, 2006, vol. 39, no. 26, pp. 8487–8505.
6. Zanardi P., Paris G.A.M., Venuti L.C. Quantum criticality as a resource for quantum estimation. Phys. Rev. A, 2008, vol. 78, no. 4. art. 042105, pp. 1–7. doi: 10.1103/PhysRevA.78.042105.
7. Bradshaw M., Assad S.M., Lam P.K. A tight Cramér-Rao bound for joint parameter estimation with a pure two-mode squeezed probe. Phys. Lett. A, 2017, vol. 381, no. 32, pp. 2598–2607. doi: 10.1016/j.physleta.2017.06.024.
8. Abe S. Estimation of the thermal degree of freedom in thermo field dynamics. Phys. Lett. A, 1999, vol. 254, nos. 3–4, pp. 149–153. doi: 10.1016/S0375-9601(99)00061-4.
9. Asymptotic Theory of Quantum Statistical Inference: Selected Papers. Hayashi M. (Ed.). Singapore, World Sci. Publ., 2005. 560 p. doi: 10.1142/5630.
10. Wiseman H.M., Milburn G.J. Quantum Measurement and Control. Cambridge, Cambridge Univ. Press, 2010. 460 p.
11. Boixo S., Monras A. Operational interpretation for global multipartite entanglement. Phys. Rev. Lett., 2008, vol. 100, no. 10, art. 100503, pp. 1–4. doi: 10.1103/PhysRevLett.100.100503.
12. Genoni M.G., Giorda P., Paris M.G.A. Optimal estimation of entanglement. Phys. Rev. A, 2008, vol. 78, no. 3, art. 032303, pp. 1–9. doi: 10.1103/PhysRevA.78.032303.
13. Werner R.F. Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A, 1989, vol. 40, no. 8, pp. 4277–4281. doi: 10.1103/PhysRevA.40.4277.
14. Popescu R.F. Bell's inequalities versus teleportation: What is nonlocality? Phys. Rev. Lett., 1994, vol. 72, no. 6, pp. 797–799. doi: 10.1103/PhysRevLett.72.797.
15. Zhang Y.-S., Huang Y.-F., Li Ch.-F., Guo G.-C. Experimental preparation of the Werner state via spontaneous parametric down-conversion. Phys. Rev. A, 2002, vol. 66, no. 6, art. 062315, pp. 1–4. doi: 10.1103/PhysRevA.66.062315.
16. Barbieri M., De Martini F., Di Nepi G., Mataloni P., D'Ariano G.M., Macchiavello C. Detection of entanglement with polarized photons: Experimental realization of an entanglement witness. Phys. Rev. Lett., 2003, vol. 91, no. 22, art. 227901, pp. 1–4. doi: 10.1103/PhysRevLett.91.227901.
17. Jakóbczyk L. Generation of Werner-like stationary states of two qubits in a thermal reservoir. J. Phys. B: At. Mol. Opt. Phys., 2010, vol. 43, no. 1, art. 015502, pp. 1–7. doi: 10.1088/0953-4075/43/1/015502.
18. Peres A. Separability criterion for density matrices. Phys. Rev. Lett., 1996, vol. 77, no. 8, pp. 1413–1415. doi: 10.1103/PhysRevLett.77.1413.
19. Horodecki M., Horodecki P., Horodecki R. Separability of mixed states: Necessary and sufficient conditions. Phys. Lett. A, 1996, vol. 223, no. 1–2, pp. 1–8. doi: 10.1016/S0375-9601(96)00706-2.
20. Abe S., Rajagopal A.K. Nonadditive conditional entropy and its significance for local realism. Phys. A, 2001, vol. 289, nos. 1–2, pp. 157–164. doi: 10.1016/S0378-4371(00)00476-3.
21. Pittenger A.O., Rubin M.H. Separability and Fourier representations of density matrices. Phys. Rev. A, 2000, vol. 62, no. 3, art. 032313, pp. 1–9. doi: 10.1103/PhysRevA.62.032313.
22. Pittenger A.O., Rubin M.H. Note on separability of the Werner states in arbitrary dimensions. Opt. Commun., 2000, vol. 179, no. 1–6, pp. 447–449. doi: 10.1016/S0030-4018(00)00612-X.
23. Abe S. Nonadditive information measure and quantum entanglement in a class of mixed states of an 
system. Phys. Rev. A, 2002, vol. 65, art. 052323, pp. 1–6. doi: 10.1103/PhysRevA.65.052323.
24. Nayak A.S., Sudha, Usha Devi A.R. Rajagopal A.K. One parameter family of 
-qudit Werner–Popescu states: Bipartite separability using conditional quantum relative Tsallis entropy. J. Quantum Inf. Sci., 2018, vol. 8, no. 1, pp. 12–23. doi: 10.4236/jqis.2018.81002.
25. Jozsa R. Fidelity for mixed quantum states. J. Mod. Opt., 1994, vol. 41, no. 12, pp. 2315–2323. doi: 10.1080/09500349414552171.
26. Schumacher B. Quantum coding. Phys. Rev. A, 1995, vol. 51, no. 4, pp. 2738–2747. doi: 10.1103/PhysRevA.51.2738.
27. Abe S. Nonadditive generalization of the quantum Kullback–Leibler divergence for measuring the degree of purification. Phys. Rev. A, 2003, vol. 68, no. 3, art. 032302, pp. 1–3. doi: 10.1103/PhysRevA.68.032302.
28. Rajagopal A.K., Usha Devi A.R., Rendell R.W. Kraus representation of quantum evolution and fidelity as manifestations of Markovian and non-Markovian forms. Phys. Rev. A, 2010, vol. 82, no. 4, art. 042107, pp. 1–7. doi: 10.1103/PhysRevA.82.042107.
For citation: Abe S. Estimating entanglement in a class of N-qudit states. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 213–219.
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