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Quantum metrology is a subject of studying quantum measurement and quantum statistical deduction, and the precision of parameter estimation can be enhanced by quantum properties. In general, the process of parameter estimation includes four steps:preparation of probe state, parameterization process, measurement, and data processing. Of these four steps, the preparation of probe state is the most crucial. However, in practical applications, in the process of preparing quantum probe state, the probe system will couple to its environment, which will inevitably cause the quantum properties of the probe system to deteriorate, and thus reducing the precision of quantum parameter estimation. The dynamics of quantum Fisher information (QFI) for W state and Greenberger-Horne-Zeilinger (GHZ) state have been studied in decoherence channels. Because W state and GHZ state have different entanglement properties, the studies of the dynamics of QFI for the superposition of W state and GHZ state are of practical significance in quantum metrology field. In this paper, the dynamics of QFIs for the superposition of W state and GHZ state in three typical decoherence channels (depolarization channel, amplitude damping channel and phase damping channel) are studied. In the four steps of quantum parameter estimation, our major attention is paid to the first step (i.e., the preparation of probe state). For comparison, the QFIs of different probe states are studied, with the other three steps fixed, i.e., all the probe states will undergo the same parameterization, measurement and estimation process. The parameterization process involved here is a quantum spin operation (specified by the spin rotation direction), which is chosen to maximize the QFI of the probe state. The initial probe states under consideration are the superpositions of W state and GHZ state of three-particle and five-particle systems, and the QFI dynamics of those probe states are studied in the three different typical decoherence channels. By using the operator-sum (Kraus) representation of those three typical decoherence channels, the QFI dynamics of the probe state can be analytically derived in three different decoherence channels. The results show that in the depolarization channel, the maximum QFI of the probe state decreases with the decoherence evolving to zero in the end; in the amplitude damping channel, the QFI of the probe state decreases to the minimum with the decoherence evolution and then increases to the shot noise limit; in the phase damping channel, the QFI of the probe state decreases with the evolution of decoherence, but the final stable value is not zero. Further analyses show that W state component of the superposition plays a role in resisting phase damping and the GHZ state component plays a role in resisting amplitude damping. These results can help us to choose the optimal probe state for maximizing the estimation precision in practice.
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Keywords:
- quantum metrology /
- quantum Fisher information /
- decoherence channel /
- W state /
- GHZ state /
- superposition
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[2] Jin G R, Kim S W 2007 Phys. Rev. A 76 043621
[3] Hyllus P, Laskowski W, Krischek P, Schwemmer C, Wieczorek W, Weinfurter H, Pezzé L, Smerzi A 2012 Phys. Rev. A 85 022321
[4] Liu W F, Zhang L H, Li C J 2010 Int. J. Theor. Phys. 49 2463
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[6] Yao Y, Xiao X, Ge L, Wang X G, Sun C P 2014 Phys. Rev. A 89 042336
[7] Giovannetti V, Lloyd S, Maccone L 2006 Phys. Rev. Lett. 96 010401
[8] Ozaydin F 2014 Phys. Lett. A 378 3161
[9] Ozaydin F, Altintas A A, Bugu S, Yesilyurt C 2014 Acta Phys. Pol. A 125 606
[10] Luati A 2004 Ann. Stat. 32 1770
[11] Jing X X, Liu J, Xiong H N, Wang X G 2015 Phys. Rev. A 92 012312
[12] Pezzé L, Smerzi A 2009 Phys. Rev. Lett. 102 100401
[13] Escher B M, Filho R L D M, Davidovich L 2011 Nat. Phys. 7 406
[14] Demokowicz-Dobrzański R, Kolodyński J, Gutǎ M 2012 Nat. Commun. 3 1063
[15] Roy S M, Braunstein S L 2008 Phys. Rev. Lett. 100 220501
[16] Greenberger D M, Horne M A, Shimony A, Zeilinger A 1990 Am. J. Phys. 58 1131
[17] Dr W, Vidal G, Cirac J I 2000 Phys. Rev. A 62 062314
[18] Briegel H J, Raussendorf R 2001 Phys. Rev. Lett. 86 910
[19] Ma J, Huang Y X, Wang X G, Sun C P 2011 Phys. Rev. A 84 022302
[20] Ozaydin F, Altintas A A, Bugu S, Yesilyurt C 2013 Int. J. Theor. Phys. 52 2977
[21] Ozaydin F, Altintas A A, Bugu S, Yesilyurt C 2014 Int. J. Theor. Phys. 53 3219
[22] Yi X J, Huang G Q, Wang J M 2012 Int. J. Theor. Phys. 51 3458
[23] Erol V 2017 Int. J. Theor. Phys. 56 3202
[24] Erol V 2017 arXiv: 1704.07367 (preprints)
[25] Helstrom C W 1976 Quantum Detection and Estimation Theory (New York: Academic Press) pp56-57
[26] Holevo A S 1982 Probabilistic and Statistical Aspects of Quantum Theory (Amsterdam: North-Holland) pp102-104
[27] Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439
[28] Paris M G A 2009 Int. J. Quantum Inf. 07 125
[29] Ma J, Huang Y X, Wang X G, Sun C P 2011 Phys. Rev. A 84 022302
[30] Pang S S, Brun T A 2014 Phys. Rev. A 90 022117
[31] Liu J, Jing X X, Wang X G 2014 Sci. Rep. 5 8565
[32] Wang X, Shi X 2015 Phys. Rev. A 92 042318
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[1] Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439
[2] Jin G R, Kim S W 2007 Phys. Rev. A 76 043621
[3] Hyllus P, Laskowski W, Krischek P, Schwemmer C, Wieczorek W, Weinfurter H, Pezzé L, Smerzi A 2012 Phys. Rev. A 85 022321
[4] Liu W F, Zhang L H, Li C J 2010 Int. J. Theor. Phys. 49 2463
[5] Liu J, Xiong H N, Song F, Wang X G 2014 Physica A 410 167
[6] Yao Y, Xiao X, Ge L, Wang X G, Sun C P 2014 Phys. Rev. A 89 042336
[7] Giovannetti V, Lloyd S, Maccone L 2006 Phys. Rev. Lett. 96 010401
[8] Ozaydin F 2014 Phys. Lett. A 378 3161
[9] Ozaydin F, Altintas A A, Bugu S, Yesilyurt C 2014 Acta Phys. Pol. A 125 606
[10] Luati A 2004 Ann. Stat. 32 1770
[11] Jing X X, Liu J, Xiong H N, Wang X G 2015 Phys. Rev. A 92 012312
[12] Pezzé L, Smerzi A 2009 Phys. Rev. Lett. 102 100401
[13] Escher B M, Filho R L D M, Davidovich L 2011 Nat. Phys. 7 406
[14] Demokowicz-Dobrzański R, Kolodyński J, Gutǎ M 2012 Nat. Commun. 3 1063
[15] Roy S M, Braunstein S L 2008 Phys. Rev. Lett. 100 220501
[16] Greenberger D M, Horne M A, Shimony A, Zeilinger A 1990 Am. J. Phys. 58 1131
[17] Dr W, Vidal G, Cirac J I 2000 Phys. Rev. A 62 062314
[18] Briegel H J, Raussendorf R 2001 Phys. Rev. Lett. 86 910
[19] Ma J, Huang Y X, Wang X G, Sun C P 2011 Phys. Rev. A 84 022302
[20] Ozaydin F, Altintas A A, Bugu S, Yesilyurt C 2013 Int. J. Theor. Phys. 52 2977
[21] Ozaydin F, Altintas A A, Bugu S, Yesilyurt C 2014 Int. J. Theor. Phys. 53 3219
[22] Yi X J, Huang G Q, Wang J M 2012 Int. J. Theor. Phys. 51 3458
[23] Erol V 2017 Int. J. Theor. Phys. 56 3202
[24] Erol V 2017 arXiv: 1704.07367 (preprints)
[25] Helstrom C W 1976 Quantum Detection and Estimation Theory (New York: Academic Press) pp56-57
[26] Holevo A S 1982 Probabilistic and Statistical Aspects of Quantum Theory (Amsterdam: North-Holland) pp102-104
[27] Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439
[28] Paris M G A 2009 Int. J. Quantum Inf. 07 125
[29] Ma J, Huang Y X, Wang X G, Sun C P 2011 Phys. Rev. A 84 022302
[30] Pang S S, Brun T A 2014 Phys. Rev. A 90 022117
[31] Liu J, Jing X X, Wang X G 2014 Sci. Rep. 5 8565
[32] Wang X, Shi X 2015 Phys. Rev. A 92 042318
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