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量子Fisher信息在参数估计理论和量子精密测量领域扮演着非常重要的角色, 不仅可以用来标定量子系统的测量精度极限, 还可用于有益量子计量的纠缠态判定. 本文从量子测量的角度出发, 对多比特WV态(
$\alpha \left\vert W_N \right\rangle + $ $ \sqrt{1-\alpha^2}\left\vert 00...0\right\rangle$ )进行研究, 通过计算其在局域操作下和Lipkin-Meshkov-Glick (LMG)非局域操作模型下的量子Fisher信息, 分析其在精密测量方面的表现. 研究发现: 在局域操作下, 随着参数α由0到1的变化, 多比特WV态的量子Fisher信息逐渐变大, 表明其量子纠缠程度在不断增加, 也体现出更强的量子测量能力. 在LMG非局域操作下, 随着相互作用强度γ的增大,$N=3$ 量子比特WV态的量子Fisher信息值趋于稳定, 几乎不受参数α的影响, 而当体系量子比特数$N>3$ 时, 量子Fisher信息值会随参数α的变大而变大; 当参数α固定时, 多比特WV态的量子Fisher信息会随着相互作用强度γ的增强而变大, 呈现出相互作用强度越大, WV态的量子测量能力越强.-
关键词:
- 量子Fisher信息 /
- WV纠缠态 /
- Lipkin-Meshkov-Glick模型 /
- 精密测量
As an important quantity in the field of parameter estimation theory and quantum precision measurement, quantum Fisher information (QFI) can not only be used to set the theoretical limit of measurement precision in quantum system, but also be exploited to witness metrological useful quantum entanglement. Recently, it has also been broadly used in many aspects of quantum information science, including quantum metrology, multipartite entanglement structure detection, quantum phase transition, quantum chaos, quantum computation and etc. In this work, from the perspective of quantum measurement, we study the quantum Fisher information of an N-qubit WV state ($\alpha \left\vert W_N \right\rangle +\sqrt{1-\alpha^2}\left\vert 00\cdots0\right\rangle$ ) under local operation and Lipkin-Meshkov-Glick (LMG) model. Furthermore, with the general Cramér-Rao lower bound (CRLB) we analyze its performance in high-precision phase measurement. The results show that, under the local operation, the QFI of an N-qubit WV state becomes larger with the increase of parameter α. This not only means the enhanced quantum entanglement, but also implies the powerful ability in high-precision quantum measurement. In the LMG model, as the increase of interactional strength γ the QFI of$N=3$ qubits WV state gradually tends to be stable and almost not be affected by parameter α, which relaxes the requirement in the preparation of target state and indicates a great potential in achieving the relatively stable measurement precision. When the number of qubits from WV state is larger than 3, the QFI of WV state increases with the increase of parameter α. In the case of fixed parameter α, we investigate the QFI of an N-qubit WV state with respect to interaction strength γ. It is found that the QFI of WV state will increase with the increasing interaction strength, which implies that the greater the interaction strength, the stronger the quantum measurement ability of the WV state. Our work will promote the development of high-precision quantum metrology and especially the interaction-enhanced quantum measurement, and further provide new insights in quantum information processing.-
Keywords:
- quantum Fisher information /
- WV state /
- Lipkin-Meshkov-Glick model /
- precision measurement
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图 1 N量子比特WV态在局域操作下的量子Fisher信息随参数α的变化. 从下(黑色实线)到上(浅粉色实线)分别表示3量子比特WV态到8量子比特WV态的结果
Fig. 1. Quantum Fisher information of an N-qubit WV state with respect to α under local operation. From bottom (black line) to top (light pink line) it respectively denotes the result from 3-qubit WV state to 8-qubit WV state.
图 2 量子比特WV态在LMG模型下的量子Fisher信息随参数α的变化 (a) 3比特; (b) 4比特; (c) 5比特; (d) 6比特
Fig. 2. Quantum Fisher information of qubit WV state with respect to α in the LMG model: (a) 3-qubit; (b) 4-qubit; (c) 5-qubit; (d) 6-qubit.
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[1] Fisher R A 1912 Messenger of Mathematics 41 155
[2] Helstrom C W 1976 Quantum Detection and Estimation Theory (New York: Academic Press) pp235–293
[3] Holevo A S 1982 Probabilistic and Statistical Aspects of Quantum Theory (Amsterdam: North Holland) pp52–96
[4] Kreutz C, Timmer J 2013 Optimal Experiment Design, Fisher Information. In: Dubitzky W, Wolkenhauer O, Cho K H, Yokota H, editors. Encyclopedia of Systems Biology (New York: Springer) pp1576–1579
[5] Ly A, Marsman M, Verhagen J, Grasman R P, and Wagenmakers E J 2017 J. Math. Psychol. 80 40
Google Scholar
[6] Giovannetti V, Lloyd S, Maccone L 2006 Phys. Rev. Lett. 96 010401
Google Scholar
[7] Li Y, Pezzè L, Li W D, Smerzi A 2019 Phys. Rev. A 99 022324
Google Scholar
[8] Fiderer L J, E. Fraïsse J M, Braun D 2019 Phys. Rev. Lett. 123 250502
Google Scholar
[9] Li Y, Ren Z H 2023 Phys. Rev. A 107 012403
Google Scholar
[10] Yin P, Zhao X B, Yang Y X, Guo Y, Zhang W H, Li G C, Han Y J, Liu B H, Xu J S, Chiribella G, Chen G, Li C F, Guo G C 2023 Nat. Phys. 19 1122
Google Scholar
[11] Li Y, Pezze L, Gessner M, Ren Z H, Li W D, Smerzi A 2018 Entopy 20 628
Google Scholar
[12] Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439
Google Scholar
[13] Wootters W K 1981 Phys. Rev. D 23 357
Google Scholar
[14] Pezzè L, Smerzi A 2009 Phys. Rev. Lett. 102 100401
Google Scholar
[15] Hyllus P, Laskowski W, Krischek R, Schwemmer C, Wieczorek W, Weinfurter H, Pezzè L, Smerzi A 2012 Phys. Rev. A 85 022321
Google Scholar
[16] Pezzè L, Smerzi A, Oberthaler M K, Schmied R, Treutlein P 2018 Rev. Mod. Phys. 90 035005
Google Scholar
[17] 任志红, 李岩, 李艳娜, 李卫东 2019 物理学报 68 040601
Google Scholar
Ren Z H, Li Y, Li Y N, Li W D 2019 Acta Phys. Sin. 68 040601
Google Scholar
[18] Zanardi P, Paris M G A, Venuti L C 2008 Phys. Rev. A 78 042105
Google Scholar
[19] Hauke P, Heyl M, Tagliacozzo L, Zoller P 2016 Nat. Phys. 12 778
Google Scholar
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Google Scholar
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Google Scholar
[22] Gühne O, Tòth G 2009 Phys. Rep. 474 1
Google Scholar
[23] Pezzè L, Li Y, Li W D, Smerzi A 2016 Proc. Natl. Acad. Sci. 113 11459
Google Scholar
[24] Li Y and Li P F 2020 Phys. Lett. A 384 126413
Google Scholar
[25] Ren Z H, Li W D, Smerzi A, Gessner M 2021 Phys. Rev. Lett. 126 080502
Google Scholar
[26] Li Y, Ren Z H 2022 Physica A 596 127137
Google Scholar
[27] 宋立军, 严冬, 刘烨 2011 物理学报 60 120302
Google Scholar
Song L J, Yan D, Liu Y 2011 Acta. Phys. Sin. 60 120302
Google Scholar
[28] Wang X Q, Ma J, Zhang X H, Wang X G 2011 Chin. Phys. B 20 050510
Google Scholar
[29] Berrada K, Abdel-Khalek S, Khalil E M, Alkaoud A, Eleuch H 2022 Chaos, Solitons Fractals 164 112621
Google Scholar
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Google Scholar
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Google Scholar
[32] Jin H C, Jeong S K 2015 Phys. Rev. A 92 042307
Google Scholar
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Google Scholar
[34] Shi X 2020 Phys. Lett. A 384 126392
Google Scholar
[35] Lai L M, Fei S M, Wang Z X 2021 J. Phys. A 54 425301
Google Scholar
[36] Jarzyna M, Demkowicz-Dobrzanski R 2015 New J. Phys. 17 013010
Google Scholar
[37] Lipkin H J, Meshkov N, Glick A 1965 Nucl. Phys. 62 188
Google Scholar
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