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In the field of quantum metrology, an important application is quantum parameter estimation. As the fundamental theory of quantum parameter estimation, quantum Cramér-Rao inequality shows that the variance of parameter estimation is determined by the inverse of quantum Fisher information. Higher quantum Fisher information corresponds to a lower variance, thereby improving the precision of parameter estimation. Quantum Fisher information has been extensively investigated in many aspects of non-relativistic quantum mechanics, including entanglement structure detection, quantum teleportation, quantum phase transition, quantum chaos, and quantum computation. However, there are few researches considering the influence of relativistic effect on quantum Fisher information, and therefore, we attempt to investigate this topic in this work. The relativistic transformation of particle states is employed, and the quantum Fisher information about amplitude parameter
$ \theta $ and phase parameter$\varphi $ are investigated in moving reference frame. In this work, the parameters to be estimated are encoded into the spin degree of freedom, and the pure single-qubit state and the pure two-qubit state are both considered. The quantum Fisher information about$ \theta $ and$\varphi $ of single-qubit state and two-qubit state in moving reference frame are numerically calculated, respectively. It can be observed that the quantum Fisher information is associated with rapidity, amplitude parameter, and the ratio of the width to the particle mass${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$ . The quantum Fisher information of the estimated parameters decreases with rapidity increasing for both single-qubit state and two-qubit state. As rapidity approaches infinity, i.e. increases to the speed of light, the quantum Fisher information reaches to a constant which decreases as the ratio${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$ increases. More importantly, for the phase parameter$ \varphi $ , it is observed that the quantum Fisher information of two-qubit state reduces more significantly than that of single-qubit state. While, for the amplitude parameter$\theta $ , the quantum Fisher information of two-qubit state is greater than that of single-qubit state. These results are useful and valuable for improving the precision of parameter estimation under the influence of relativistic effect.[1] 钟伟 2014 博士学位论文(杭州: 浙江大学)
Zhong W 2014 Ph. D. Dissertation (Hangzhou: Zhejiang University
[2] Lu X M, Wang X G 2021 Phys. Rev. Lett. 126 120503
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[3] Matteo G A P 2009 Int. J. Quant. Inf. 7 125
Google Scholar
[4] Helstrom C W 1967 Phys. Lett. A 25 101
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[5] Holevo A S 1982 Probabilistic and Statistical Aspects of Quantum Theory (North Holland Amsterdam) pp52–96, 160–168
[6] Yuen H P, Lax M 1973 IEEE Trans. Inf. Th. 19 740
Google Scholar
[7] Braunstein S, Caves C 1994 Phys. Rev. Lett. 72 3439
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[8] Braunstein S, Caves C, Milburn G 1996 Ann. Phys. 247 135
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[9] Giovannetti V, Lloyd S, Maccone L 2004 Science 306 1330
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[12] 任志红, 李岩, 李艳娜, 李卫东 2019 物理学报 68 040601
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Ren Z H, Li Y, Li Y N, Li W D 2019 Acta Phys. Sin. 68 040601
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[13] Liu R, Chen Y, Jiang M, Yang X D, Wu Z, Li Y C, Yuan H D, Peng X H, Du J F 2021 Npj. Quantum. Inf. 7 170
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[14] 牛明丽, 王月明, 李志坚 2022 物理学报 71 090601
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Niu M L, Wang Y M, Li Z J 2022 Acta. Phys. Sin. 71 090601
Google Scholar
[15] Fisher R A 1922 Philosoph. Trans. Roy. Soc. London 222 309
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[16] Fisher R A 1925 Proc. Camb. Phil. Soc. 22 700
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[17] Helstrom C W 1969 J. Stat. Phys. 1 231
Google Scholar
[18] Holevo A S 2001 Statistical Structure of Quantum Theory (Berlin, Heidelberg: Springer) pp45–70
[19] Pezze L, Smerzi A, Oberthaler M K, Schmied R, Treutlein P 2018 Rev. Mod. Phys. 90 035005
Google Scholar
[20] 井晓幸 2016 博士学位论文(杭州: 浙江大学)
Jing X X 2016 Ph. D. Dissertation (Hangzhou: Zhejiang University
[21] Boixo S, Flammia S T, Caves C M, Geremia J M 2007 Phys. Rev. Lett. 98 090401
Google Scholar
[22] Liu W F, Zhang L H, Li C J 2010 Int J. Theor. Phys. 49 2463
Google Scholar
[23] 刘然, 吴泽, 李宇晨, 陈昱全, 彭新华 2023 物理学报 72 110305
Google Scholar
Liu R, Wu Z, Li Y C, Chen Y Q, Peng X H 2023 Acta Phys. Sin. 72 110305
Google Scholar
[24] Li N, Luo S L 2013 Phys. Rev. A 88 014301
Google Scholar
[25] Kourbolagh Y A, Azhdargalam M 2019 Phys. Rev. A 99 012304
Google Scholar
[26] 李岩2019 博士学位论文(太原: 山西大学)
Li Y 2019 Ph. D. Dissertation (Taiyuan: Shanxi University
[27] 武莹, 李锦芳, 刘金明 2018 物理学报 67 140304
Google Scholar
Wu Y, Li J F, Liu J M 2018 Acta Phys. Sin. 67 140304
Google Scholar
[28] Zhang X Y, Lu X M, Liu J, Ding W K, Wang X G 2023 Phys. Rev. A 107 012414
Google Scholar
[29] Li J N, Cui D Z 2023 Phys. Rev. A 108 012419
Google Scholar
[30] Sone A, Cerezo M, Beckey J L, Coles P J 2021 Phys. Rev. A 104 062602
Google Scholar
[31] Jafarzadeh M, Rangani J H, Amniat-Talab M. 2020 Proc. R. Soc. A 476 20200378
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[33] Yao Y, Xing X, Li G, Wang X G, Sun C P 2014 Phys. Rev. A 89 042336
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[34] Hosler D, Kok P 2013 Phys. Rev. A 88 052112
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[35] Zhou T, Cui J X, Cao Y 2013 Quantum. Inf. Process 12 747
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[36] Gingrich R M and Adami C 2002 Phys. Rev. Lett. 89 270402
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[37] Gingrich R M, Bergou A J, Adami C 2003 Phys. Rev. A 68 042102
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[38] Luo S L 2000 Lett. Math. Phys. 53 243
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[39] Ren J R, Song S X 2010 Int J. Theor. Phys. 49 1317
Google Scholar
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图 1 运动坐标系中单粒子态的QFI (a) $ \theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2} $, ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1,2,3,4 $时, $F\left( \varphi \right)$随$\xi $的变化; (b) $ \theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2} $, $ {{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1,2,3,4 $时, $F\left( \theta \right)$随$\xi $的变化; (c) ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$, $F\left( \varphi \right)$随$\theta $, $\xi $的变化; (d) ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$, $F\left( \theta \right)$随$\theta $, $\xi $的变化
Figure 1. Quantum Fisher information for one-particle state in moving frames: (a) $F\left( \varphi \right)$ is plotted as a function of $\xi $ for $ \theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2} $ and $ {{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1,2,3,4 $ ; (b) $F\left( \theta \right)$ is plotted as a function of $\xi $ for $ \theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2} $and $ {{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1,2,3,4 $; (c) $F\left( \varphi \right)$ is plotted as a function of $\theta $ and $\xi $ for ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$; (d) $F\left( \theta \right)$ is plotted as a function of $\theta $ and $\xi $ for ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$.
图 2 运动坐标系中单粒子态与双粒子态的QFI对比 (a) ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$, $\theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2}$时, $F\left( \varphi \right)$随$\xi $的变化对比; (b) ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$, $\theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2}$时, $F\left( \theta \right)$随$\xi $的变化对比
Figure 2. Quantum Fisher information for one-particle state versus that for two-particle state in moving frames: (a) $F\left( \varphi \right)$ is plotted as a function of $\xi $ for ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$ and $\theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2}$; (b) $F\left( \theta \right)$ is plotted as a function of $\xi $ for ${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m} = 1$ and $\theta = {{\text{π}} \mathord{\left/ {\vphantom {{\text{π}} 2}} \right. } 2}$.
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[1] 钟伟 2014 博士学位论文(杭州: 浙江大学)
Zhong W 2014 Ph. D. Dissertation (Hangzhou: Zhejiang University
[2] Lu X M, Wang X G 2021 Phys. Rev. Lett. 126 120503
Google Scholar
[3] Matteo G A P 2009 Int. J. Quant. Inf. 7 125
Google Scholar
[4] Helstrom C W 1967 Phys. Lett. A 25 101
Google Scholar
[5] Holevo A S 1982 Probabilistic and Statistical Aspects of Quantum Theory (North Holland Amsterdam) pp52–96, 160–168
[6] Yuen H P, Lax M 1973 IEEE Trans. Inf. Th. 19 740
Google Scholar
[7] Braunstein S, Caves C 1994 Phys. Rev. Lett. 72 3439
Google Scholar
[8] Braunstein S, Caves C, Milburn G 1996 Ann. Phys. 247 135
Google Scholar
[9] Giovannetti V, Lloyd S, Maccone L 2004 Science 306 1330
Google Scholar
[10] Giovannetti V, Lloyd S, Maccone L 2006 Phys. Rev. Lett. 96 010401
Google Scholar
[11] Giovannetti V, Lloyd S, Maccone L 2011 Nat. Photonics 5 222
Google Scholar
[12] 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
[13] Liu R, Chen Y, Jiang M, Yang X D, Wu Z, Li Y C, Yuan H D, Peng X H, Du J F 2021 Npj. Quantum. Inf. 7 170
Google Scholar
[14] 牛明丽, 王月明, 李志坚 2022 物理学报 71 090601
Google Scholar
Niu M L, Wang Y M, Li Z J 2022 Acta. Phys. Sin. 71 090601
Google Scholar
[15] Fisher R A 1922 Philosoph. Trans. Roy. Soc. London 222 309
Google Scholar
[16] Fisher R A 1925 Proc. Camb. Phil. Soc. 22 700
Google Scholar
[17] Helstrom C W 1969 J. Stat. Phys. 1 231
Google Scholar
[18] Holevo A S 2001 Statistical Structure of Quantum Theory (Berlin, Heidelberg: Springer) pp45–70
[19] Pezze L, Smerzi A, Oberthaler M K, Schmied R, Treutlein P 2018 Rev. Mod. Phys. 90 035005
Google Scholar
[20] 井晓幸 2016 博士学位论文(杭州: 浙江大学)
Jing X X 2016 Ph. D. Dissertation (Hangzhou: Zhejiang University
[21] Boixo S, Flammia S T, Caves C M, Geremia J M 2007 Phys. Rev. Lett. 98 090401
Google Scholar
[22] Liu W F, Zhang L H, Li C J 2010 Int J. Theor. Phys. 49 2463
Google Scholar
[23] 刘然, 吴泽, 李宇晨, 陈昱全, 彭新华 2023 物理学报 72 110305
Google Scholar
Liu R, Wu Z, Li Y C, Chen Y Q, Peng X H 2023 Acta Phys. Sin. 72 110305
Google Scholar
[24] Li N, Luo S L 2013 Phys. Rev. A 88 014301
Google Scholar
[25] Kourbolagh Y A, Azhdargalam M 2019 Phys. Rev. A 99 012304
Google Scholar
[26] 李岩2019 博士学位论文(太原: 山西大学)
Li Y 2019 Ph. D. Dissertation (Taiyuan: Shanxi University
[27] 武莹, 李锦芳, 刘金明 2018 物理学报 67 140304
Google Scholar
Wu Y, Li J F, Liu J M 2018 Acta Phys. Sin. 67 140304
Google Scholar
[28] Zhang X Y, Lu X M, Liu J, Ding W K, Wang X G 2023 Phys. Rev. A 107 012414
Google Scholar
[29] Li J N, Cui D Z 2023 Phys. Rev. A 108 012419
Google Scholar
[30] Sone A, Cerezo M, Beckey J L, Coles P J 2021 Phys. Rev. A 104 062602
Google Scholar
[31] Jafarzadeh M, Rangani J H, Amniat-Talab M. 2020 Proc. R. Soc. A 476 20200378
Google Scholar
[32] Liu X B, Jing J L, Tian Z H, Yao W P 2021 Phys. Rev. D 103 125025
Google Scholar
[33] Yao Y, Xing X, Li G, Wang X G, Sun C P 2014 Phys. Rev. A 89 042336
Google Scholar
[34] Hosler D, Kok P 2013 Phys. Rev. A 88 052112
Google Scholar
[35] Zhou T, Cui J X, Cao Y 2013 Quantum. Inf. Process 12 747
Google Scholar
[36] Gingrich R M and Adami C 2002 Phys. Rev. Lett. 89 270402
Google Scholar
[37] Gingrich R M, Bergou A J, Adami C 2003 Phys. Rev. A 68 042102
Google Scholar
[38] Luo S L 2000 Lett. Math. Phys. 53 243
Google Scholar
[39] Ren J R, Song S X 2010 Int J. Theor. Phys. 49 1317
Google Scholar
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