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Quantum Fisher information plays a vital role in the field of quantum metrology and quantum information, because it not only quantifies the ultimate precision bound of parameter estimation but also provides criteria for entanglement detection. Nevertheless, experimentally extracting quantum Fisher information is intractable. Quantum state tomography is a typical approach to obtaining the complete information about a quantum system and extract quantum Fisher information. However it becomes infeasible for large-scale quantum systems owing to the exponentially growing complexity. In this paper, we present a general relationship between quantum Fisher information and the overlap of quantum states. Specifically, we show that for pure states, the quantum Fisher information can be exactly extracted from the overlap, whereas for mixed states, only the lower bound can be obtained. We also develop a protocol for measuring the overlap of quantum states, which only requires one additional auxiliary qubit and a single measurement for pure state. Our protocol is more efficient and scalable than previous approaches because it requires less time and fewer measurements. We use this protocol to characterize the multiparticle entanglement in a three-body interaction system undergoing adiabatic quantum phase transition, and experimentally demonstrate its feasibility for the first time in a nuclear magnetic resonance quantum system. We conduct our experiment on a 4-qubit nuclear magnetic resonance quantum simulator, three of which are used to simulate the quantum phase transition in a three-body interaction system, and the remaining one is used as the auxiliary qubit to detect the overlap of the quantum state. We use gradient ascent pulse engineering pulses to implement the process of evolution. By measuring the auxiliary qubit, the experimental results of quantum Fisher information are obtained and match well with the theoretical predictions, thus successfully characterizing the multiparticle entanglement in a practical quantum system. We further confirm our results by performing quantum state tomography on some quantum states in the adiabatic process. The experimentally reconstructed quantum states are close to the corresponding instantaneous ground states.
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图 1 测量量子态重叠的量子线路 (a) 当待测量子系统S是混态时, 系统中需要添加一个辅助量子比特A和额外的待测系统S的复制来测量
${\mathscr{D}}$ ; (b) 当待测量子系统S是纯态时, 系统中仅需要添加一个额外的辅助量子比特A来测量${\mathscr{D}}$ Figure 1. Quantum circuit for measuring the overlap
${\mathscr{D}}$ of quanum states: (a) An auxiliary qubit A and an additional copy of the system S are added into the system for measuring${\mathscr{D}}$ when S is mixed; (b) only an auxiliary qubit A are added into the system for measuring${\mathscr{D}}$ when S is pure.
图 2 利用量子Fisher信息实现实验多体纠缠刻画示意图 (a) 基于平均量子Fisher信息的多体纠缠判据; (b) 用于实验模拟耦合了辅助量子比特的三自旋相互作用系统13C-iodotriuroethylene样品分子结构及其他相关参数. 对角部分与非对角部分分别表示化学位移与J耦合大小(单位均为Hz)
Figure 2. Schematic diagram of experimentally characterizing the multiparticle entanglement in three-body interaction system with quantum Fisher information: (a) Criteria for multiparticle entanglement based on the average of quantum Fisher information; (b) molecular structure and the relevant parameters of 13C-iodotriuroethylene for simulating the three-body interaction system coupling with an auxiliary qubit. The diagonal and off-diagonal elements represent chemical shifts and J-couplings (all in Hz), respectively.
图 3 利用量子态重叠提取到的平均量子Fisher信息的实验结果 (a), (b), (c)分别对应哈密顿量
$H_{zz}, H_{zzz}, H_{zzz}^\prime$ 的结果Figure 3. Experimental result of the average of quantum Fisher information extracted from overlap: (a), (b), (c) correspond to the result of Hamiltonian
$H_{zz}, H_{zzz}, H_{zzz}^\prime$ , respectively.
图 4 绝热过程制备的基态密度矩阵的实验重构结果 (a), (b), (c)分别对应哈密顿量
$H_{zz}, H_{zzz}, H_{zzz}^\prime$ Figure 4. Experimentally reconstructed density matrices of ground states prepared by adiabatic process: (a), (b), (c) correspond to the result of Hamiltonian
$H_{zz}, H_{zzz}, H_{zzz}^\prime$ , respectively. -
[1] Giovannetti V, Lloyd S, Maccone L 2006 Phys. Rev. Lett. 96 010401
Google Scholar
[2] Giovannetti V, Lloyd S, Maccone L 2011 Nat. Photon. 5 222
Google Scholar
[3] 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
[4] 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
[5] Niu M L, Wang Y M, Li Z J 2022 Acta. Phys. Sin. 71 090601
Google Scholar
[6] Kaubruegger R, Silvi P, Kokail C, van Bijnen R, Rey A M, Ye J, Kaufman A M, Zoller P 2019 Phys. Rev. Lett. 123 260505
Google Scholar
[7] Yang X D, Thompson J, Wu Z, Gu M L, Peng X H, Du J F 2020 Npj Quantum Inf. 6 62
Google Scholar
[8] Koczor B, Endo S, Jones T, Matsuzaki Y, Benjamin S C 2020 New J. Phys. 22 083038
Google Scholar
[9] Kaubruegger R, Vasilyev D V, Schulte M, Hammerer K, Zoller P 2021 Phys. Rev. X. 11 041045
[10] 陈然一鎏, 赵犇池, 宋旨欣, 赵炫强, 王琨, 王鑫 2021 物理学报 70 210302
Google Scholar
Chen R Y L, Zhao B C, Song Z X, Zhao X Q, Wang K, Wang X 2021 Acta. Phys. Sin. 70 210302
Google Scholar
[11] Marciniak C D, Feldker T, Pogorelov I, Kaubruegger R, Vasilyev D V, van Bijnen R, Schindler P, Zoller P, Blatt R, Monz T 2022 Nature 603 604
Google Scholar
[12] Hyllus P, Laskowski W, Krischek R, Schwemmer C, Wieczorek W, Weinfurter H, Pezze L, Smerzi A 2012 Phys. Rev. A 85 022321
Google Scholar
[13] Tóth G 2012 Phys. Rev. A 85 022322
Google Scholar
[14] Hauke P, Heyl M, Tagliacozzo L, Zoller P 2016 Nat. Phys. 12 778
Google Scholar
[15] Li N, Luo S L 2013 Phys. Rev. A 88 014301
Google Scholar
[16] Hong Y, Luo S, Song H 2015 Phys. Rev. A 91 042313
Google Scholar
[17] Mirkhalaf S S, Smerzi A 2017 Phys. Rev. A 95 022302
Google Scholar
[18] Yu M, Li D X, Wang J C, Chu Y M, Yang P C, Gong M S, Goldman N, Cai J M 2021 Phys. Rev. Research 3 043122
Google Scholar
[19] Rath A, Branciard C, Minguzzi A, Vermersch B 2021 Phys. Rev. Lett. 127 260501
Google Scholar
[20] Garttner M, Hauke P, Rey A M 2018 Phys. Rev. Lett. 120 040402
Google Scholar
[21] Fiderer L J, Fraisse J M E, Braun D 2019 Phys. Rev. Lett. 123 250502
Google Scholar
[22] Pezzè L, Smerzi A, Oberthaler M K, Schmied R, Treutlein P 2018 Rev. Mod. Phys. 90 035005
Google Scholar
[23] Yuan H 2016 Phys. Rev. Lett. 117 160801
Google Scholar
[24] Macrì T, Smerzi A, Pezzè L 2016 Phys. Rev. A 94 010102
Google Scholar
[25] Ekert A K, Alves C M, Oi D K, Horodecki M, Horodecki P, Kwek L C 2002 Phys. Rev. Lett. 88 217901
Google Scholar
[26] Quan H T, Song Z, Liu X F, Zanardi P, Sun C P 2006 Phys. Rev. Lett. 96 140604
Google Scholar
[27] Zhang J, Peng X, Rajendran N, Suter D 2008 Phys. Rev. Lett. 100 100501
Google Scholar
[28] Peng X H, Zhang J F, Du J F, Suter D 2010 Phys. Rev. A 81 042327
Google Scholar
[29] Ding Z, Liu R, Radhakrishnan C, Ma W C, Peng X H, Wang Y, Byrnes T, Shi F Z, Du J F 2021 Npj Quantum Inf. 7 145
Google Scholar
[30] Peng X H, Zhu X W, Fang X M, Feng M, Gao K L, Yang X D, Liu M L 2001 Chem. Phys. Lett. 340 509
Google Scholar
[31] Albash T, Lidar D A 2018 Rev. Mod. Phys. 90 015002
Google Scholar
[32] Khaneja N, Reiss T, Kehlet C, Schulte-Herbruggen T, Glaser S J 2005 J. Magn. Reson. 172 296
Google Scholar
[33] 李俊, 崔江煜, 杨晓东, 罗智煌, 潘健, 余琦, 李兆凯, 彭新华, 杜江峰 2018 物理学报 64 167601
Google Scholar
Li J, Cui J Y, Yang X D, Luo Z H, Pan J, Yu Q, Li Z K, Peng X H, Du J F 2018 Acta. Phys. Sin. 64 167601
Google Scholar
[34] 孔祥宇, 朱垣晔, 闻经纬, 辛涛, 李可仁, 龙桂鲁 2018 物理学报 67 220301
Google Scholar
Kong X Y, Zhu Y Y, Wen J W, Xin T, Li K R, Long G L 2018 Acta. Phys. Sin. 67 220301
Google Scholar
[35] Wang T L, Wu L N, Yang W, Jin G R, Lambert N, Nori F 2014 New J. Phys. 16 063039
Google Scholar
[36] Yin S Y, Song J, Zhang Y J, Liu S T 2019 Phys. Rev. B 100 184417
Google Scholar
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