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量子Fisher信息给出参数估计的最优精度极限, 在量子度量学中有重要的应用. 近年来, 在量子系统中实现非厄米哈密顿量的理论与实验研究受到广泛关注. 本文研究基于非厄米哈密顿量本征态的参数估计, 给出其中单参数与两参数估计的量子Fisher信息及其量子Cramér-Rao下界, 计算与分析非互易、具有增益-耗散的Su-Schrieffer-Heeger模型, 非厄米量子Ising链、拓扑陈绝缘体模型和二能级系统中动量及外场参数估计的量子Fisher信息. 结果表明: 在这几个非厄米模型中, 对于单参数估计, 量子Fisher信息在能隙闭合区域和例外点附近显著增大, 从而提高参数估计的精度极限; 对于两参数估计, 量子Fisher信息矩阵的行列式在能隙闭合和例外点附近同样明显增大, 拓扑区域比平庸区域的整体评估精度更高, 且由陈数确定两参数估计误差的拓扑下界.
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关键词:
- 量子Fisher信息 /
- 参数估计 /
- 非厄米系统 /
- 拓扑态
Quantum Fisher information bounds the ultimate precision limit in the parameter estimation and has important applications in quantum metrology. In recent years, the theoretical and experimental studies of non-Hermitian Hamiltonians realized in quantum systems have attracted wide attention. Here, the parameter estimation based on eigenstates of non-Hermitian Hamiltonians is investigated, and the corresponding quantum Fisher information and quantum Cramér-Rao bound for the single-parameter and two-parameter estimations are given. In particular, the quantum Fisher information about estimating intrinsic momentum and external parameters in the non-reciprocal and gain-and-loss Su-Schrieffer-Heeger models, and non-Hermitian versions of the quantum Ising chain, Chern-insulator model and two-level system are calculated and analyzed. For these non-Hermitian models, the results show that in the case of single-parameter estimation in these non-Hermitian models, the quantum Fisher information increases significantly in the gapless regime and near the exceptional points, which can improve the accuracy limit of parameter estimation. For the two-parameter estimation, the determinant of the quantum Fisher information matrix also increases obviously near the gapless and exceptional points. In addition, a higher overall accuracy can be achieved in the topological regime than in the trivial regime, and the topological bound in two-parameter estimation can be determined by the Chern number.-
Keywords:
- quantum Fisher information /
- parameter estimation /
- non-Hermitian systems /
- topological states
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图 1 基于非互易SSH模型的单参数估计 (a)能隙
$ \Delta E $ 随t和δ的变化; (b)$ \delta=0.2 $ 时利用右本征矢$ |\psi^{{\rm{R}}}\rangle $ 评估k的量子Fisher信息$ F_k $ 随k和t的变化(上图)及其积分$ M_k $ 随t的变化(下图实线); (c)利用$ |\psi^{{\rm{R}}}\rangle $ 评估δ时$ M_\delta $ 随δ的变化(实线); (d)利用$ |\psi^{{\rm{R}}}\rangle $ 评估t时$ M_t $ 随t的变化(实线). 图(b)—(d)中的数据点表示利用左本征矢$ |\psi^{{\rm{L}}}\rangle $ 评估k, t或δ时相应的数值结果. 图中$ t^\prime=1 $ ,$ F_k $ 和$ M_\mu $ 做对数处理Fig. 1. Single-parameter estimation based on the non-reciprocal SSH model: (a) Energy gap
$ \Delta E $ as functions of t and δ; (b)$ F_k $ by the right eigenstate$ |\psi^{{\rm{R}}}\rangle $ as functions of k and t for estimating k (top) and the integration$ M_k $ by$ |\psi^{{\rm{R}}}\rangle $ as a function of t (solid line in the bottom)with$ \delta=0.2 $ ; (c) the integration$ M_\delta $ by using$ |\psi^{{\rm{R}}}\rangle $ (solid line) as a function of δ for estimating δ; (d) the integration$ M_t $ by using$ |\psi^{{\rm{R}}}\rangle $ (solid line) as a function of t for estimating t. The data points in panels (b)–(d) denote the corresponding numerical results for estimating k, t or δ by using the left eigenstate$ |\psi^{{\rm{L}}}\rangle $ .$ t^\prime=1 $ is set and$ F_k $ and$ M_\mu $ are logarithmically plotted in the picture.
图 2 原胞数
$ N=20 $ 的非互易SSH模型在不同边界耦合常数Γ下的单参数估计 (a)和(b)分别是开边界情况$ \varGamma = 0 $ 时本征能量的实部与虚部随参数t的变化; (c)为开边界条件下利用中间能态$ |\psi_{{\rm{mid}}}\rangle $ 和基态$ |\psi_{{\rm{ground}}}\rangle $ 评估参数t的量子Fisher信息$ F_{t} $ 随t的变化, 图中EP表示例外点, GP表示能隙闭合点; (d)和(e)分别是$ \varGamma = 0.1, 0.6 $ 时本征能量实部随t的变化; (f)和(g)为不同边界耦合常数Γ下$ F_{t} $ 随t的变化. 图中$ t^\prime=1,\; \delta=2/3 $ ,$ F_{t} $ 做对数处理Fig. 2. Single-parameter estimation based on the non-reciprocal SSH model with different boundary coupling coefficients Γ and the unit cell of
$ N=20 $ : (a) The real part and (b) the imaginary part of the eigen-spectrum as functions of t under open boundary condition with$ \varGamma = 0 $ ; (c)$ F_t $ as a function of t by the mid-spectrum eigenstate$ |\psi_{{\rm{mid}}}\rangle $ and the ground state$ |\psi_{{\rm{ground}}}\rangle $ for estimating t with$ \varGamma = 0 $ . Here EP and GP denote exceptional point and gapless point, respectively; (d) and (e) the real part of energy as a function of t with$ \varGamma = 0.1, 0.6 $ , respectively; (f) and (g)$ F_t $ as a function of t different values of Γ. In the figure,$ t^\prime=1,\; \delta=2/3 $ , and$ F_{t} $ is logarithmically plotted.
图 3 基于具有增益-耗散的SSH模型((a), (b))和非厄米量子Ising链((c), (d))的单参数估计 (a)能隙
$ \Delta E $ 随t和γ的变化, 有能隙区域能谱为实, 无能隙区域能谱为复且存在EP点; (b)$ \gamma=0.5 $ 和$ t=\{0.3, 1, 1.7\} $ 时评估k的量子Fisher信息$ F_k $ 随k的变化; (c)能隙$ \Delta E $ 随λ和h的变化, 能隙关闭处为复能量的铁磁态和顺磁态的相边界; (d)评估λ时的$ M_\lambda $ 随λ的变化. 图中$ t^\prime=1 $ 和$ J=1 $ Fig. 3. Single-parameter estimation based on the gain-and-loss SSH model ((a), (b)) and the non-Hermtian quantum Ising chain ((c), (d)): (a) Energy gap
$ \Delta E $ as functions of t and γ, and the gapped (gapless) region contains real (complex) eigen-spectrum (with exceptional points); (b)$ F_k $ as a function of k for estimating k with$ \gamma=0.5 $ and$ t=\{0.3, 1, 1.7\} $ ; (c) energy gap$ \Delta E $ as functions of λ and h, and the gapless line denotes the phase boundary between the ferromagnetic and paramagnetic states with complex energies; (d)$ M_\lambda $ as a function of λ for estimating λ.$ t^\prime=1 $ and$ J=1 $ are set.
图 4 基于非厄米陈绝缘体模型的两参数估计 (a)拓扑相图, 包括有能隙的拓扑和平庸区域, 分别对应陈数
$ C=1 $ 和$ C=0 $ , 以及无能隙区域; (b)$ \delta=0.2 $ 和$ t=\{1.4, 2, 2.6\} $ (依次从上到下)时评估$ \{k_x, k_y\} $ 的量子Fisher信息矩阵行列式$ {\rm{det}}{\cal{F}}{_{k_xk_y}} $ 随$ k_x, k_y $ 的变化; (c)$ \delta=0.2 $ 时评估$ \{k_x, k_y\} $ 的$ M_{k_xk_y} $ 和评估$ \{t, \delta\} $ 的$ M_{t\delta} $ 随t的变化. 图(c)中$ M_{\mu\nu} $ 和图(b)中间图$ {\rm{det}}{\cal{F}}{_{k_xk_y}} $ 做对数处理Fig. 4. Two-parameter estimation based on the non-Hermtian Chern-insulator model: (a) Topological phase diagram with gapped topological (
$ C=1 $ ), trivial ($ C=0 $ ), and gapless regions; (b) determinant of quantum Fisher information matrix$ {\rm{det}}{\cal{F}}{_{k_xk_y}} $ as functions of$ k_x $ and$ k_y $ for estimating$ \{k_x, k_y\} $ with$ \delta=0.2 $ and$ t=\{1.4, 2, 2.6\} $ (from top to bottom); (c) the integration$ M_{k_xk_y} $ for estimating$ \{k_x, k_y\} $ and$ M_{t\delta} $ for estimating$ \{t, \delta\} $ as a function of t with$ \delta=0.2 $ .$ M_{\mu\nu} $ in panel (c) and$ {\rm{det}}{\cal{F}}{_{k_xk_y}} $ in the middle of panels (b) are logarithmically plotted.
图 5 基于非厄米二能级系统的两参数估计评估
$ \{k_x, k_y\} $ 时, (a)$ M_{k_xk_y} $ 和(b)V随r和δ的变化. 图(a)中$ M_{k_xk_y} $ 做对数处理Fig. 5. Two-parameter estimation based on the non-Hermitian two-level system. (a)
$ M_{k_xk_y} $ and (b) V as functions of r and δ for estimating$ \{k_x, k_y\} $ .$ M_{k_xk_y} $ in panel (a) is logarithmically plotted. -
[1] Helstrom C W 1969 J. Stat. Phys. 1 231
Google Scholar
[2] Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439
Google Scholar
[3] Liu J, Yuan H D, Lu X M, Wang X G 2020 J. Phys. A: Math. Theor. 53 023001
Google Scholar
[4] Sidhu J S, Kok P 2020 AVS Quantum Sci. 2 014701
Google Scholar
[5] 任志红, 李岩, 李艳娜, 李卫东 2019 68 040601
Google Scholar
Ren Z H, Li Y, Li Y N, Li W D 2019 Acta Phys. Sin. 68 040601
Google Scholar
[6] Provost J P, Vallee G 1980 Commun. Math. Phys. 76 289
Google Scholar
[7] Mera B, Zhang A W, Goldman N 2022 SciPost Phys. 12 018
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[49] Yuce C 2018 Phys. Rev. A 97 042118
Google Scholar
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Google Scholar
[51] Guo Z X, Yu X J, Hu X D, Li Z 2022 Phys. Rev. A 105 053311
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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