Universal shot-noise limit for quantum metrology with local Hamiltonians

Abstract

Quantum many-body interactions can induce quantum entanglement among particles, rendering them valuable resources for quantum-enhanced sensing. In this work, we derive a universal and fundamental bound for the growth of the quantum Fisher information. We apply our bound to the metrological protocol requiring only separable initial states, which can be readily prepared in experiments. By establishing a link between our bound and the Lieb-Robinson bound, which characterizes the operator growth in locally interacting quantum many-body systems, we prove that the precision cannot surpass the shot noise limit at all times in locally interacting quantum systems. This conclusion also holds for an initial state that is the non-degenerate ground state of a local and gapped Hamiltonian. These findings strongly hint that when one can only prepare separable initial states, nonlocal and long-range interactions are essential resources for surpassing the shot noise limit. This observation is confirmed through numerical analysis on the long-range Ising model. Our results bridge the field of many-body quantum sensing and operator growth in many-body quantum systems and open the possibility to investigate the interplay between quantum sensing and control, many-body physics and information scrambling

Figures (3)

• Figure 1: Comparison between our protocol (a) with the protocol in Ref. chu2023strongquantum (b). In our protocol (a), the information of the estimation parameter is encoded into the many-body quantum states through the many-body dynamics $U_{\lambda}(t)=e^{-\text{i}(\lambda\sum_{i}h_{X_{i}}+H_{1})t}$ while in Ref. chu2023strongquantum, the encoding dynamics given by $U_{\lambda}=e^{-\text{i}\lambda\sum_{i}h_{X_{i}}}$ with $X_{j}=\{j\}$. In our protocol, the initial state is chosen to be either a separable state or the nondegenerate ground states of a gapped and local Hamiltonian while in Ref. chu2023strongquantum the initial state is prepared through the many-body dynamics $U_{0}(t)$.
• Figure 2: (a) Numerical calculation of the operator diffusion in the TFI chain with $N\!=\!10$. (b) Coefficient $|\eta_{ij}^{(1)}|$ characterizing the decay of the two-body interactions. (c)-(f) Scaling of the QFI with respect to the number of spins at different times for differential initial separable spin coherent states $\ket{\psi_{0}}\!=\!\bigotimes_{i=1}^{N}[\cos(\theta/2)\ket{\uparrow}_{i}+\sin(\theta/2)e^{{\rm i\phi}}\ket{\downarrow}_{i}]$. Here numerical data are obtained by directly diagonalizing the Hamiltonian of the TFI model, while theoretical data are derived using results by mapping the TFI model to the free fermion model. The analytical result refers to Eq. (\ref{['eq:ANA']}). Other parameters used for the calculations are $J\!=\!2,$$\lambda\!=\!5$, and $\phi\!=\!0$.
• Figure 3: Numerical calculation of the operator diffusion in (a) the CI model (b) the LRI model with $N\!=\!10$. The scaling of the QFI with respect to the number of spins at different times for differential initial separable spin coherent states $\ket{\psi_{0}}\!=\!\bigotimes_{i=1}^{N}[\cos(\theta/2)\ket{\uparrow}_{i}\!+\!\sin(\theta/2)e^{{\rm i\phi}}\ket{\downarrow}_{i}]$ in (c)-(f) the CI model and (g) and (h) the LRI model. Other parameters used for the calculations are $J\!=\!\lambda\!=\!h\!=\!1,$$\phi\!=\!0$ in the CI model, and $J\!=\!1$, $\lambda\!=\!0.5,$$\alpha\!=\!3$ in the LRI model.