Universal Sensitivity Bound for Thermal Quantum Dynamic Sensing

Abstract

This work unifies the equilibrium and non-equilibrium frameworks of quantum metrology within the context of many-body systems. We investigate dynamic sensing schemes to derive an upper bound on the quantum Fisher information for probe states in thermal equilibrium with their environment. We establish that the dynamic quantum Fisher information for a thermal probe state is upper bounded by the degree of non-commutation between the transformed local generator and the Hamiltonian for the thermal state. Furthermore, we show that this upper bound scales as the square of the product of the inverse temperature and the evolution time. In the low-temperature limit, we establish an additional upper bound expressed as the seminorm of the commutator divided by the energy gap. We apply this thermal dynamic sensing scheme to various models, demonstrating that the dynamic quantum Fisher information satisfies the established upper bounds.

Figures (3)

• Figure 1: Schematics of the thermal dynamic sensing scheme. Gibbs state, $\rho_0=e^{-\beta H}/Z$ with $H=\sum_{k=1}^N H^{(k)}$, is used as the probe state and the parameter encoding process $U_\lambda$ is unitary. The parameter-dependent state $\rho_\lambda$ is still a Gibbs state, and we can obtain a general upper bound for the quantum Fisher information which is determined by the seminorm of the commutator $||i[H,h_\lambda]||$, where the transformed local generator $h_\lambda$ naturally appears.
• Figure 2: QFI as a function of $P=\tanh{\frac{\beta}{2}}$ for the thermal state $\rho_0=e^{-\beta J_z}/Z$. (a) QFI and its upper bound for a non-linear parameter encoding process $U_\lambda=e^{-i\lambda t J_x^2}$. (b) Comparison of the dynamic QFI of the non-linear parameter encoding process $U_\lambda=e^{-i\lambda t J_x^2}$ and the linear parameter encoding process $U_\lambda=e^{-i\lambda t J_x}$.
• Figure 3: QFI for the LMG model. (a) QFI and its bound as a function of evolution time, for a fixed inverse temperature $\beta=1.1$. (b) QFI and its bound as a function of inverse temperature $\beta$, for a fixed evolution time $t=3.14$.