Information Geometry and Parameter Sensitivity of Non-Hermitian Hamiltonians

Information geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the Fisher-Rao metric with the non-Hermitian systems. By approximating the Lindblad master equation in the non-Hermitian Hamiltonian, we calculate the time evolution of the quantum geometric metric. Finally, we give an example of the quantum spin Ising model of the imaginary magnetic field, explore the energy spectrum of $\mathcal{PT}$-symmetric Hamiltonian and the evolution of geometric metric, and discuss that the dissipative effect of the imaginary magnetic field can be eliminated under the condition of adding the control Hamiltonian, so as to improve the accuracy of parameter estimation.