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. this paper, we explore the application of information geometry in the realm of non-Hermitian quantum systems, focusing on the Fisher-Rao metric as a measure of parameter sensitivity. We approximate the Lindblad master equation for non-Hermitian Hamiltonians to analyze the temporal evolution of the quantum geometric metric. Utilizing the quantum spin Ising model with an imaginary magnetic field as an exemplar, we investigate energy spectrum and geometric metric evolution within PT-symmetry Hamiltonians. We demonstrate that detrimental effects of dissipation can be counteracted by introducing a control Hamiltonian, leading to improved accuracy in parameter estimation. Our work provides insights into the role of quantum control in mitigating dissipative impacts and enhancing the precision of quantum metrological tasks.