Precision measurement for open systems by non-hermitian linear response

The lower bound of estimated precision for a coherent parameter unitarily encoded in closed systems has been obtained, and such a lower bound is inversely proportional to the fluctuation of the encoding operator. In this paper, we first derive some general results regarding the lower bound of estimated precision for a dissipative parameter, which is non-unitarily encoded in open systems, by combining the law of error propagation and the non-hermitian linear response theory. This lower bound is related to the correlation of the encoding dissipative operator and the evolution time. We next demonstrate the utility of our general results by considering three different kinds of non-unitary encoding processes, including particle loss, relaxation, and dephasing. We finally compare the lower bound with the quantum Fisher information obtained by tomography and find they are consistent in the regime where the non-hermitian linear response applies. This lower bound can guide us to find the optimal initial states and detecting operators to significantly simplify the measurement process.