Quantum parameter estimation of non-Hermitian systems with optimal measurements

Quantum parameter estimation with Hermitian systems has been applied in various fields, but there are relatively few results concerning non-Hermitian systems. Here, we study the quantum parameter estimation for general non-Hermitian Hamiltonians and derive an intuitive expression of quantum Fisher information (QFI) for pure states. Furthermore, we propose the condition for optimal measurements, which is applicable to both Hermitian and non-Hermitian Hamiltonians. To illustrate these results, we calculate and study the QFI of a specific parity-time (PT) -symmetric non-Hermitian Hamiltonian and give the optimal measurement. Surprisingly, we find some interesting properties of this PT-symmetric Hamiltonian QFI, such as the mutations in QFI at exceptional point. Moreover, we also compare the variance of estimation generated by the optimal measurement with the theoretical precision bound to verify the condition for optimal measurements we proposed.

Automated summary

In this study, we investigate quantum parameter estimation for general non-Hermitian Hamiltonians and derive an intuitive expression of quantum Fisher information for pure states. We also propose a condition for optimal measurements that is applicable to both Hermitian and non-Hermitian Hamiltonians. To demonstrate these findings, we analyze the quantum Fisher information and optimal measurement of a specific parity-time (PT) symmetric non-Hermitian Hamiltonian, revealing interesting properties such as mutations in the quantum Fisher information at exceptional points. Furthermore, we compare the variance of estimation generated by the optimal measurement with the theoretical precision bound to verify our proposed condition for optimal measurements.