Quantum parameter estimation of non-Hermitian systems with optimal measurements

TL;DR

This paper extends quantum parameter estimation techniques to non-Hermitian systems, deriving an expression for quantum Fisher information and identifying optimal measurements, with applications to $ ext{PT}$-symmetric Hamiltonians and their unique properties.

Contribution

It introduces a general framework for quantum parameter estimation in non-Hermitian systems, including an expression for QFI and conditions for optimal measurements applicable to both Hermitian and non-Hermitian Hamiltonians.

Findings

Derived an intuitive expression for QFI in non-Hermitian systems.

Identified conditions for optimal measurements in non-Hermitian quantum systems.

Analyzed QFI behavior at exceptional points in $ ext{PT}$-symmetric Hamiltonians.

Abstract

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 $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian, and give the optimal measurement. Surprisingly, we find some interesting properties of this $\mathcal{PT}$-symmetric Hamiltonian QFI, such as the mutations in QFI at EP. 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.