Non-Hermitian effects on the quantum parameter estimation in pseudo-Hermitian systems

TL;DR

This paper develops a framework for calculating Quantum Fisher Information in non-Hermitian quantum systems, revealing how non-Hermiticity influences parameter estimation and enabling advancements in quantum metrology.

Contribution

It introduces an explicit QFI expression for non-Hermitian systems using spectral decomposition and validates it through pseudo-Hermitian and PT-symmetric models.

Findings

Derived a QFI formula applicable to non-Hermitian systems.

Validated the formula with pseudo-Hermitian and PT-symmetric systems.

Showed non-Hermiticity can enhance quantum parameter estimation.

Abstract

Quantum Fisher Information (QFI) is a fundamental quantity in quantum parameter estimation theory, characterizing the ultimate precision bound of parameter estimation. In this work, we investigate QFI for quantum states in non-Hermitian systems. By employing the projected Hilbert space method and spectral decomposition, we derive an explicit expression for the QFI in terms of the density matrix and parameter generators. This formulation not only recovers the well-known results in the Hermitian case but also captures the non-Hermitian effects induced by the time-dependent norm of the state. To validate our theoretical framework, we analyze a single-qubit pseudo-Hermitian system and apply Naimark dilation theory to embed it into an equivalent Hermitian system. The comparison between the original and dilated systems demonstrates the consistency and applicability of the proposed QFI formula in non-Hermitian settings. In addition, we investigate a $\mathcal{PT}$-symmetric system to further explore the influence of non-Hermiticity on QFI. Our findings offer a new perspective for analyzing and enhancing QFI in non-Hermitian systems, paving the way for promising applications in non-Hermitian quantum metrology and sensing.