Precision assessment in non-Hermitian systems: a comparative study of three formalisms

TL;DR

This paper compares three approaches for assessing quantum Fisher information in non-Hermitian systems, highlighting their differences in physical interpretation and emphasizing the metric formalism's advantages for consistent quantum metrology.

Contribution

It provides a comparative analysis of three probability-conserving formalisms for QFI in non-Hermitian quantum systems, clarifying their physical implications and recommending the metric formalism.

Findings

All three formalisms conserve probability.

The simple normalization can lead to unphysical results.

The metric formalism offers a coherent framework for quantum metrology.

Abstract

Quantifying measurement precision in quantum systems is vital for advancing quantum technologies such as sensing, communication, and computation. The quantum Fisher information (QFI) sets the ultimate precision bound in Hermitian systems; however, extending this concept to non-Hermitian systems, even those with real spectra, poses conceptual challenges due to their non-unitary dynamics. We compare three probability-conserving approaches for evaluating QFI in such systems: (i) simple normalization, (ii) metric formalism, and (iii) master-equation framework. Although all three ensure probability conservation, they differ in physical interpretation and in how they quantify estimation precision. Our study is particularly motivated by previous studies that have shown that the simple normalization method for non-Hermitian Hamiltonian generated dynamics may lead to misleading or even unphysical conclusions for certain quantum information theoretic tasks. We emphasize, in this article, that the metric formalism naturally enables the use of standard Hermitian metrology tools in cases where it provides a coherent and physically consistent framework for non-Hermitian systems.