Evaluating the quantum Ziv-Zakai bound in noisy environments

TL;DR

This paper derives the quantum Ziv-Zakai bound in noisy environments, analyzing its performance with different Gaussian states under photon loss and phase diffusion, providing insights for noisy quantum parameter estimation.

Contribution

It presents a general form of the QZZB in noisy environments and compares the estimation performance of various Gaussian states under noise.

Findings

QZZB for coherent states outperforms others under photon loss.

TMSVS can outperform CS in phase diffusion environments.

Provides a framework for noisy quantum parameter estimation analysis.

Abstract

In the highly non-Gaussian regime, the quantum Ziv-Zakai bound (QZZB) provides a lower bound on the available precision, demonstrating the better performance compared with the quantum Cram\'er-Rao bound. However, evaluating the impact of a noisy environment on the QZZB without applying certain approximations proposed by Tsang [Phys. Rev. Lett. 108, 230401 (2012)] remains a difficult challenge. In this paper, we not only derive the general form of the QZZB with the photon loss and the phase diffusion by invoking the technique of integration within an ordered product of operators, but also show its estimation performance for several different Gaussian resources, such as a coherent state (CS), a single-mode squeezed vacuum state (SMSVS) and a two-mode squeezed vacuum state (TMSVS). Our results indicate that compared with the SMSVS and the TMSVS, the QZZB for the CS always shows the better estimation performance under the photon-loss environment. More interestingly, for the phase-diffusion environment, the estimation performance of the QZZB for the TMSVS can be better than that for the CS throughout a wide range of phase-diffusion strength. Our findings will provide a useful guidance for investigating the noisy quantum parameter estimation.