Evaluating the quantum Ziv–Zakai bound for phase estimation in noisy environments

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 Cramér-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 ) 10.1103/PhysRevLett.108.230401 ] remains a difficult challenge. In this paper, we not only derive the asymptotically tight QZZB for phase estimation with the photon loss and the phase diffusion by invoking the variational method and 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). In this asymptotically tight situation, 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 an useful guidance for investigating the noisy quantum parameter estimation.