Tighter Heisenberg limits for phase estimation in noisy environments based on the quantum Ziv-Zakai bound

In exploring the precision limits of quantum metrology, the quest for a tighter Heisenberg limit in real-world environments becomes a key challenge. For this reason, in this paper, we propose a tighter Heisenberg limit for phase estimation, called the photon loss (PL)-type bound and the photon diffusion (PD)-type bound in realistic scenarios under the framework of quantum Ziv-Zakai bound (QZZB). In order to demonstrate the superiority of the proposed Heisenberg limits for phase estimation, as a comparison, we also introduce the Margolus-Levitin (ML)-type and Mandelstam-Tamm (MT)-type bounds [Phys. Rev. A90, 043818 (2014)10.1103/PhysRevA.90.043818] based on the QZZB framework when considering the same initial states, i.e., a coherent state, a superposition state of the vacuum and general Fock states, and an even coherent state. The simulation results show that for photon-loss scenario, the PL-type bounds for all given initial states are closer to the QZZB than the ML-type and MT-type bounds, thereby exhibiting a tighter Heisenberg limit. In contrast, for the phase-diffusion scenario, when the phase diffusion strength exceeds a certain threshold, the tightness of PD-type bounds for the coherent state and superposition state can present better than that of the ML-type and MT-type bounds. Furthermore, the tighter QZZB over the QCRB can be achieved using the superposition state in photon loss or the even coherent state in phase diffusion.