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Chinese Journal of Physics

Quantum Ziv-Zakai Bound for Displacement Parameter Estimation in Photon Loss Environment

Wei Ye, Songtao Zhang, Ting Zhu, Ying Xia, Huan Zhang, Shixun You, Wenwen Hu, Jiujiu Chen, Jing Peng, Daisheng Zhang, Shoukang Chang

Published Aug 1, 2026• 0 citations
Publisher
DOI
10.1016/j.cjph.2026.08.032

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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.
In this work, we address the central problem about how to effectively find the available precision limit of unknown parameters. In the framework of the quantum Ziv-Zakai bound (QZZB), we employ noiseless linear amplification (NLA)techniques to an initial coherent state (CS) as the probe state, and focus on whether the phase estimation performance is improved significantly in noisy scenarios, involving the photon-loss and phase-diffusion cases. More importantly, we also obtain two kinds of Heisenberg error limits of the QZZB with the NLA-based CS in these noisy scenarios, making comparisons with both the Margolus-Levitin (ML) type bound and the Mandelstam-Tamm (MT) type bound. Our analytical results show that in cases of photon loss and phase diffusion, the phase estimation performance of the QZZB can be improved remarkably by increasing the NLA gain factor. Particularly, the improvement is more pronounced with severe photon losses. Furthermore in minimal photon losses, our Heisenberg error limit shows better compactness than the cases of the ML-type and MT-type bounds. Our findings will provide an useful guidance for accomplishing more complex quantum information processing tasks.