Entanglement-constrained quantum metrology: Rapid low-entanglement gains and tapered high-level growth
Phys. Rev. A 113, 012403 – Published 2 January, 2026
DOI: https://doi.org/10.1103/96bz-51yp
Abstract
A specific type of quantum correlated state achieves optimal precision in parameter estimation under unitary encoding. However, a general, quantitative understanding of how the available entanglement resource fundamentally limits metrological performance has remained largely unexplored. We address this by considering pure probe states undergoing unitary encoding and analyze the achievable precision for a fixed amount of initial entanglement, thereby establishing a direct relation between the optimal precision and the available input entanglement in both bipartite and multipartite scenarios. For two-qubit probes, we analytically derive an exact relationship between the entanglement-constrained optimal quantum Fisher information and the limited initial entanglement, measured via both generalized geometric measure and entanglement entropy. We demonstrate that this fundamental relationship persists across the same range of the entanglement measures even when higher-dimensional bipartite probes are considered. Furthermore, we identify the specific states that realize maximum precision in these scenarios. Additionally, by considering the geometric measure of entanglement, we extend our approach to multiqubit probes. We find that in every case, the optimal quantum Fisher information exhibits a universal behavior: a steep increase in the low-entanglement regime, followed by a gradual and nearly saturated improvement as the probe entanglement approaches values close to those required for achieving the Heisenberg limit.
