Entanglement-Constrained Quantum Metrology: Rapid Low-Entanglement Gains, Tapered High-Level Growth

It is a specific type of quantum correlated state that achieves optimal precision in parameterestimation under unitary encoding. We consider the potential experimental limitation on probe entanglement, and find a relation between achievable precision and initial probe 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.