Optimal Local Measurements in Single-Parameter Quantum Metrology

Quantum measurement plays a crucial role in quantum metrology. Due to the limitations of experimental capabilities, collectively measuring multiple copies of probing systems can present significant challenges. Therefore, the concept of locality in quantum measurements must be considered. In this work, we investigate the possibility of achieving the Quantum Cramér-Rao Bound (QCRB) through local measurements (LM). We first demonstrate that if there exists a LM to saturate the QCRB for qubit systems, then we can construct another rank-1 local projective measurement to saturate the QCRB. In this sense, rank-1 local projective measurements are sufficient to analyze the problem of saturating the QCRB. For pure qubits, we propose two necessary and sufficient methods to determine whether and how a given parameter estimation model can achieve QCRB through LM. The first method, dubbed iterative matrix partition method (IMP) and based on unitary transformations that render the diagonal entries of a tracless matrix vanish, elucidates the underlying mathematical structure of LM as well as the local measurements with classical communications (LMCC), generalizing the result by [Zhou et al Quantum Sci. Technol. 5, 025005 (2020)], which only holds for the later case. We clarify that the saturation of QCRB through LM for the GHZ-encoded states is actually due to the self-similar structure in this approach. The second method, dubbed hierarchy of orthogonality conditions (HOC) and based on the parametrization of rank-1 measurements for qubit systems, allows us to construct several examples of saturating QCRB, including the three-qubit W states and $N$-qubit W states ($N \geq 3$). Our findings offer insights into achieving optimal performance in quantum metrology when measurement resources are limited.