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 (LMs). 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 the QCRB through LMs. The first method, dubbed the iterative matrix partition method and based on unitary transformations that render the diagonal entries of a traceless matrix vanish, elucidates the underlying mathematical structure of LMs as well as the local measurements with classical communications (LMCC), generalizing the result by Zhou . [], which only holds for the later case. We clarify that the saturation of the QCRB through LMs for the Greenberger–Horne–Zeilinger-encoded states is actually due to the self-similar structure in this approach. The second method, dubbed the hierarchy of orthogonality conditions and based on the parametrization of rank-1 measurements for qubit systems, allows us to construct several examples of saturating QCRBs, including the three-qubit <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>W</a:mi></a:math> states and the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>N</b:mi></b:math>-qubit <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>W</c:mi></c:math> states (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>N</d:mi><d:mo>≥</d:mo><d:mn>3</d:mn></d:mrow></d:math>). Our findings offer insights into achieving optimal performance in quantum metrology when measurement resources are limited.