Higher-order corrections to the quantum Cramér-Rao bound

Quantum Fisher information and the associated quantum Cramér-Rao bound (QCRB) are fundamental tools in frequentist quantum metrology, offering both analytical simplicity and practical precision limits for parameter estimation. The QCRB sets a lower bound on the mean square error (MSE) in the idealized limit of infinite measurement trials (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>ν</a:mi><a:mo>→</a:mo><a:mi>∞</a:mi></a:mrow></a:math>). Here we perform a systematic expansion in powers of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mn>1</b:mn><b:mo>/</b:mo><b:mi>ν</b:mi></b:mrow></b:math> to refine MSE estimates in realistic, finite-resource scenarios. These corrections reveal differences between measurements that appear equally optimal under the QCRB. They also help to distinguish among multiple optimal state families for estimating an unknown unitary transformation. Additionally, we explore the Bhattacharyya bound and its quantum counterpart, which constrain these corrections. Our results are relevant for preasymptotic metrology, enabling optimized protocols with limited resources without reliance on numerical simulations.