Characterizing the Robustness of POVMs for Metrology in the Presence of Noise

For a given input state used in sensing, the quantum Cramér-Rao bound (QCRB) sets a fundamental limit on the precision achievable by an estimator of an unknown parameter, determined by the inverse of the quantum Fisher information (QFI). The QFI serves as an upper bound on the classical Fisher information (CFI), representing the maximum extractable information about the unknown parameter from measurements on a physical system. Thus, a central goal in quantum estimation is to find a measure, described by a POVM, that saturates the QFI (achieve maximum CFI), and thereby achieving the QCRB. Although multiple POVMs can yield CFI values equal to the QFI, we show that not all POVMs are equally robust to noise in the probe state. Furthermore, we develop a framework for understanding why certain measurements are less robust, and we quantify how fragile these measurement bases are to noise. Our findings reveal that some measurements, which achieve maximum CFI for a pure state, can exhibit a dramatic reduction in CFI, approaching to zero, even in the presence of very weak noise.