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The quantum Cram\'er-Rao bound (QCRB) sets a fundamental limit for themeasurement of classical signals with detectors operating in the quantumregime. Using linear-response theory and the Heisenberg uncertainty relation,we derive a general condition for achieving such a fundamental limit. Whenapplied to classical displacement measurements with a test mass, this conditionleads to an explicit connection between the QCRB and the Standard Quantum Limitwhich arises from a tradeoff between the measurement imprecision and quantumbackaction; the QCRB can be viewed as an outcome of a quantum non-demolitionmeasurement with the backaction evaded. Additionally, we show that the testmass is more a resource for improving measurement sensitivity than a victim ofthe quantum backaction, which suggests a new approach to enhancing thesensitivity of a broad class of sensors. We illustrate these points with laserinterferometric gravitational wave detectors.

Towards the Fundamental Quantum Limit of Linear Measurements of Classical Signals