  As laser interferometer gravitational wave (GW) detectors become quantum
noise dominated, understanding the fundamental limit on measurement sensitivity
imposed by quantum uncertainty is crucial to guide the search for further noise
reduction. Recent efforts have included applying ideas from quantum information
theory to GW detection -- specifically the quantum Cramer Rao bound, which is a
minimum bound on error in parameter estimation using a quantum state and is
determined by the state's quantum Fisher information (QFI) with respect to the
parameter. Identifying the QFI requires knowing the interaction between the
quantum measurement device and the signal, which was rigorously derived for GW
interferometer detectors in [Phys. Rev. D 98, 124006]. In this paper, we
calculate the QFI and fundamental quantum limit (FQL) for GW detection, and
furthermore derive explicit reciprocity relations involving the QFI which
summarize information exchange between the detector and a surrounding weak
quantum GW field. Specifically, we show that the GW power radiation by the
detector's quantum fluctuations are proportional to the QFI, and therefore
inversely proportional to its FQL. Similarly, the detector's decoherence rate
in a white noise GW bath can be explicitly related to the QFI/FQL. These
relations are fundamental and appear generalizable to a broader class of
quantum measurement systems.
