Optimal waveform estimation of stochastic quantum signals

In experiments like the upcoming Gravity from Quantum Entanglement of Space-Time (GQuEST), the variance instead of the mean of a Gaussian quantum state encodes the parameter of interest. For GQuEST, this is the scale factor of the Verlinde-Zurek theory of Quantum Gravity. Although the Quantum Fisher information of Gaussian states is well-studied, the optimal protocol (initial quantum state and measurement scheme) for waveform estimation of a stochastic signal is unknown. We derive the optimal protocol and suggest how it could improve the search for Quantum Gravity. We also discuss the related problem of simultaneous estimation of mean and variance for sensing unmodelled physics and performing tests of general relativity using gravitational-wave detectors like LIGO.