Fundamental limits to quantum metrology with noncommuting generators*

Precision metrology across many applications, e.g. gravitational-wave detection, has reached or is fast approaching the quantum limit. In the quantum regime, the fundamental limit on parameter estimation is set by the information-theoretic Quantum Cramer-Rao Bound, e.g. the Energetic Quantum Limit/Mizuno Theorem for gravitational-wave interferometers. Although this limit can be saturated in single-variable cases, for multiple and continuous parameter estimation it is missed by up to a factor of a square-root of two in the signal-to-noise ratio if the probe observables (the generators of the unitary transformation) do not commute. This is the case for detuned gravitational-wave interferometers where the amplitude quadrature of the intra-cavity light does not auto-commute at different times. In this work, we explore how the missing factor can be restored and the sensitivity improved. We also consider the effects of losses.