Parameter estimation in the presence of the most general Gaussian dissipative reservoir

We analyze the performance of quantum parameter estimation in the presence of the most general Gaussian dissipative reservoir. We derive lower bounds on the precision of phase estimation and a closely related problem of frequency estimation. For both problems we show that it is impossible to achieve the Heisenberg limit asymptotically in the presence of such a reservoir. However, we also find that for any fixed number of probes used in the setup there exists a Gaussian dissipative reservoir, which, in principle, allows for the Heisenberg-limited performance for that number of probes. We discuss a realistic implementation of a frequency estimation scheme in the presence of a Gaussian dissipative reservoir in a cavity system.