  Going beyond the standard quantum limit in noisy quantum metrology is an
important and challenging task. Here we show how Dicke states can be used to
surpass the standard quantum limit and achieve the Heisenberg limit in open
quantum systems. The system we study has qubits symmetrically coupled to a
resonator and our objective is to estimate the coupling between the qubits and
the resonator. The time-dependent quantum Fisher information with respect to
the coupling is studied for this open quantum system where the same decay rates
are assumed on all qubits. We show that when the system is initialized to a
Dicke state with an optimal excitation number one can go beyond the standard
quantum limit and achieve the Heisenberg limit even for finite values of the
decays on the qubit and the resonator, particularly when the qubits and
resonator are strongly coupled. We compare our results against the highly
entangled GHZ state and a completely separable state and show that the GHZ
state performs quite poorly whereas under certain noise conditions the
separable state is able to go beyond the standard quantum limit due to
subsequent interactions with a resonator.
