Nonlinear Quantum Metrology of Many-Body Open Systems

We introduce general bounds for the parameter estimation error in nonlinear quantum metrology of many-body open systems in the Markovian limit. Given a k-body Hamiltonian and p-body Lindblad operators, the estimation error of a Hamiltonian parameter using a Greenberger-Horne-Zeilinger state as a probe is shown to scale as N-[k-(p/2)], surpassing the shot-noise limit for 2k > p + 1. Metrology equivalence between initial product states and maximally entangled states is established for p >= 1. We further show that one can estimate the system-environment coupling parameter with precision N-(p/2), while many-body decoherence enhances the precision to N-k in the noise-amplitude estimation of a fluctuating k- body Hamiltonian. For the long-range Ising model, we show that the precision of this parameter beats the shot-noise limit when the range of interactions is below a threshold value.