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 $p\ge 1$. 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.

Figure 1

Quantum metrology protocol of a many-body open quantum system. A protocol with $N$ two-level systems evolving under a long-range dissipator with two-body Lindblad operators accounting for the role of a monitoring environment (represented by the eyes) is illustrated. After repeating the process $\nu$ times, the value of a parameter $x$ is estimated with precision $\delta x$ that scales as $N^{-\beta }$.