Quantum-limited metrology with product states

We study the performance of initial product states of $n$-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear $k$-body $(k\ensuremath{\ll}n)$ Hamiltonian. We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initial states, the lower bound scales as $1/{n}^{k}$, and for initial product states, it scales as $1/{n}^{k\ensuremath{-}1/2}$. We show that the latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of a quadratic Hamiltonian $(k=2)$, implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the $O({n}^{\ensuremath{-}3/2})$ scaling for $k=2$; the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in the enhanced sensitivity scaling. We show that phase decoherence does not affect the $O({n}^{\ensuremath{-}3/2})$ sensitivity scaling for initial product states.