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 < 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-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(-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(-3/2)) sensitivity scaling for initial product states.