  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.
