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 (kn) 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/nᵏ, 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.