Quantum-limited metrology with product states

Abstract

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 $\left(k\ll n\right)$ 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 $\left(k=2\right)$, implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the $O\left(n^{-3/2}\right)$ 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 $O\left(n^{-3/2}\right)$ sensitivity scaling for initial product states.