Quantum-limited metrology with product states

TL;DR

This paper investigates the limits of quantum metrology using initial product states in nonlinear Hamiltonian systems, demonstrating that specific scaling of measurement precision is achievable without entanglement, especially in quadratic Hamiltonian scenarios.

Contribution

It derives the theoretical lower bounds on measurement uncertainty for product states in nonlinear quantum metrology and shows these bounds are attainable with simple measurements, challenging the necessity of entanglement.

Findings

Lower bound scales as 1/n^k for arbitrary states

Product states achieve a 1/n^(k-1/2) scaling

Quadratic Hamiltonian case achieves 1/n^(3/2) scaling without entanglement

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 (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.