Ultimate precision of joint quadrature parameter estimation with a Gaussian probe

TL;DR

This paper develops a semidefinite programming approach to compute the Holevo Cramér-Rao bound for joint quadrature parameter estimation in Gaussian states, demonstrating optimal measurements vary with entanglement.

Contribution

It introduces a method to calculate the Holevo Cramér-Rao bound for Gaussian states and shows how optimal measurements depend on entanglement.

Findings

The bound is tight and achievable with Gaussian measurements.

Optimal measurement strategies differ for entangled versus separable states.

The method applies to symmetric two-mode squeezed thermal states.

Abstract

The Holevo Cram\'er Rao bound is a lower bound on the sum of the mean square error of estimates for parameters of a state. We provide a method for calculating the Holevo Cram\'er-Rao bound for estimation of quadrature mean parameters of a Gaussian state by formulating the problem as a semidefinite program. In this case, the bound is tight; it is attained by purely Guassian measurements. We consider the example of a symmetric two-mode squeezed thermal state undergoing an unknown displacement on one mode. We calculate the Holevo Cram\'er-Rao bound for joint estimation of the conjugate parameters for this displacement. The optimal measurement is different depending on whether the state is entangled or separable.