Upper bounds on the Holevo Cram\'er-Rao bound for multiparameter quantum parametric and semiparametric estimation

TL;DR

This paper explores bounds on quantum estimation precision, showing the relationships between different bounds and providing tighter bounds, especially for Gaussian states, advancing understanding in quantum multiparameter estimation.

Contribution

It generalizes the Helstrom CRB to semiparametric settings, establishes an upper limit on the Holevo CRB relative to the generalized Helstrom CRB, and identifies optimal measurements for Gaussian states.

Findings

Holevo CRB cannot exceed twice the generalized Helstrom CRB

A tighter intermediate bound is proposed

Gaussian measurements can achieve half the quantum Fisher information for Gaussian states

Abstract

We formulate multiparameter quantum estimation in the parametric and semiparametric setting. While the Holevo Cram\'er-Rao bound (CRB) requires no substantial modifications in moving from the former to the latter, we generalize the Helstrom CRB appropriately. We show that the Holevo CRB cannot be greater than twice the generalized Helstrom CRB. We also present a tighter, intermediate, bound. Finally, we show that for parameters encoded in the first moments of a Gaussian state there always exists a Gaussian measurement that gives a classical Fisher information matrix that is one-half of the quantum Fisher information matrix.