Efficient computation of the Nagaoka--Hayashi bound for multi-parameter estimation with separable measurements

TL;DR

This paper introduces a new, efficiently computable bound for multi-parameter quantum estimation with separable measurements, providing a more experimentally accessible limit on estimation precision.

Contribution

It presents a tighter, computable bound for multi-parameter quantum estimation with separable measurements, bridging the gap between theoretical limits and experimental feasibility.

Findings

The bound can be efficiently computed via semidefinite programming.

Examples demonstrate the bound's applicability to finite-copy measurements.

Implications for saturating the Holevo bound are discussed.

Abstract

Finding the optimal attainable precisions in quantum multiparameter metrology is a non trivial problem. One approach to tackling this problem involves the computation of bounds which impose limits on how accurately we can estimate certain physical quantities. One such bound is the Holevo Cramer Rao bound on the trace of the mean squared error matrix. The Holevo bound is an asymptotically achievable bound when one allows for any measurement strategy, including collective measurements on many copies of the probe. In this work we introduce a tighter bound for estimating multiple parameters simultaneously when performing separable measurements on finite copies of the probe. This makes it more relevant in terms of experimental accessibility. We show that this bound can be efficiently computed by casting it as a semidefinite program. We illustrate our bound with several examples of collective measurements on finite copies of the probe. These results have implications for the necessary requirements to saturate the Holevo bound.