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

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 a finite number of 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 programme. 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.

Automated summary

Finding the optimal attainable precisions in quantum multiparameter metrology is a non-trivial problem, often tackled by computing bounds on the accuracy of estimating physical quantities. A tighter bound for estimating multiple parameters simultaneously, particularly relevant in terms of experimental accessibility, has been introduced in this work. Efficiently computed as a semidefinite programme, this bound has implications for achieving the Holevo bound in quantum metrology experiments.