We derive a family of optimal protocols, in the sense of saturating the quantum Cramér-Rao bound, for measuring a linear combination of $d$ field amplitudes with quantum sensor networks, a key subprotocol of general quantum sensor network applications. We demonstrate how to select different protocols from this family under various constraints. Focusing primarily on entanglement-based constraints, we prove the surprising result that highly entangled states are not necessary to achieve optimality in many cases. Specifically, we prove necessary and sufficient conditions for the existence of optimal protocols using at most $k$-partite entanglement. We prove that the protocols which satisfy these conditions use the minimum amount of entanglement possible, even when given access to arbitrary controls and ancillas. Our protocols require some amount of time-dependent control, and we show that a related class of time-independent protocols fail to achieve optimal scaling for generic functions.

• Received 3 September 2022
• Revised 22 August 2023
• Accepted 6 September 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033228

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society