Minimum Entanglement Protocols for Function Estimation

TL;DR

This paper develops optimal quantum sensor network protocols for measuring linear combinations of field amplitudes, showing that minimal entanglement suffices for optimality and highlighting the importance of time-dependent control.

Contribution

It introduces a family of optimal measurement protocols saturating the quantum Cramér-Rao bound, characterizes entanglement requirements, and compares time-dependent and independent strategies.

Findings

Highly entangled states are not always necessary for optimality.

Protocols with minimal entanglement can achieve optimal measurement precision.

Time-dependent control is essential for optimal scaling in generic functions.

Abstract

We derive a family of optimal protocols, in the sense of saturating the quantum Cram\'{e}r-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 ancilla. 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.