  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.
