  Phase-insensitive optical amplifiers uniformly amplify each quadrature of an
input field and are of both fundamental and technological importance. We find
the quantum limit on the precision of estimating the gain of a quantum-limited
phase-insensitive optical amplifier using a multimode probe that may also be
entangled with an ancilla system. In stark contrast to the sensing of loss
parameters, the average photon number $N$ and number of input modes $M$ of the
probe are found to be equivalent and interchangeable resources for optimal gain
sensing. All pure-state probes whose reduced state on the input modes to the
amplifier is diagonal in the multimode number basis are proven to be
quantum-optimal under the same gain-independent measurement. We compare the
best precision achievable using classical probes to the performance of an
explicit photon-counting-based estimator on quantum probes and show that an
advantage exists even for single-photon probes and inefficient photodetection.
A closed-form expression for the energy-constrained Bures distance between two
product amplifier channels is also derived.
