  The problem of estimating multiple loss parameters of an optical system using
the most general ancilla-assisted parallel strategy is solved under energy
constraints. An upper bound on the quantum Fisher information matrix is derived
assuming that the environment modes involved in the loss interaction can be
accessed. Any pure-state probe that is number-diagonal in the modes interacting
with the loss elements is shown to exactly achieve this upper bound even if the
environment modes are inaccessible, as is usually the case in practice. We
explain this surprising phenomenon, and show that measuring the Schmidt bases
of the probe is a parameter-independent optimal measurement. Our results imply
that multiple copies of two-mode squeezed vacuum probes with an arbitrarily
small nonzero degree of squeezing, or probes prepared using single-photon
states and linear optics can achieve quantum-optimal performance in conjunction
with on-off detection. We also calculate explicitly the energy-constrained
Bures distance between any two product loss channels. Our results are relevant
to standoff image sensing, biological imaging, absorption spectroscopy, and
photodetector calibration.
