  Quantum estimation of parameters defining open-system dynamics may be
enhanced by using ancillas that are entangled with the probe but are not
submitted to the dynamics. Here we consider the important problem of estimation
of transmission of light by a sample, with losses due to absorption and
scattering. We show, through the determination of the quantum Fisher
information, that the ancilla strategy leads to the best possible precision in
single-mode estimation, the one obtained for a Fock state input, through joint
photon-counting of probe and ancilla, which are modes of a bimodal squeezed
state produced by an optical parametric amplifier. This proposal overcomes the
challenge of producing and detecting high photon-number Fock states, and it is
quite robust against additional noise: we show that it is immune to phase noise
and the precision does not change if the incoming state gets disentangled.
Furthermore, the quantum gain is still present under moderate photon losses of
the input beams. We also discuss an alternative to joint photon counting, which
is readily implementable with present technology, and approaches the quantum
Fisher information result for weak absorption, even with moderate photons
losses of the input beams before the sample is probed: a time-reversal
procedure, placing the sample between two optical parametric amplifiers, with
the second undoing the squeezing produced by the first one. The precision of
estimation of the loss parameter is obtained from the average outgoing total
photon number and its variance. In both procedures, the state of the probe and
the detection procedure are independent of the value of the parameter.
