  Enhancing the precision of a measurement requires maximizing the information
that can be gained about the quantity of interest from probing a system. For
optical based measurements, such an enhancement can be achieved through two
approaches, increasing the number of photons used to interrogate the system and
using quantum states of light to increase the amount of quantum Fisher
information gained per photon. Here we consider the use of quantum states of
light with a large number of photons, namely the bright single-mode and
two-mode squeezed states, that take advantage of both of these approaches for
the problem of transmission estimation. We show that, in the limit of large
squeezing, these states approach the maximum possible quantum Fisher
information per photon for transmission estimation that is achieved with the
Fock state and the vacuum two-mode squeezed state. Since the bright states we
consider can be generated at much higher powers than the quantum states that
achieve the maximum quantum Fisher information per photon, they can achieve an
much higher absolute precision as quantified by the quantum Cram\'er-Rao bound.
We discuss the effects of losses external to the system on the precision of
transmission estimation and identify simple measurements techniques that can
saturate the quantum Cram\'er-Rao bound for the bright squeezed states even in
the presence of such external losses.
