  In spite of their simple description in terms of rotations or symplectic
transformations in phase space, quadratic Hamiltonians such as those modeling
the most common Gaussian operations on bosonic modes remain poorly understood
in terms of entropy production. For instance, determining the von Neumann
entropy produced by a Bogoliubov transformation is notably a hard problem, with
generally no known analytical solution. Here, we overcome this difficulty by
using the replica method, a tool borrowed from statistical physics and quantum
field theory. We exhibit a first application of this method to the field of
quantum optics, where it enables accessing entropies in a two-mode squeezer or
optical parametric amplifier. As an illustration, we determine the entropy
generated by amplifying a binary superposition of the vacuum and an arbitrary
Fock state, which yields a surprisingly simple, yet unknown analytical
expression.
