  Bosonic Gaussian states are a special class of quantum states in an infinite
dimensional Hilbert space that are relevant to universal continuous-variable
quantum computation as well as to near-term quantum sampling tasks such as
Gaussian Boson Sampling. In this work, we study entanglement within a set of
squeezed modes that have been evolved by a random linear optical unitary. We
first derive formulas that are asymptotically exact in the number of modes for
the R\'enyi-2 Page curve (the average R\'enyi-2 entropy of a subsystem of a
pure bosonic Gaussian state) and the corresponding Page correction (the average
information of the subsystem) in certain squeezing regimes. We then prove
various results on the typicality of entanglement as measured by the R\'enyi-2
entropy by studying its variance. Using the aforementioned results for the
R\'enyi-2 entropy, we upper and lower bound the von Neumann entropy Page curve
and prove certain regimes of entanglement typicality as measured by the von
Neumann entropy. Our main proofs make use of a symmetry property obeyed by the
average and the variance of the entropy that dramatically simplifies the
averaging over unitaries. In this light, we propose future research directions
where this symmetry might also be exploited. We conclude by discussing
potential applications of our results and their generalizations to Gaussian
Boson Sampling and to illuminating the relationship between entanglement and
computational complexity.
