### A.2 Proof of Eq. (376), (377) and (378)

In the lossless case [see Eq. (380)], the noises (double-sided) power spectral densities (376), (377) and
(378) are equal to
were
and
Therefore,
Note that the squeezing factor has gone.
Taking into account that

we obtain that
and
It is easy to show by direct calculation that
which proves Eq. (379).