  Finding pure states in an enlarged Hilbert space that encode the mixed state
of a quantum field theory as a partial trace is necessarily a challenging task.
Nevertheless, such purifications play the key role in characterizing quantum
information-theoretic properties of mixed states via entanglement and
complexity of purifications. In this article, we analyze these quantities for
two intervals in the vacuum of free bosonic and Ising conformal field theories
using, for the first time, the~most general Gaussian purifications. We provide
a comprehensive comparison with existing results and identify universal
properties. We further discuss important subtleties in our setup: the massless
limit of the free bosonic theory and the corresponding behaviour of the mutual
information, as well as the Hilbert space structure under the Jordan-Wigner
mapping in the spin chain model of the Ising conformal field theory.
