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Finding pure states in an enlarged Hilbert space that encode the mixed stateof a quantum field theory as a partial trace is necessarily a challenging task.Nevertheless, such purifications play the key role in characterizing quantuminformation-theoretic properties of mixed states via entanglement andcomplexity of purifications. In this article, we analyze these quantities fortwo intervals in the vacuum of free bosonic and Ising conformal field theoriesusing, for the first time, the~most general Gaussian purifications. We providea comprehensive comparison with existing results and identify universalproperties. We further discuss important subtleties in our setup: the masslesslimit of the free bosonic theory and the corresponding behaviour of the mutualinformation, as well as the Hilbert space structure under the Jordan-Wignermapping in the spin chain model of the Ising conformal field theory.

Entanglement and Complexity of Purification in (1+1)-dimensional free Conformal Field Theories