  We study a general corresponding principle between discrete-variable quantum
states and continuous-variable (especially, restricted on Gaussian) states via
quantum purification method. In the previous work, we have already investigated
an information-theoretic correspondence between the Gaussian maximally mixed
states (GMMSs) and their purifications known as Gaussian maximally entangled
states (GMESs) in [Phys. Lett. A {\bf 380}, 3607 (2016)]. We here compare an
$N\times N$-dimensional maximally entangled state to the GMES we proposed
previously, through an explicit calculation of quantum fidelity between those
entangled states. By exploiting the results, we naturally conclude that our
GMES is more suitable to the concept of \emph{maximally entangled} state in
Gaussian quantum information, and thus it might be useful or applicable for
quantum information tasks than the two-mode squeezed vacuum (TMSV) state in the
Gaussian regime.
