  We propose a systematic scheme to reach the properties of two-dimensional
(2D) statistical and quantum systems by studying the effective
(1+1)-dimensional theory that is constructed from the tensor network
representation. On on hand, we discover that the degeneracy of the 2D system
can be determined by the purity of the boundary thermal state, which is the
density operator of the effective theory at zero (effective) temperature. On
the other hand, we find that the gapped (or critical) 2D system leads to a
gapped (or critical) effective (1+1)-dimensional theory whose criticality can
be accessed by the entanglement entropy $S$ of its ground state dubbed as
boundary pure state. We also uncover that for the critical systems, $S$ obeys
the same logarithmic law as that found in the critical 1D quantum chains, which
reads $S = (\kappa c/6)\log_2 D + const.$, with $c$ the central charge and
$\kappa$ a constant related to the scaling property of the correlation length
$\xi$ as $\xi \sim D^{\kappa}$. Such a scaling law presents an efficient way to
characterize the critical universality class of the original 2D systems. An
important implication of our work is that many well-established theories for 1D
quantum chains become available for studying 2D systems with the help of the
proposed lower dimensional correspondence.
