  In quantum spin chains at criticality, two types of scaling for the
entanglement entropy exist: one comes from conformal field theory (CFT), and
the other is for entanglement support of matrix product state (MPS)
approximation. They indicates that the matrix dimension of the MPS represents a
length scale of spin correlation. On the other hand, the quantum spin-chain
models can be mapped onto two-dimensional (2D) classical ones. Motivated by the
scaling and the mapping, we introduce new entanglement entropy for 2D classical
spin configuration as well as entanglement spectrum, and examine their basic
properties in Ising and 3-state Potts models on the square lattice. They are
defined by the singular values of the reduced density matrix for a Monte Carlo
snapshot. We find scaling relations concerned with length scales in the
snapshot at $T_{c}$. There, the spin configuration is fractal, and various
sizes of ordered clusters coexist. Then, the singular values automatically
decompose the original snapshot into a set of images with different length
scale. This is the origin of the scaling. In contrast to the MPS scaling,
long-range spin correlation can be described by only few singular values.
Furthermore, we find multiple gaps in the entanglement spectrum, and in
contrast to standard topological phases, the low-lying entanglement levels
below the gap represent spontaneous symmetry breaking. Based on these
observations, we discuss about the amount of information contained in one
snapshot in a viewpoint of the CFT scaling.
