  In statistical physics, if we successively divide an equilibrium system into
two parts, we will face a situation that, within a certain length $\xi$, the
physics of a subsystem is no longer the same as the original system. Then the
extensive properties of the thermal entropy $S($AB$)= S($A$)+S($B$)$ is
violated. This observation motivates us to introduce the concept of correlation
entropy between two points, as measured by mutual information in the
information theory, to study the critical phenomena. A rigorous relation is
established to display some drastic features of the non-vanishing correlation
entropy of the subsystem formed by any two distant particles with long-range
correlation. This relation actually indicates the universal role of the
correlation entropy in understanding critical phenomena. We also verify these
analytical studies in terms of two well-studied models for both the thermal and
quantum phase transitions: two-dimensional Ising model and one-dimensional
transverse field Ising model. Therefore, the correlation entropy provides us
with a new physical intuition in critical phenomena from the point of view of
the information theory.
