  This review paper is intended for scholars with different backgrounds,
possibly in only one of the subjects covered, and therefore little background
knowledge is assumed. The first part is an introduction to classical and
quantum information theory (CIT, QIT): basic definitions and tools of CIT are
introduced, such as the information content of a random variable, the typical
set, and some principles of data compression. Some concepts and results of QIT
are then introduced, such as the qubit, the pure and mixed states, the Holevo
theorem, the no-cloning theorem, and the quantum complementarity. In the second
part, two applications of QIT to open problems in theoretical physics are
discussed. The black hole (BH) information paradox is related to the phenomenon
of the Hawking radiation (HR). Consid- ering a BH starting in a pure state,
after its complete evaporation only the Hawking radiation will remain, which is
shown to be in a mixed state. This either describes a non-unitary evolution of
an isolated system, contradicting the evolution postulate of quantum mechanics
and violating the no-cloning theorem, or it implies that the initial
information content can escape the BH, therefore contradicting general
relativity. The progress toward the solution of the paradox is discussed. The
renormalization group (RG) aims at the extraction of the macroscopic
description of a physical system from its microscopic description. This passage
from microscopic to macroscopic can be described in terms of several steps from
one scale to another, and is therefore formalized as the action of a group. The
c-theorem proves the existence, under certain conditions, of a function which
is monotonically decreasing along the group transformations. This result
suggests an interpretation of this function as entropy, and its use to study
the information flow along the RG transformations.
