### 9.6 Non-unitary time evolution

An interesting issue discussed in the literature is the time evolution of the entanglement
entropy. It was suggested in [26] that the eigenvalues of a reduced density matrix depend on time
. This is not possible if the time evolution of the density matrix is described by a unitary
operator. Thus, the time evolution should be nonunitary. In particular, the entanglement entropy
should depend on time . Similar conclusions have been made in [30] and [158, 159, 160],
where, in particular, it was shown that the entanglement entropy is an increasing function of
time. These observations may have interesting applications for black holes. As was proposed
by Hawking [129] the evolution in time of a black hole should be nonunitary, so that a pure
initial state may evolve into a mixed state. From the entanglement point of view, this behavior
appears not to be in contradiction with the principles of quantum mechanics, rather it is a simple
consequence of the entangled nature of the system. The irreversible loss of information due to
entanglement is also seen from the evolution of the entropy under the Renorm-Group (RG)
flow [203, 164].