### 2.1 Definition

Consider a pure vacuum state of a quantum system defined inside a space-like region and
suppose that the degrees of freedom in the system can be considered as located inside certain sub-regions of
. A simple example of this sort is a system of coupled oscillators placed in the sites of a space-like
lattice. Then, for an arbitrary imaginary surface , which separates the region into two
complementary sub-regions and , the system in question can be represented as a union of two
sub-systems. The wave function of the global system is given by a linear combination of the product of
quantum states of each sub-system, . The states are formed by the
degrees of freedom localized in the region , while the states are formed by those, which are
defined in region . The density matrix that corresponds to a pure quantum state
has zero entropy. By tracing over the degrees of freedom in region we obtain a density matrix
with elements . The statistical entropy, defined for this density matrix by the standard
formula
is by definition the entanglement entropy associated with the surface . We could have traced over the
degrees of freedom located in region and formed the density matrix . It is clear
that
for
any integer . Thus, we conclude that the entropy (3) is the same for both density matrices and
,
This property indicates that the entanglement entropy for a system in a pure quantum state is not an
extensive quantity. In particular, it does not depend on the size of each region or and thus is only
determined by the geometry of .