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
.