2.1 Definition

Consider a pure vacuum state |ψ > of a quantum system defined inside a space-like region 𝒪 and suppose that the degrees of freedom in the system can be considered as located inside certain sub-regions of 𝒪. 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 Σ, which separates the region 𝒪 into two complementary sub-regions A and B, 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, ∑ |ψ >= i,aψia|A >i |B >a. The states |A >i are formed by the degrees of freedom localized in the region A, while the states |B >a are formed by those, which are defined in region B. The density matrix that corresponds to a pure quantum state |ψ >
ρ0(A,B ) = |ψ > < ψ | (1 )
has zero entropy. By tracing over the degrees of freedom in region A we obtain a density matrix
ρB = TrA ρ0(A,B ), (2 )
with elements (ρB )ab = (ψ ψ†)ab. The statistical entropy, defined for this density matrix by the standard formula
SB = − TrρB ln ρB (3 )
is by definition the entanglement entropy associated with the surface Σ. We could have traced over the degrees of freedom located in region B and formed the density matrix (ρA )ij = (ψT ψ∗)ij. It is clear that1
k k Tr ρA = Trρ B
for any integer k. Thus, we conclude that the entropy (3View Equation) is the same for both density matrices ρA and ρ B,
S = S . (4 ) A B
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 A or B and thus is only determined by the geometry of Σ.
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