  An entanglement measure for a bipartite quantum system is a state functional
that vanishes on separable states and that does not increase under separable
(local) operations. It is well-known that for pure states, essentially all
entanglement measures are equal to the v. Neumann entropy of the reduced state,
but for mixed states, this uniqueness is lost. In quantum field theory,
bipartite systems are associated with causally disjoint regions. There are no
separable (normal) states to begin with when the regions touch each other, so
one must leave a finite "safety-corridor". Due to this corridor, the normal
states of bipartite systems are necessarily mixed, and the v. Neumann entropy
is not a good entanglement measure in the above sense.
  In this paper, we study various entanglement measures which vanish on
separable states, do not increase under separable (local) operations, and have
other desirable properties. In particular, we study the relative entanglement
entropy, defined as the minimum relative entropy between the given state and an
arbitrary separable state. We establish rigorous upper and lower bounds in
various quantum field theoretic (QFT) models, as well as also model-independent
ones. The former include free fields on static spacetime manifolds in general
dimensions, or integrable models with factorizing $S$-matrix in 1+1 dimensions.
The latter include bounds on ground states in general conformal QFTs, charged
states (including charges with braid-group statistics) or thermal states in
theories satisfying a "nuclearity condition". Typically, the bounds show a
divergent behavior when the systems get close to each other--sometimes of the
form of a generalized "area law"--and decay when the systems are far apart. Our
main technical tools are of operator algebraic nature.
