  The split property of a pure state for a certain cut of a quantum spin system
can be understood as the entanglement between the two subsystems being weak.
From this point of view, we may say that if it is not possible to transform a
state $\omega$ via quasi-local automorphisms into a state satisfying the split
property, then the state $\omega$ has a long-range entanglement. It is well
known that in 1D, gapped ground states have the split property with respect to
cutting the system into left and right half-chains. In 2D, however, the split
property fails to hold for interesting models such as Kitaev's toric code. In
fact, we will show that this failure is related to the existence of anyons.
  There is a folklore saying that the existence of anyons, like in the toric
code model, implies long-range entanglement of the state. In this paper, we
prove this folklore in an infinite dimensional setting. More precisely, we
introduce superselection sectors which we regard as anyons. We show that the
existence of a non-trivial superselection sector (=anyon) of a state $\omega$
implies that the state $\omega$ is long-range entangled.
  Whilst the split property fails to hold for Kitaev's toric code, it does
satisfy the weaker approximate split property. Contrary to the fact that
quasi-local automorphism can destroy the strict split property in general for
2D, we show this approximate version is stable under it.
