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The split property of a pure state for a certain cut of a quantum spin systemcan 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 astate $\omega$ via sufficiently local automorphisms (in a sense that we willmake precise) into a state satisfying the split property, then the state$\omega$ has a long-range entanglement. It is well known that in 1D, gappedground states have the split property with respect to cutting the system intoleft and right half-chains. In 2D, however, the split property fails to holdfor interesting models such as Kitaev's toric code. In fact, we will show thatthis failure is the reason that anyons can exist in that model.  There is a folklore saying that the existence of anyons, like in the toriccode model, implies long-range entanglement of the state. In this paper, weprove this folklore in an infinite dimensional setting. More precisely, we showthat long-range entanglement, in a way that we will define precisely, is anecessary condition to have non-trivial superselection sectors. Anyons inparticular give rise to such non-trivial sectors. States with the splitproperty for cones, on the other hand, do not admit non-trivial sectors.  A key technical ingredient of our proof is that under suitable assumptions onlocality, the automorphisms generated by local interactions can be'approximately factorized': they can be written as the tensor product ofautomorphisms localized in a cone and its complement respectively, followed byan automorphism acting near the 'boundary' of $\Lambda$, and conjugation with aunitary. This result may be of independent interest. This technique also allowsus to prove that the approximate split property, a weaker version of the splitproperty that is satisfied in e.g. the toric code, is stable under applyingsuch automorphisms.

The split and approximate split property in 2D systems: stability and absence of superselection sectors