  Topological entanglement entropy (TEE), the sub-leading term in the
entanglement entropy of topological order, is the direct evidence of the
long-range entanglement. While effective in characterizing topological orders
on closed manifolds, TEE is model-dependent when entanglement cuts intersect
with physical gapped boundaries. In this paper, we study the origin of this
model-dependence by introducing a model-independent picture of partitioning the
topological orders with gapped boundaries. In our picture, the entanglement
boundaries (EBs), i.e. the virtual boundaries of each subsystem induced by the
entanglement cuts, are assumed to be gapped boundaries with boundary defects.
At this model-independent stage, there are two choices one has to make manually
in defining the bi-partition: the boundary condition on the EBs, and the
coherence between certain boundary states. We show that TEE appears because of
a constraint on the defect configurations on the EBs, which is choice-dependent
in the cases where the EBs touch gapped boundaries. This choice-dependence is
known as the ambiguity in entanglement entropy. Different models intrinsically
employ different choices, rendering TEE model-dependent. For Z2 toric code, the
ambiguity can be fully characterized by two parameters that respectively
quantifies the EB condition and the coherence. In particular, calculations
compatible with the folding trick naturally choose EB conditions that respect
electric-magnetic duality and set specific parameter values.
