The paper by Alexei Kitaev and John Preskill formulates a universal characterization of quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. They consider a disk in the plane with a smooth boundary of length \( L \), large compared to the correlation length. By tracing out the degrees of freedom outside the disk, they obtain a marginal density operator \( \rho \) for the interior degrees of freedom. The von Neumann entropy \( S(\rho) \) of this density operator, which measures the entanglement between the interior and exterior variables, has the form \( S(\rho) = \alpha L - \gamma + \cdots \), where \( \alpha \) is nonuniversal and ultraviolet divergent, while \( -\gamma \) is a universal additive constant characterizing the global feature of the entanglement in the ground state. They call \( -\gamma \) the *topological entanglement entropy*. Using topological quantum field theory (TQFT) methods, they derive a formula for \( \gamma \) in terms of the properties of the superselection sectors of the medium. Specifically, \( \gamma \) is given by \( \gamma = \log \mathcal{D} \), where \( \mathcal{D} \geq 1 \) is the *total quantum dimension* of the medium, defined as \( \mathcal{D} = \sqrt{\sum_a d_a^2} \), with \( d_a \) being the *quantum dimension* of a particle with charge \( a \). The paper also discusses the topological invariance and universality of \( S_{\text{topo}} \), the topological entanglement entropy, under smooth deformations of the Hamiltonian. They provide a detailed derivation of \( S_{\text{topo}} \) using TQFT and a conformal field theory (CFT) approach, showing that \( S_{\text{topo}} = -\log \mathcal{D} \). The results highlight the connection between topological order and entanglement entropy in two-dimensional systems, providing insights into the nature of topological order and its quantification.The paper by Alexei Kitaev and John Preskill formulates a universal characterization of quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. They consider a disk in the plane with a smooth boundary of length \( L \), large compared to the correlation length. By tracing out the degrees of freedom outside the disk, they obtain a marginal density operator \( \rho \) for the interior degrees of freedom. The von Neumann entropy \( S(\rho) \) of this density operator, which measures the entanglement between the interior and exterior variables, has the form \( S(\rho) = \alpha L - \gamma + \cdots \), where \( \alpha \) is nonuniversal and ultraviolet divergent, while \( -\gamma \) is a universal additive constant characterizing the global feature of the entanglement in the ground state. They call \( -\gamma \) the *topological entanglement entropy*. Using topological quantum field theory (TQFT) methods, they derive a formula for \( \gamma \) in terms of the properties of the superselection sectors of the medium. Specifically, \( \gamma \) is given by \( \gamma = \log \mathcal{D} \), where \( \mathcal{D} \geq 1 \) is the *total quantum dimension* of the medium, defined as \( \mathcal{D} = \sqrt{\sum_a d_a^2} \), with \( d_a \) being the *quantum dimension* of a particle with charge \( a \). The paper also discusses the topological invariance and universality of \( S_{\text{topo}} \), the topological entanglement entropy, under smooth deformations of the Hamiltonian. They provide a detailed derivation of \( S_{\text{topo}} \) using TQFT and a conformal field theory (CFT) approach, showing that \( S_{\text{topo}} = -\log \mathcal{D} \). The results highlight the connection between topological order and entanglement entropy in two-dimensional systems, providing insights into the nature of topological order and its quantification.