We derive some of the axioms of the algebraic theory of anyon [A. Kitaev, Ann. Phys., 321, 2 (2006)] from a conjectured form of entanglement area law for two-dimensional gapped systems. We derive the fusion rules of topological charges and show that the multiplicity of the fusion rules satisfy these axioms. Moreover, even though we make no assumption about the exact value of the constant sub-leading term of the entanglement entropy, this term is shown to be equal to ln D, where D is the total quantum dimension of the underlying anyon theory. These derivations are rigorous and follow from the entanglement area law alone. More precisely, our framework starts from two local entropic constraints which are implied by the area law. They allow us to prove what we refer to as the isomorphism theorem, which enables us to define superselection sectors and fusion multiplicities without a Hamiltonian. These objects and the axioms of the anyon theory are shown to emerge from the structure and the internal self-consistency relations of an object known as the information convex.2. I(A : C) σ = 0 for the partition in Fig. 23.

A special feature of the ground state in a topologically ordered phase is the existence of large scale correlations depending only on the topology of the regions. These correlations can be detected by the topological entanglement entropy or by a measure called irreducible correlation. We show that these two measures coincide for states obeying an area law and having zero-correlation length. Moreover, we provide an operational meaning for these measures by proving its equivalence to the optimal rate of a particular class of secret sharing protocols. This establishes an information-theoretical approach to multipartite correlations in topologically ordered systems.

We prove that any one-dimensional (1D) quantum state with small quantum conditional mutual information in all certain tripartite splits of the system, which we call a quantum approximate Markov chain, can be well-approximated by a Gibbs state of a short-range quantum Hamiltonian. Conversely, we also derive an upper bound on the (quantum) conditional mutual information of Gibbs states of 1D short-range quantum Hamiltonians. We show that the conditional mutual information between two regions A and C conditioned on the middle region B decays exponentially with the square root of the length of B.These two results constitute a variant of the Hammersley-Clifford theorem (which characterizes Markov networks, i.e. probability distributions which have vanishing conditional mutual information, as Gibbs states of classical short-range Hamiltonians) for 1D quantum systems. The result can be seen as a strengthening -for 1D systems -of the mutual information area law for thermal states. It directly implies an efficient preparation of any 1D Gibbs state at finite temperature by a constant-depth quantum circuit.