Long-Range Entanglement Is Necessary for a Topological Storage of Quantum Information

A general inequality between entanglement entropy and a number of topologically ordered states is derived, even without using the properties of the parent Hamiltonian or the formalism of topological quantum field theory. Given a quantum state |ψ⟩, we obtain an upper bound on the number of distinct states that are locally indistinguishable from |ψ⟩. The upper bound is determined only by the entanglement entropy of some local subsystems. As an example, we show that log⁡N≤2γ for a large class of topologically ordered systems on a torus, where N is the number of topologically protected states and γ is the constant subcorrection term of the entanglement entropy. We discuss applications to quantum many-body systems that do not have any low-energy topological quantum field theory description, as well as tradeoff bounds for general quantum error correcting codes