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 $\ket{\psi}$, we obtain an upper bound on the number of distinct states that are locally indistinguishable from $\ket{\psi}$. The upper bound is determined only by the entanglement entropy of some local subsystems. As an example, we show that $\log N \leq 2\gamma$ for a large class of topologically ordered systems on a torus, where $N$ is the number of topologically protected states and $\gamma$ 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.