Long-range entanglement is necessary for a topological storage of quantum information

Understand

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.