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 $|\psi ⟩$, we obtain an upper bound on the number of distinct states that are locally indistinguishable from $|\psi ⟩$. The upper bound is determined only by the entanglement entropy of some local subsystems. As an example, we show that $\log N\le 2\gamma$ 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.

Figure 1

Each of the diagrams represents the subsystems $A_i$, $B_i$, $C_i$, $i=1$, 2, 3, with the constraint that $A_i{B}_i{C}_i$ is equal to $A_{i+1}{B}_{i+1}$. Because of this construction, the entanglement entropy of $A_3{B}_3{C}_3$ is the only nonlocal contribution to the sum $\sum _{i=1}^{3}I(A_i\mathrm{\text{:}}{C}_i|B_i)$. The entanglement entropy of $A_3{B}_3{C}_3$ becomes ${log}_{2}N$ for a maximally mixed state over $N$ locally indistinguishable states. The rest of the contributions can be computed from the formula for the entanglement entropy of local subsystems, i.e., Eq. (2).