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

TL;DR

This paper establishes a fundamental inequality linking entanglement entropy to the number of topologically protected quantum states, providing bounds applicable even without a topological quantum field theory framework.

Contribution

It derives a general inequality relating entanglement entropy to the count of topologically indistinguishable states without relying on specific Hamiltonian properties or TQFT formalism.

Findings

Upper bound on the number of topologically protected states based on entanglement entropy

Demonstration that log N   2gamma for certain topological systems

Application of bounds to quantum error correction and many-body systems

Abstract

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.