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

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 $|\psi ⟩$, 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.

5 pages, 1 figure, minor changes. For more details, see the first version. After submitting this work to the journal, I have found a related work by T. Osborne; see arXiv:0806.2962

References in corpus (8)

• A class of quantum many-body states that can be efficiently simulated
• Local stabilizer codes in three dimensions without string logical operators
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Ground state entanglement and geometric entropy in the Kitaev's model
• Entanglement and topological entropy of the toric code at finite temperature
• Topological order in a 3D toric code at finite temperature
• Algorithms for the Markov Entropy Decomposition

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• Verlinde formula from entanglement
• Ground state entanglement constrains low-energy excitations
• Complexity and order in approximate quantum error-correcting codes
• Thermalization of Topological Entropy after a Quantum Quench
• An analog of topological entanglement entropy for mixed states
• Locality of Edge States and Entanglement Spectrum from Strong Subadditivity
• Entropic topological invariant for a gapped one-dimensional system
• Hidden hyperspace geometry and long-distance quantum coupling
• Topological self-organization of strongly interacting particles
• Entanglement and edge effects in superpositions of many-body Fock states with spatial constraints
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• Long-range bipartite entanglement in XXZ spin chains with the exponential and power-law long-range interactions
• Knots and entanglement