Area law in one dimension: Degenerate ground states and Renyi entanglement entropy

Abstract

An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of $n$ spins the ground states of a local Hamiltonian with energy gap $\epsilon$ are constant-fold degenerate. Then, the Renyi entanglement entropy $R_{\alpha }(0<\alpha <1)$ of any ground state across any cut is upper bounded by $\tilde{O}({\alpha }^{-3}/\epsilon )$, and any ground state can be well approximated by a matrix product state of subpolynomial bond dimension $2^{\tilde{O}({\epsilon }^{-1/4}{\log }^{3/4}n)}$.

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References in corpus (14)

• Area laws for the entanglement entropy - a review
• An Area Law for One Dimensional Quantum Systems
• Matrix product states represent ground states faithfully
• Lieb-Schultz-Mattis in Higher Dimensions
• Entropy scaling and simulability by Matrix Product States
• Hamiltonian complexity
• Exponential Decay of Correlations Implies Area Law
• An area law for entanglement from exponential decay of correlations
• Universal quantum computation with little entanglement
• Stochastic Error Cancellation in Analog Quantum Simulation
• An improved 1D area law for frustration-free systems
• Entanglement vs. gap for one-dimensional spin systems
• Connecting global and local energy distributions in quantum spin models on a lattice
• Ground State Entanglement in One Dimensional Translationally Invariant Quantum Systems

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• Computing the Degenerate Ground Space of Gapped Spin Chains in Polynomial Time
• Computing energy density in one dimension
• A polynomial-time algorithm for the ground state of one-dimensional gapped Hamiltonians
• Approximating local properties by tensor network states with constant bond dimension
• Two-dimensional local Hamiltonian problem with area laws is QMA-complete
• Thermal Area Law for Lattice Bosons
• Entanglement Dynamics From Random Product States: Deviation From Maximal Entanglement
• Sharp implications of AGSPs for degenerate ground spaces