Entropy and Entanglement in Quantum Ground States

Abstract

We consider the relationship between correlations and entanglement in gapped quantum systems, with application to matrix product state representations. We prove that there exist gapped one-dimensional local Hamiltonians such that the entropy is exponentially large in the correlation length, and we present strong evidence supporting a conjecture that there exist such systems with arbitrarily large entropy. However, we then show that, under an assumption on the density of states which is believed to be satisfied by many physical systems such as the fractional quantum Hall effect, that an efficient matrix product state representation of the ground state exists in any dimension. Finally, we comment on the implications for numerical simulation.

7 pages, no figures

References in corpus (11)

• Anyons in an exactly solved model and beyond
• Entanglement in quantum critical phenomena
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Matrix product states represent ground states faithfully
• Lieb-Schultz-Mattis in Higher Dimensions
• Criticality, the area law, and the computational power of PEPS
• Spectral Gap and Exponential Decay of Correlations
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Randomizing quantum states: Constructions and applications
• Solving Gapped Hamiltonians Locally
• Locality in Quantum and Markov Dynamics on Lattices and Networks

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