Entropy and Entanglement in Quantum Ground States

TL;DR

The paper analyzes how correlations relate to entanglement in gapped local quantum systems and the viability of matrix product state (MPS) representations. It first constructs a 1D gapped Hamiltonian whose ground state is an MPS, demonstrating that the entanglement entropy across a cut can scale as $S=\log k$ while the correlation length $\xi$ stays bounded, using an expander-graph-based design to ensure a spectral gap. It then shows that, under a density-of-states assumption satisfied by many physical systems (e.g., free fermions and lattice fractional quantum Hall analogs), the ground state can be efficiently represented as a higher-dimensional matrix product state or matrix product density operator, with bond dimension scaling as $\alpha_{\max}\sim D^{l_{\mathrm{proj}}^d\beta/J}$ and $l_{\mathrm{proj}}\sim R \log(V\beta/J\epsilon)$. The discussion interprets these results physically, noting that the assumption corresponds to non-glassy behavior at temperatures $\sim \Delta E/\log(V)$ and acknowledging a Terhal–DiVincenzo-type counterexample, which motivates potential Hamiltonian modifications to retain efficient MPS representations in practice.

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.