An Area Law for One Dimensional Quantum Systems (0705.2024v4)

Abstract: We prove an area law for the entanglement entropy in gapped one dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on completely positive maps which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on R\'{e}nyi entropy for sufficiently large $\alpha<1$ and implies the ability to approximate the ground state by a matrix product state.

• The paper establishes a rigorous proof that entanglement entropy in gapped 1D quantum systems adheres to an area law.
• It leverages bounds on Von Neumann and Rényi entropies to demonstrate accurate ground state approximations using matrix product states.
• The findings highlight implications for quantum expanders and suggest alternative proof approaches via completely positive maps.

An Area Law for One-Dimensional Quantum Systems: An Overview

The paper by M. B. Hastings demonstrates a proof for an area law concerning the entanglement entropy in one-dimensional quantum systems that exhibit a spectral gap. The focus of the investigation is on establishing that the entanglement entropy is bounded in these systems, implying that it scales proportional to the boundary area as opposed to the volume—a principle anticipated by the so-called "area law." Previous theoretical efforts and calculations in fields such as conformal field theory hinted at this result, particularly in one dimension, as the entanglement entropy tends to diverge only near critical points, scaling with the correlation length. This research sought to furnish a rigorous proof for this fact.

Key Contributions and Results

1. Proof of the Area Law: The paper establishes a proof of the area law for gapped one-dimensional quantum systems. Despite the definitive result, the derived bound on entanglement entropy is subject to exponential growth with respect to the correlation length, which is unexpectedly rapid. The proof leverages bounds on the Von Neumann entropy and explores the intricate link between Von Neumann entropy, R\'enyi entropy, and the error margins when approximating ground states using matrix product states (MPS).
2. Implications for Quantum Expanders: Hastings discusses the implications of these results regarding quantum expanders—a category of special matrix product states characterized by exponential decay in correlation functions coupled with high entanglement. The existence of strong entanglement in such states provides insight into the difficulty of establishing general proofs for the area law.
3. Applications to Matrix Product States: Through the application of the derived bounds, the paper elucidates how the entanglement bounds can assist in approximating ground states with matrix product states accurately. For a system with a spectral gap, it postulates that there is a limit on approximation error dependent on the bond dimension of the matrix product state.
4. Conjecture on Completely Positive Maps: An intriguing conjecture posed in the paper suggests a link between the properties of completely positive maps and the entropy bound, which could serve as an alternate proof approach for the area law. If validated, this conjecture would potentially provide a comprehensive understanding of the behavior of bipartite quantum states and their correlations in gapped systems.

Theoretical and Practical Implications

The mathematical elucidation of an area law provides key insights into how quantum entanglement behaves in condensed matter systems, specifically those with an energy gap. This theoretical understanding is crucial for advancing methods in quantum information and computation, where controlling quantum correlations is paramount. Notably, the results aid the numerical simulation of one-dimensional quantum systems, revealing that such systems can be efficiently approximated by MPS, assuming the persistence of an energy gap.

Future Directions

Future research may focus on strengthening the derived bounds and exploring similar relations in higher-dimensional systems, where the area law's applicability remains an unresolved question. Additionally, the validation of the conjecture on completely positive maps could pave the way for novel methodological schemas in characterizing quantum entropy and extend these findings to broader classes of quantum models.

In conclusion, Hastings' research offers a robust framework for understanding the grounding principles of entanglement entropy in gapped one-dimensional quantum systems while presenting potential avenues for further exploration in the field of quantum many-body physics.