Entanglement in quantum spin chains is strictly finite at any temperature

TL;DR

This paper proves that quantum spin chains at any finite temperature have strictly finite entanglement, demonstrated through an explicit matrix product state decomposition with an efficient sampling algorithm.

Contribution

It introduces an exact decomposition of thermal states into matrix product states with finite bond dimension, revealing finite entanglement in quantum spin chains at all temperatures.

Findings

Gibbs states can be decomposed into matrix product states with finite bond dimension.

Schmidt number remains finite at any finite temperature, even in the thermodynamic limit.

Provides an efficient classical algorithm for sampling the decomposed states.

Abstract

Entanglement is the hallmark of quantum physics, yet its characterization in interacting many-body systems at thermal equilibrium remains one of the most important challenges in quantum statistical physics. We prove that the Gibbs state of any quantum spin chain can be exactly decomposed into a mixture of matrix product states with a bond dimension that is independent of the system size, at any finite temperature. As a consequence, the Schmidt number, arguably the most stringent measure of bipartite entanglement, is strictly finite for thermal states, even in the thermodynamic limit. Our decomposition is explicit and is accompanied by an efficient classical algorithm to sample the resulting matrix product states.