# Symmetry enforces entanglement at high temperatures

## TL;DR

This paper proves that certain symmetric quantum states can maintain entanglement at arbitrarily high temperatures, challenging the common notion that thermal states become classical and unentangled.

## Contribution

It demonstrates that symmetry constraints can prevent the sudden death of entanglement in thermal states, extending previous understanding to fermionic systems and superselection rules.

## Key findings

- Strongly symmetric thermal states remain entangled at high temperatures.
- Fermionic Gibbs states exhibit persistent fermionic negativity at high temperatures.
- Symmetry correlations can preserve quantum entanglement despite thermal decoherence.

## Abstract

Many-body quantum systems with local interactions undergo ``sudden death of entanglement" at high temperatures, whereby thermal states become classical mixtures of product states. We investigate whether symmetry constraints can prevent this phenomenon. We prove that strongly symmetric thermal states (canonical ensemble) of generic Hamiltonians with on-site Abelian symmetries remain entangled with non-zero entanglement negativity at arbitrarily high temperatures, under mild conditions on the symmetry actions and the charge sector of the strong symmetry. Our results extend to weakly symmetric thermal states (Gibbs ensemble) under superselection rules, which restrict state decompositions to be symmetric. In particular, we show that fermionic Gibbs states evade sudden death of entanglement and have persistent fermionic negativity at high temperatures, proving along the way some existing conjectures about fermionic entanglement. These findings demonstrate that global symmetry correlations can preserve quantum entanglement despite thermal decoherence, providing new insights into the interplay between symmetry and quantum information in thermal equilibrium.