Abstract
We study the entanglement structure of the Greenberger-Horne-Zeilinger (GHZ) state as it thermalizes under a strongly-symmetric quantum channel describing the Metropolis-Hastings dynamics for the d-dimensional classical Ising model at inverse temperature β. This channel outputs the classical Gibbs state when acting on a product state in the computational basis. When applying this channel to a GHZ state in spatial dimension d>1, the resulting mixed state changes character at the Ising phase transition temperature from being long-range entangled to short-range-entangled as temperature increases. Nevertheless, we show that the topological entanglement negativity of a large region is insensitive to this transition and takes the same value as that of the pure GHZ state at any finite temperature β>0. We establish this result by devising a local operations and classical communication (LOCC) ``decoder" that provides matching lower and upper bounds on the negativity in the thermodynamic limit which may be of independent interest. This perspective connects the negativity to an error-correction problem on the (d-1)-dimensional bipartitioning surface and explains the persistent negativity in certain correlated noise models found in previous studies. Numerical results confirm our analysis.