Persistent Topological Negativity in a High-Temperature Mixed-State

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 \(\beta\). 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 \(\beta>0\). 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.