Persistent Topological Negativity in a High-Temperature Mixed-State

TL;DR

This paper investigates how the topological entanglement negativity of a GHZ state remains constant across the Ising phase transition when thermalized by a symmetric quantum channel, revealing persistent quantum correlations in mixed states.

Contribution

It demonstrates that the topological negativity of a thermalized GHZ state remains unchanged across the phase transition, using a novel LOCC decoder and connecting negativity to an error-correction problem.

Findings

Negativity remains constant across the phase transition.

A local LOCC decoder bounds the negativity in the thermodynamic limit.

Numerical results support the theoretical analysis.

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 $\beta$. 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.