Entanglement cost in topological stabilizer models at finite temperature

The notion of entanglement has been useful for charactering quantum many-body systems. From the perspective of quantum information theory, it is tempting to ask whether their entanglement structures possess any operational meanings, e.g., measuring the cost of preparing an entangled system using only local operations and classical communication (LOCC). While the answer is affirmitive for pure states in that entanglement entropy coincides with entanglement cost, the case for mixed-states such as systems described by thermal Gibbs states is less understood. To this end, we address this question by studying a recently proposed quantity dubbed κ-entanglement, which not only measures entanglement for mixed states but also quantifies the entanglement cost under positive-partial-transpose (PPT) preserving operations. In particular, we focus on Gibbs states of d-dimensional toric code models for d = 2, 3, 4, and show that their κ-entanglement coincides with entanglement negativity, which has been known to diagnose topological order at finite temperature. Our finding therefore provides an operational meaning for their long-range entanglement structure.