  The notion of entanglement has been useful for characterizing universal
properties of quantum phases of matter. From the perspective of quantum
information theory, it is tempting to ask whether their entanglement structures
possess any operational meanings, e.g., quantifying the cost of preparing an
entangled system via free operations such as the local operations and classical
communication (LOCC). While the answer is affirmative for pure states in that
entanglement entropy coincides with entanglement cost, the case for mixed
states is less understood. To this end, we study the entanglement cost required
to prepare the thermal Gibbs states of certain many-body systems under
positive-partial-transpose (PPT) preserving operations, a set of free
operations that include LOCC. Specifically, we show that for the Gibbs states
of $d$-dimensional toric code models for $d = 2, 3, 4$, the PPT entanglement
cost exactly equals entanglement negativity, a measure of mixed-state
entanglement that has been known to diagnose topological order at finite
temperature.
