Entanglement cost in topological stabilizer models at finite temperature

TL;DR

This paper investigates the entanglement cost of thermal states in topological stabilizer models, revealing that for certain toric code models, this cost equals the entanglement negativity, linking entanglement measures to topological order at finite temperature.

Contribution

It establishes that the PPT entanglement cost for Gibbs states in specific topological models equals entanglement negativity, providing operational meaning to mixed-state entanglement measures.

Findings

PPT entanglement cost equals entanglement negativity for 2D, 3D, 4D toric codes.

Entanglement negativity diagnoses topological order at finite temperature.

Operational interpretation of mixed-state entanglement measures in topological phases.

Abstract

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.