Characterizing long-range entanglement in a mixed state through an emergent order on the entangling surface

Topologically-ordered phases of matter at nonzero temperature are conjectured to exhibit universal patterns long-range entanglement, which can be detected using the entanglement negativity, a mixed-state measure. In this paper, we show that negativity in certain topological orders understood through properties an emergent symmetry-protected (SPT) order is localized on bipartition. This connection leads understanding (i) contributions diagnose finite-temperature and (ii) behavior across phase transitions thermal fluctuations eventually destroy bipartition surface. Within correspondence, related stability SPT against symmetry-breaking field. protected by higher-form symmetries---which arise, for example, description ${\mathbb{Z}}_{2}$ $d=4$ spatial dimensions---remain robust even presence weak perturbation, leading orders.