Holographic entanglement as nonlocal magnetism

Abstract

The Ryu-Takayanagi prescription can be cast in terms of a set of microscopic threads that help visualize holographic entanglement in terms of distillation of EPR pairs. While this framework has been exploited for regions with a high degree of symmetry, we take the first steps towards understanding general entangling regions, focusing on AdS4. Inspired by simple constructions achieved for the case of disks and the half-plane, we reformulate bit threads in terms of a magnetic-like field generated by a current flowing through the boundary of the entangling region. The construction is possible for these highly symmetric settings, leading us to a modified Biot-Savart law in curved space that fully characterizes the entanglement structure of the state. For general entangling regions, the prescription breaks down as the corresponding modular Hamiltonians become inherently nonlocal. We develop a formalism for general shape deformations and derive a flow equation that accounts for these effects as a systematic expansion. We solve this equation for a complete set of small deformations and show that the structure of the expansion explicitly codifies the expected nonlocalities. Our findings are consistent with numerical results existing in the literature, and shed light on the fundamental nature of quantum entanglement as a nonlocal phenomenon.

Lorentzian threads and generalized complexity

“…Due to the fact that dψ is future-directed and timelike, these Σ(c) are slices. 10 Moreover, as ψ| A = −1/2 and ψ| B = 1/2, they all must also be homologous to A, since all the level sets must have σ A = ∂A as boundary.…”

“…This approach has been a fruitful avenue of research in past years. Bit threads have been used to study multipartite entanglement [1][2][3][4][5], holographic monogamy of mutual information [6,7], the hypergraph entropy cone [8], metric reconstruction [9], describe the entanglement structure of a state in terms of a modified Biot-Savart law [10], etc. Furthermore, recent developments extending the thread formulation to Lorentzian manifolds [11] showed that quantities other than entanglement can be formulated in this language.…”

“…Note that we demand this number to be larger but not equal to one since we are checking Slater's condition, which demands the existence of a feasible solution where the inequalities are strictly satisfied 9. We provide the proof of this theorem in appendix A 10. A slice is defined[17] as a compact codimension-one hypersurface-with-boundary in M which is everywhere light or spacelike with future directed normal and whose interior is embedded in M.…”

Geometrizing the partial entanglement entropy: from PEE threads to bit threads

“…Bit threads have been also generalized to its quantum corrected version [45,46] and the covariant version [47]. For other recent progresses on bit threads, please refer to [48][49][50][51][52][53][54].…”