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arXiv:2006.02899 · Journal of High Energy Physics

Multipartite entanglement and topology in holography

Jonathan Harper

Published Mar 1, 2021• 10 citations
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Abstract

Abstract Starting from the entanglement wedge of a multipartite mixed state we describe a purification procedure which involves the gluing of several copies. The resulting geometry has non-trivial topology and a single oriented boundary for each original boundary region. In the purified geometry the original multipartite entanglement wedge cross section is mapped to a minimal surface of a particular non-trivial homology class. In contrast, each original bipartite entanglement wedge cross section is mapped to the minimal wormhole throat around each boundary. Using the bit thread formalism we show how maximal flows for the bipartite and multipartite entanglement wedge cross section can be glued together to form maximal multiflows in the purified geometry. The defining feature differentiating the flows is given by the existence of threads which cross between different copies of the original entanglement wedge. Together these demonstrate a possible connection between multipartite entanglement and the topology of holographic spacetimes.

DOI
10.1007/jhep03(2021)116

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Abstract We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured E P = E W relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.
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Generalizing the bit thread formalism, we use convex duality to derive dual flow programs to the bipartite and multipartite holographic entanglement of purification proposals and then prove several inequalities using these constructions. In the multipartite case we find the flows exhibit novel behavior which allows for a constrained flux on the boundary of the homology region. We show this flux can be made distinct from bi-partite terms and reflects the truly multipartite portion of the holographic entanglement of purification.
We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured EP = EW relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.
Abstract We generalize bit threads to hyperthreads in the context of holographic spacetimes. We define a “k-thread” to be a hyperthread which connects k different boundary regions and posit that it may be considered as a unit of k-party entanglement. Using this new object, we show that the contribution of hyperthreads to calculations of holographic entanglement entropy are generically finite. This is accomplished by constructing a surface whose area determines their maximum allowed contribution. We also identify surfaces whose area is proportional to the maximum number of k-threads, motivating a possible measure of multipartite entanglement. We use this to make connections to the current understanding of multipartite entanglement in holographic spacetimes.
A bstract We investigate the geometry of entanglement wedges for asymptotic n -to- n scattering configurations in asymptotically AdS 3 spacetimes. Extending the 2-to-2 Connected Wedge Theorem, we establish a strictly weaker sufficient condition for the input entanglement wedge E(V1∪⋯∪Vn) \mathcal{E}\left({V}_1\cup \cdots \cup {V}_n\right) E(V1​∪⋯∪Vn​) E V 1 ∪ ⋯ ∪ V n to be connected: the existence of a single pair of input regions satisfying a 2-to-all causal intersection condition already forces full multipartite wedge connectivity. We also derive novel necessary conditions, showing that when the input wedge is connected, certain output ridges must enter the input wedge, and we organize these consequences into a layered reduction on the boundary lattice. Furthermore, we analyze the generalized bulk scattering region SE=E(V1∪⋯∪Vn)∩E(W1∪⋯∪Wn) {\mathcal{S}}_E=\mathcal{E}\left({V}_1\cup \cdots \cup {V}_n\right)\cap \mathcal{E}\left({W}_1\cup \cdots \cup {W}_n\right) SE​=E(V1​∪⋯∪Vn​)∩E(W1​∪⋯∪Wn​) S E = E V 1 ∪ ⋯ ∪ V n ∩ E W 1 ∪ ⋯ ∪ W n and obtain necessary conditions for it to be nonempty; for n > 2 these conditions are stronger than mere wedge connectedness. Our results provide new geometric restrictions on multipartite entanglement in holography and clarify the holographic dictionary for multi-partite scattering processes, while also highlighting intrinsic limitations for n > 2.