Multipartite entanglement and topology in holography

Published for SISSA by Springer Received: August 4, 2020 Revised: February 2, 2021 Accepted: February 3, 2021 Published: March 10, 2021 Jonathan Harper Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02453, U.S.A. E-mail: 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. Keywords: AdS-CFT Correspondence, Black Holes, Differential and Algebraic Geometry ArXiv ePrint: 2006.02899 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP03(2021)116 JHEP03(2021)116 Multipartite entanglement and topology in holography Contents 1 2 Holographic multipartite entanglement wedge cross section 2.1 Geometric construction 2.2 Bit thread duals 2 3 3 3 Biparite mixed states, purification, and antiloop dualization 3.1 Purifying bipartite mixed states 3.1.1 Thermofield double 3.1.2 Bipartite entanglement wedge cross section 3.2 Dualization using antiloops 5 6 6 7 7 4 Multipartite entanglement wedge cross section 4.1 Constructing the purified geometry P 4.2 Minimal surfaces on P and constructed thread configurations 4.3 Optimized bit thread configurations on P 4.4 The degenerate case 4.5 More boundary regions 4.6 Covariant and higher dimensions 10 10 12 14 15 16 19 5 Discussion 19 1 Introduction The connection between geometry and entanglement has been a fruitful tool of modern research. This is most clearly evidenced by the Ryu-Takayanagi (RT) formula [1] and its covariant generalization [2] which allows one to relate the area of a bulk minimal surface and the entanglement entropy of the boundary of a holographic CFT. More recently, efforts have been made to generalize this to other interesting surfaces in the bulk, most notably the entanglement wedge cross section which is known to contain information about quantum and classical correlations and acts as a measure of mixed state entanglement [3–5]. The entanglement wedge can be purified by gluing a copy of the CPT conjugate and gluing along the common boundaries. In the purification the entanglement wedge cross section is related to the entanglement entropy between the two sides of the constructed wormhole. An important generalization can be performed by considering more than two boundary regions. One then considers the multipartite entanglement wedge cross section whose area should be thought of as a measure of multipartite entanglement in the CFT [6, 7]. The question remains how one should think of purifications in this case. A simple doubling of –1– JHEP03(2021)116 1 Introduction 2 Holographic multipartite entanglement wedge cross section In this section we review the geometric construction of the multipartite entanglement wedge cross section and corresponding bit thread programs. These surfaces and their dual flows will be used later to motivate our procedure. Throughout this paper we will be primarily working with static slices of pure AdS3 . However, both the entanglement wedge cross –2– JHEP03(2021)116 the geometry does not have the effect of relating the topological data, the area of minimal surfaces in each homology class of the manifold, to that of the multipartite entanglement wedge cross section. In this paper we will do just this: using several copies of the original entanglement wedge we will illustrate a gluing procedure which will result in a particular purification where the multipartite entanglement wedge cross section can be related to a minimal surface in a particular homology class of the manifold. This restores the relationship between topology and entanglement present in the bipartite case. We will find it useful to use the bit thread formalism [8, 9] which allows one to relate geometric surfaces to maximal, normed, divergenceless vector fields, or flows, in the bulk. It was shown that both the bipartite [10, 11] and multipartite [10] entanglement wedge cross section can be written in bit thread language. Using these flows as building blocks we can use them to create a new flow on the purified geometry. Importantly, these flows originally can end on all boundaries of the entanglement wedge. Since these are glued to create the purified manifold we will find these flows can be joined across the boundaries. Using the tools of convex optimization we will describe how to calculate these flows directly using the topology of the manifold. In section 2 we give some background detailing the multipartite entanglement wedge cross section and bit thread duals. Next, in section 3 we describe how mixed states generally contain bit threads which end on boundaries and show how gluing the geometry to form a pure state allows the threads on each copy to be identified to form a single flow. We also show how convex duality, which relates geometrically surfaces to flows, can be performed using knowledge of the homology class of the surface. Then, in section 4 we put these together to describe a purification procedure for the entanglement wedge of a multipartite mixed state. Using our construction we will find the multipartite entanglement wedge cross section can be described as a minimal surface in a particular homology class of the manifold and thus is related to topological data. Using the tools developed we will describe flows of bit threads which calculate this quantity and are built from the original flows which calculated the multipartite entanglement wedge cross section. Finally, in section 5 we describe some connections between our construction and multipartite reflected entropy, tensor networks, and multiboundary wormholes. To conclude we emphasize a possible connection between measures of multipartite entanglement and the topology of holographic spacetimes. While in preparation the papers [12, 13] appeared which describe different approaches for constructing purifications of the multipartite entanglement wedge cross section. Notably, both understood the need for multiple copies of the original geometry in these constructions. <latexit sha1_base64="ftA+jsq9D (...truncated)