Hyperthreads in holographic spacetimes

Published for SISSA by Springer Received: August 2, Revised: September 1, Accepted: September 5, Published: September 20, 2021 2021 2021 2021 Jonathan Harper Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02453, U.S.A. E-mail: 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. Keywords: AdS-CFT Correspondence, Classical Theories of Gravity ArXiv ePrint: 2107.10276 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP09(2021)118 JHEP09(2021)118 Hyperthreads in holographic spacetimes Contents 1 Introduction 1 2 Bit threads 3 7 9 11 12 14 15 4 Discussion 4.1 Multipartite distillation of CFT states 4.2 MMI violating “geometries” 4.3 Signed threads 18 18 20 22 1 Introduction The connection between geometry and quantum information has a long and rich history in the context of holographic systems. Per the Ryu-Takayanagi (RT) formula [1], the entanglement entropy of a part of a holographic CFT state is given by the area of the minimum surface homologous to the region of interest A in a constant time slice of the dual spacetime S(A) = min area(m). (1.1) m∼A Tools from convex optimization theory [2] allow this problem to be recast as a dual maximization problem which asks for the maximum number of “bit threads” [3, 4] which connect A to its compliment while satisfying a local norm bound S(A) = max µ v Z √ A hnµ v µ , s.t. ∇µ v µ = 0, |v µ | ≤ 1. (1.2) √ Here h is the induced metric on the boundary and nµ the unit normal. The integral curves of the vector field v µ are the individual bit threads which can be thought of as a distilled bell pair of two qubits residing at the endpoints of the thread. The constraints of v µ dictate that threads cannot begin or end in the bulk and take up a finite amount of space in the geometry. As such the minimal surface functions as a bottleneck, limiting the number of bit threads permitted through. As a result the maximum number of threads is given precisely by the entanglement entropy as demanded by the duality. –1– JHEP09(2021)118 3 Hyperthreads 3.1 Maximal hyperthread configurations on H 3.2 Maximal hyperthread configurations on Hk 3.2.1 Vacuum AdS3 3.2.2 Black hole states 3.2.3 Multiboundary wormholes max n X k=2 Z k Hk dµ(h) s.t. ∀x ∈ Σ, Z H dµ(h)∆(x, h) ≤ 1. (1.3) Here Hk ⊂ H is the space of all hyperthreads connecting k different boundary regions (“kthreads”), with each term weighted by the number of qubits (endpoints) it contributes. The optimization is over the possible measures µ of H with the constraint that in the manifold the local density of hyperthreads is at most 1. The measure is essentially an assignment of weights to various configurations of hyperthreads. We will show that this program is dual to min ν(x) s.t. ∀h ∈ Hk , Z dν(x)∆(x, h) ≥ k Σ (1.4) which asks for the smallest possible configuration or barriers such that any k-thread will cross barriers with a total weight of at least k. The 2-threads turn out to provide the strictest constraint so that the solution is given precisely by the union of the minimal surfaces associated to the single party entanglement entropies. That is, given a partition of the boundary into n regions ν∗ = n X k=2 kµ∗ (Hk ) = n X area(mAi ) = i n X S(Ai ), (1.5) i where ν ∗ , µ∗ are the optimal measures. We will further show that the maximum contribution of hyperthreads k ≥ 3 to the entanglement entropy is finite and demonstrate that for AdS3 it is given by the area of a particular surface which partitions the interior region Σ \ ∪i r(Ai ). Here r(Ai ) is the homology region of Ai which interpolates between Ai and mAi . Though the precise consequences of this to the structure of entanglement of CFT states is unclear, this surface could possibly quantify a limitation to the distillation of multipartite entanglement. This would potentially provide another example of holography encoding key properties of quantum information in geometric surfaces. We will also establish programs which ask for a maximal configuration of only k-threads for a fixed value of k such that 2 < k ≤ n HPk (A) = max kµ(Hk ) s.t. ∀x ∈ Σ, = min ν(x) s.t. ∀h ∈ Hk , –2– Z Z H dµ(h)∆(x, h) ≤ 1 dν(x)∆(x, h) ≥ k. Σ (1.6) JHEP09(2021)118 A better understanding of the role of multipartite entanglement in holographic systems is a longstanding question in the field. We can begin to make progress towards this goal via the bit threads formalism. If a bit thread represents two entangled qubits, then it is natural to conjecture that an object which connects k-parties should be suitably thought of as a some unit of k-party entanglement between the k boundary regions [5]. Such “hyperthreads” will be the primary focus of this article. Because hyperthreads do not have a good representation as the integral curves of a vector field we are required to adopt an alternate formulation. Consider the space of all such possible hyperthreads as a measurable set H. In the context of this set, we can use tools of measure theory to define the optimization program 2 Bit threads To begin, we will first review bipartite 2-threads in holographic spacetimes. 2 Given a holographic spacetime M we take a constant time slice Σ and partition the boundary ∂Σ into two regions A and B. The entanglement entropy of A is given by the area of minimal surface homologous to A. This is the Ryu-Takayanagi (RT) formula [1] S(A) = min area(m) = area(mA ), m∼A (2.1) where we have called the minimizing RT surface mA . Alternatively, using strong duality the RT formula can be written as the maximum flow of bit threads S(A) = max µ v Z √ A hnµ v µ , s.t. ∇µ v µ = 0, |v µ | ≤ 1. (2.2) Our goal will be to reinterpret this program as an optimization over a measureable space. We define a thread p to be a simple curve on Σ which has one end point on A and one on B. We then define the set of all such threads to be P .3 The measure µ assigns zero or 1 The measure theory approach to bit threads presented in this paper is based on to-be published work by Matt Headrick and Veronika Hubeny [6]. We thank them for sharing an early draft and allowing us to present it here. 2 We will assume some basic knowledge of bit threads and convex optimization. For an introduction (...truncated)