Perfect tensor hyperthreads

Published for SISSA by Springer Received: May 11, 2022 Accepted: September 12, 2022 Published: September 28, 2022 Jonathan Harpera,b a Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02453, U.S.A. b Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan E-mail: Abstract: Bit threads, a dual description of the Ryu-Takyanagi formula for holographic entanglement entropy (EE), can be interpreted as a distillation of the quantum information to a collection of Bell pairs between different boundary regions. In this article we discuss a generalization to hyperthreads which can connect more than two boundary regions leading to a rich and diverse class of convex programs. By modeling the contributions of different species of hyperthreads to the EEs of perfect tensors we argue that this framework may be useful for helping us to begin to probe the multipartite entanglement of holographic systems. Furthermore, we demonstrate how this technology can potentially be used to understand holographic entropy cone inequalities and may provide an avenue to address issues of locking. Keywords: AdS-CFT Correspondence, Models of Quantum Gravity ArXiv ePrint: 2205.01140 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP09(2022)239 JHEP09(2022)239 Perfect tensor hyperthreads Contents 1 2 Preliminaries 2.1 Tools of convex optimization 2.2 Bit thread configurations 2.3 The holographic entropy cone and the K-basis 4 4 5 8 3 Perfect tensor hyperthreads 11 4 2 regions 14 5 3 regions 15 6 4 regions 6.1 Intermission: negative threads 6.2 Adding entropy inequality constraints 6.3 4 regions revisited 19 21 23 26 7 5 regions 30 8 Discussion 8.1 Main conjecture 8.2 Multipartite distillation 8.3 The positive K cone 35 35 37 39 A Locking configurations of perfect tensor hyperthreads for some 5 region extremal rays 40 1 Introduction The purpose of this article is to begin to address ways of characterizing the multipartite entanglement of holographic states. Given a holographic state |ψi with a dual classical bulk geometry M we can consider a static time slice Σ and partition the boundary ∂Σ into a boundary region A along with its purifier the complement O. In such a set up it is well known that the bipartite entanglement between A and O can be quantified by the entanglement entropy (EE) SA . In the boundary theory this is calculated as the von Neumann entropy of the reduced density matrix after a partial trace of one of the two boundary regions. One way of understanding this quantity is that it determines the –1– JHEP09(2022)239 1 Introduction O <latexit sha1_base64="fXM6hdxOrgMafVthFi+DL/4qimo=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexK0BwDXryZgHlAsoTZSW8yZnZ2mZkVQsgXePGgiFc/yZt/4yTZgyYWNBRV3XR3BYng2rjut5Pb2Nza3snvFvb2Dw6PiscnLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2Mb+d++wmV5rF8MJME/YgOJQ85o8ZKjft+seSW3QXIOvEyUoIM9X7xqzeIWRqhNExQrbuemxh/SpXhTOCs0Es1JpSN6RC7lkoaofani0Nn5MIqAxLGypY0ZKH+npjSSOtJFNjOiJqRXvXm4n9eNzVh1Z9ymaQGJVsuClNBTEzmX5MBV8iMmFhCmeL2VsJGVFFmbDYFG4K3+vI6aV2VvetypVEp1apZHHk4g3O4BA9uoAZ3UIcmMEB4hld4cx6dF+fd+Vi25pxs5hT+wPn8AagBjNI=</latexit> <latexit sha1_base64="fFCzG3d4KY9DRKEqCK6PPwOvEJ0=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYNUY4YLx4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfju7nffkKleSwfzCRBP6JDyUPOqLFS47ZfLLlldwGyTryMlCBDvV/86g1ilkYoDRNU667nJsafUmU4Ezgr9FKNCWVjOsSupZJGqP3p4tAZubDKgISxsiUNWai/J6Y00noSBbYzomakV725+J/XTU1Y9adcJqlByZaLwlQQE5P512TAFTIjJpZQpri9lbARVZQZm03BhuCtvrxOWldl77pcaVRKtWoWRx7O4BwuwYMbqME91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJLJjMQ=</latexit> Figure 1. The RT surface mA along with a maximal configuration of bit threads. The area of mA and the number of bit threads both calculate the holographic entanglement entropy SA . number of Bell pairs which can be distilled from the asymptotic limit of many copies of the holographic state by a quantum channel constructed from local unitaries (LU) |ψi −→ |AOi⊗SA (1.1) where the arrow here represents the appropriate distillation protocol. From the bulk perspective the entanglement entropy can be calculated using the RyuTakayanagi (RT) formula [1] which asks for the minimal area surface homologous to A SA = min area(m) m∼A (1.2) we call this minimizing surface mA . Alternatively, the entanglement entropy is given a maximal configuration of bit threads [2, 3]: simple curves of constant thickness connecting A to O subject to a local density bound. Tools from the theory of convex optimization can be used to show that these two descriptions: minimal RT surfaces and maximal bit thread configurations are in fact the same (see figure 1). The bit threads are often represented as a geometrical avatar of the distilled Bell pairs. That is given a configuration of bit threads we can consider a course-graining or desiccation of the geometry where we only keep the portions of Σ which bit threads cross. Such a geometry can be viewed as a collection of wormholes, one for each thread. This is in turn equivalent to a simple graph consisting of a single edge of weight SA which is equivalent to SA Bell pairs (see figure 2). This perspective provides a realization of the connection between geometry and entanglement. A natural question to then ask is what if we consider more than one boundary region. Given a partition of ∂Σ into N regions along with a purifier O: ∂Σ = {A1 · · · AN , O} there are 2N − 1 independent entanglement entropies one can consider. These include single party EEs (e.g. SA1 ) as well as multiple party EEs consisting of the union of a number of boundary regions (e.g. SA1 A2 ). It is useful to organize these into an entropy –2– JHEP09(2022)239 A .. . .. . <latexit sha1_base64="DWLqlur6+WnPhdW7QMGiRy8f9P8=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEtMeCF48V7Ae0oWw223btJht2J4US+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSMirVjDeZkkp3Amq4FDFvokDJO4nmNAokbwfju7nfnnBthIofcZpwP6LDWAwEo2ilVm8SKjT9csWtuguQdeLlpAI5Gv3yVy9ULI14jExSY7qem6CfUY2CST4r9VLDE8rGdMi7lsY04sbPFtfOyIVVQjJQ2laMZKH+nshoZMw0CmxnRHFkVr25+J/XTXFQ8zMRJynymC0XDVJJUJH56yQUmjOUU0so08LeStiIasrQBlSyIXirL6+T1lXVu6leP1xX6rU8jiKcwTlcgge3UId7aEATGDzBM7zCm6OcF+fd+Vi2Fpx85hT+wPn8Actbj0E=</latexit> <latexit sha1_base64="BtcJQYb3Cc9JlljcbQ59YQ+dQ2I=">AAAB63icbVBNSwMxEJ31s9avqkcvwSLUS9mVoj1WvHisaD+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777aytb2xubRd2irt7+weHpaPjto4SRWiLRDxS3QBrypmkLcMMp91YUSwCTjvB5DbzO09UaRbJRzONqS/wSLKQEWwy6aFyczEold2qOwdaJV5OypCjOSh99YcRSQSVhnCsdc9zY+OnWBlGOJ0V+4mmMSYTPKI9SyUWVPvp/NYZOrfKEIWRsiUNmqu/J1IstJ6KwHYKbMZ62cvE/7xeYsK6nzIZJ4ZKslgUJhyZCGWPoyFTlBg+tQQTxeytiIyxwsTYeIo2BG/55VXSvqx6V9Xafa3cqOdxFOAUzqACHlxDA+6gCS0gMIZneIU3RzgvzrvzsWhdc/KZE/gD5/MH+6SNhg==</latexit> S(A) A <latexit sha1_base64="fFCzG3d4KY9DRKEqCK6PPwOvEJ0=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYN (...truncated)