Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 3

Nonarchimedean holographic entropy from networks of perfect tensors

Pages: 591 – 721

DOI: https://dx.doi.org/10.4310/ATMP.2021.v25.n3.a2

Authors

Matthew Heydeman (Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Cal., U.S.A.)

Matilde Marcolli (Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, Cal., U.S.A.; Department of Mathematics, University of Toronto, Ontario, Canada; and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)

Sarthak Parikh (Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, Cal., U.S.A.)

Ingmar Saberi (Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Germany)

Abstract

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat–Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a $p$‑adic version of entropy which obeys a Ryu–Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$‑adic backgrounds, along with a Bekenstein–Hawking-type formula for black hole entropy.We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua.

Published 21 March 2022