Nonarchimedean Holographic Entropy from Networks of Perfect Tensors

TL;DR

This paper develops a $p$-adic holographic entropy framework using perfect tensor networks on Bruhat-Tits trees, deriving entropy formulas and inequalities, and constructing perfect tensors from semiclassical states.

Contribution

It introduces a $p$-adic holographic entropy model with perfect tensor networks on Bruhat-Tits trees, including entropy formulas, inequalities, and new tensor constructions from phase space states.

Findings

Derived a $p$-adic Ryu-Takayanagi formula for entanglement entropy.

Proved entropy inequalities such as subadditivity and monogamy of mutual information.

Constructed perfect tensors from semiclassical phase space states.

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.