University of Illinois Urbana-Champaign

On canonical purification of random tensor networks

Lin, Simon

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LIN-DISSERTATION-2023.pdf

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https://hdl.handle.net/2142/121923

Description

Title
On canonical purification of random tensor networks
Author(s)
Lin, Simon
Issue Date
2023-07-31
Director of Research (if dissertation) or Advisor (if thesis)
Faulkner, Thomas
Doctoral Committee Chair(s)
Leigh, Robert G
Committee Member(s)
  • Clark, Bryan K
  • Covey, Jacob P
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • entanglement
  • quantum information
  • AdS/CFT
  • holography
  • tensor networks
Abstract
Understanding the entanglement structure of holographic states has played a significant role in demystifying quantum gravity and the emergence of spacetime. The state of the art development in this field makes heavy use of ideas from quantum information and quantum error correction to explain various features of quantum gravity including the holographic principle, the quantum extremal surface formula and subregion-subregion duality. In particular, random tensor networks (RTNs), a toy model for holography which is understood to model fixed-area states, has been demonstrated to effectively capture the aforementioned traits shared by holographic theories. While calculations of the entanglement entropy have led to most of the insight into holography and RTNs, it is useful to consider other quantum information quantities in order to give a more complete picture of the emergence of spacetime from entanglement. Among these, the canonical purification, along with its associated quantum information measure known as the reflected entropy, is of special interest in the pursuit of such new measures. The reflected entropy is closely related to the quantification of tripartite entanglement of a quantum state and enjoys a bulk geometric dual in terms of the area of entanglement wedge cross-section. In this dissertation, we study the entanglement structure of the canonical purification of states built from holographic RTNs. For simple networks made from a single or a pair of random tensors, we analytically compute the entanglement spectra of the canonically purified density matrix. We found that the spectra effectively decomposes into different super-selection sectors, each of which can be given a gravitational interpretation of semiclassical bulk saddles with different topology. We show that our formalism can be extended to incorporate the West Coast Model, a toy model for black hole evaporation. We find that the spectrum exhibits a similar form in terms of the super-selection sectors, but with a wider window of fluctuation compared to the RTN results. For RTNs built from arbitrary networks, we prove that the problem of finding the integer Renyi reflected entropy is equivalent to finding an optimization program known as the minimal triway cut when the system is far away from phase transitions. Minimal triway cuts can be formulated as integer programs which cannot be relaxed to find a dual maximal flow description. This sheds light on the gap between the existence of tripartite entanglement in holographic states and the bipartite entanglement structure motivated by bit-threads. In particular we show that the Markov gap, defined as the difference between the reflected entropy and the mutual information, is lower bounded by the integrality gap of the program that computes the triway cut. We apply this result to prove a conjecture that relates entanglement of purification to minimal entanglement wedge cross-section on a large class of RTNs.
Graduation Semester
2023-12
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/121923
Copyright and License Information
Copyright 2023 Simon Lin

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Graduate Theses and Dissertations at Illinois