On observers in holographic maps

TL;DR

This work tackles the problem of trivial Hilbert spaces for closed universes in gravitational path integrals by comparing HUZ and AAIL observer prescriptions and introduces a non-isometric holographic code implementing the AAIL construction. The key idea is to remove the observer-acting portion of the holographic map, creating a boundary at the observer–environment interface and yielding a nontrivial fundamental space of dimension $d_{Ob} e^{2S_0}$. Tensor-network generalizations and Renyi-entropy analyses corroborate the scaling and demonstrate a general, locality-based method for including observers in holographic maps across open and closed universes. The approach clarifies how path-integral data relate to holographic maps and provides a framework that can be applied to general holographic codes.

Abstract

A straightforward gravitational path integral calculation implies that closed universes are trivial, described by a one dimensional Hilbert space. Two recent papers by Harlow-Usatyuk-Zhao and Abdalla-Antonini-Iliesiu-Levine have sought to ameliorate this issue by defining special rules to incorporate observers into the path integral. However, the proposed rules are different, leading to differing results for the Hilbert space dimension. Moreover, the former work offers a holographic map realized using a non-isometric code construction to complement their path integral result and clarify its physics. In this work, we propose a non-isometric code that implements the second construction, allowing thorough comparison. Our prescription may be thought of as simply removing the portion of the map that acts on the observer, while preserving the rest, creating an effective holographic boundary at the observer-environment interface. This proposal can be directly applied to general holographic maps for both open and closed universes of any dimension.

Figures (7)

• Figure 1: Top row: leading contributions to the inner product squared from the gravitational path integral for a closed universe. All three terms are $\mathcal{O}(1)$. Middle row: by introducing an entangled non-gravitational copy of the observer, the work of HUZ suppresses the second and third terms by a factor of $1/d_{Ob}$. Bottom row: the work of AAIL insists that an observer must "stay in their own universe" and therefore removes the second and third terms from the path integral entirely.
• Figure 2: The circuit diagram describing the model holographic map for a closed universe without an observer, given in (\ref{['eq:Harlow']}) and adapted from harlow_quantum_2025.
• Figure 3: A diagrammatic representation of $\overline{|\langle\psi|V^\dagger V|\phi\rangle|^2}$ (left side) and the three leading terms in the average over the Haar measure (right side). Red lines represent $b_1$ degrees of freedom, and green lines represent $b_2$ degrees of freedom. $f_i$ indicates the insertion of fixed states $|\psi_0\rangle_{f_i}$; these drop out in the average. Overall numerical prefactors have been omitted for convenience.
• Figure 4: A circuit diagram describing the model holographic map for a closed universe with an observer, given by removing any operators and postselection acting on the observer.
• Figure 5: A diagrammatic representation of $\overline{|\langle\psi|V_{Ob}^\dagger V_{Ob}|\phi\rangle|^2}$ (left side) and the three leading terms in the average over the Haar measure for just $O_2$ (right side). Red lines represent observer $Ob$ degrees of freedom, and the green lines represent matter $M$ degrees of freedom. $f_i$ indicates the insertion of fixed states; these drop out in the average. Overall numerical prefactors have been omitted for convenience.