Helping observers in closed universes reach their full potential

TL;DR

The paper refines the observer-based holographic map framework for closed universes by constructing observer-specific SWAP-test operators using HUZ and CO rules. It shows that the AdS observer (alpha) cannot rule out a baby universe once AR is carefully accounted, while the beta observer's predictions improve but remain limited by non-isometric maps, though exact reconstructions are possible in toy models. It introduces an improved reconstruction tilde S_AdS ⊗ tilde S_{beta′} that enhances beta's accuracy in the presence of entanglement. The results illuminate how observer complementarity can reconcile semiclassical descriptions with baby universes and point to open questions about approximate AR conditions and broader applicability.

Abstract

Recent work by Engelhardt, Gesteau, and Harlow applies proposals for incorporating observers into holographic maps to study the Antonini-Rath puzzle for closed universes. In a new form of ``observer complementarity,'' they find that an AdS bulk observer measures a SWAP test to determine that there is no closed universe in the bulk, contrary to the (limited) description given by an observer inside the closed universe. In this work, we improve the predictions of both observers by using the holographic maps to define new operators to perform this same SWAP test. With these, we show that the AdS observer cannot rule out a baby universe in the bulk, and the closed universe observer can improve the accuracy of their description.

Figures (10)

• Figure 1: The two candidate bulk states described by AR antonini_holographic_2025. The top line, denoted $|\psi_1\rangle$, depicts "description 1" prepared by the AS$^2$ construction antonini_cosmology_2023, consisting of a baby universe $b$ and two disconnected AdS spacetimes $a$. The bottom line, denoted $|\psi_2\rangle$, depicts "description 2" without the baby universe. Figure adapted from higginbotham_tests_2025.
• Figure 2: A diagram representing the five holographic maps defined in engelhardt_observer_2025 and how they relate Hilbert spaces with and without the baby universe ($b$) and observers ($\alpha$, $\beta$). The middle row contains effective Hilbert spaces, the top row contains fundamental Hilbert spaces without any observer, and the bottom row contains fundamental Hilbert spaces with one observer. Holographic maps are color coded: blue for isometries, and orange for non-isometries.
• Figure 3: Circuit diagrams for holographic maps without an observer. $V_\text{HKLL}$ (left) maps a bulk state without a baby universe to the asymptotic AdS boundary; defined from the HKLL map, it is isometric. $V$ (right) includes post-selection on the baby universe and is non-isometric. Bulk inputs have been split into subsystems for observers ($\alpha$, $\beta$) and matter ($M_a$, $M_b$) for later use in applying observer rules.
• Figure 4: A diagram representing the operators performing the SWAP test on the Hilbert spaces of figure \ref{['fig:HS_map']}. Operators on fundamental Hilbert spaces have been distinguished with a tilde. Arrows indicate how each is defined relative to $\mathcal{S}_\text{AdS}$ and are colored according to whether the resulting operator performs the SWAP test exactly (blue) or approximately (orange).
• Figure 5: Circuit diagrams for the three observer-modified holographic maps constructed using the HUZ rules. A CNOT gate is used to denote the cloning operation, but the cloning can be taken to be in any basis of choice.