Non-isometry, State Dependence and Holography

TL;DR

The paper analyzes when bulk operator reconstruction in holography can be state-independent by linking non-isometry of bulk-to-boundary maps to state-dependence of reconstruction. It introduces quantitative measures for both notions, proves bounds, and shows that under gravitational path integral assumptions, weak non-isometry yields an approximately isometric encoding with nearly state-independent bulk reconstruction, while strong non-isometry enforces state-dependent reconstruction and can signal EFT breakdown. It also connects non-isometry to causal structure, arguing that a global horizon or closed universe implies non-isometric global bulk-to-boundary maps. The results offer a precise framework for understanding when holographic codes support universal versus context-dependent operator reconstructions and highlight the role of horizons and wormhole corrections in these considerations.

Abstract

We establish an equivalence between non-isometry of quantum codes and state-dependence of operator reconstruction, and discuss implications of this equivalence for holographic duality. Specifically, we define quantitative measures of non-isometry and state-dependence and describe bounds relating these quantities. In the context of holography we show that, assuming known gravitational path integral results for overlaps between semiclassical states, non-isometric bulk-to-boundary maps with a trivial kernel are approximately isometric and bulk reconstruction approximately state-independent. In contrast, non-isometric maps with a non-empty kernel always lead to state-dependent reconstruction. We also show that if a global bulk-to-boundary map is non-isometric, then there exists a region in the bulk which is causally disconnected from the boundary. Finally, we conjecture that, under certain physical assumptions for the definition of the Hilbert space of effective field theory in AdS space, the presence of a global horizon implies a non-isometric global bulk-to-boundary map.

Figures (1)

• Figure 1: Geometric black hole microstate with a spherically-symmetric shell of pressureless matter behind the horizon. In the bottom half of the figure we show the Euclidean preparation of the microstate via the gravitational Euclidean path integral. In the top half we depict the corresponding Lorentzian geometry. We are considering here the case where the two exterior geometries are identical, but generically the two black holes can be at different temperatures Chandra:2022fwiBalasubramanian:2022gmoCliment:2024trz. Shells of different mass, evolved for accordingly different amounts of Euclidean times, give rise to identical exterior geometries but different interior geometries, and therefore give rise to different black hole microstates.

Theorems & Definitions (55)

• Definition 2.1: State-specific reconstruction
• Definition 2.2: State-independent reconstruction
• Theorem 2.1
• proof
• Definition 2.3
• Proposition 2.1
• Lemma 2.1
• proof
• Definition 2.4
• Definition 2.5