Non-isometry, State Dependence and Holography

TL;DR

This paper explores the relationship between non-isometry in quantum codes and state dependence in operator reconstruction, revealing implications for holographic duality and bulk-boundary mappings in AdS/CFT.

Contribution

It establishes a formal equivalence between non-isometry and state dependence, providing bounds and implications for holography and bulk reconstruction.

Findings

Non-isometric bulk-to-boundary maps with trivial kernel are approximately isometric.

Non-isometric maps with a non-empty kernel lead to state-dependent reconstruction.

Global non-isometry implies the existence of causally disconnected bulk regions.

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.