Entanglement is not Enough

TL;DR

Entanglement entropy alone cannot account for the sustained growth of Einstein-Rosen bridges behind black hole horizons. The paper argues that quantum complexity, not entanglement, tracks the interior's evolving geometry, proposing a TS-based growth rate and a volume–complexity correspondence anchored in maximal slices, causal patches, and tensor-network models. It develops both AdS/CFT and Schwarzschild contexts, derives universal relations for ERB volume, complexity, and their bounds, and introduces Nielsen's complexity-geometry framework to explain interior fluctuations and nonlinearity. The findings offer a conceptual bridge between quantum information complexity and spacetime connectivity, with implications for black hole complements and potential limits on observability.

Abstract

This is the written version of a lecture given at KITP in Oct 2014 on Black Holes and quantum complexity. I've included (in boldface) various questions that came up during the lecture and discussions the following day, as well as the quantitative calculations that form the basis of the arguments.

Figures (22)

• Figure 1:
• Figure 2: Penrose diagram foliated by maximal slices. The green curve is the "final slice."
• Figure 3: Embedding diagram for an ERB1.
• Figure 4: Evolution of the interior of two black holes as the horizons merge. The top row shows the ERBs growing while before the merger. The bottom row shows the evolution after the merger.
• Figure 5: Fission of a black hole into two equal daughters.