The Holography of Spread Complexity: A Story of Observers

Abstract

Building on the pioneering work of \cite{Caputa:2024sux}, we propose a holographic description of spread complexity and its rate in 2D CFTs. By exploiting $SL(2,\mathbb{R})$ symmetry, we explicitly construct the Krylov basis, expressing spread complexity as a linear combination of generator expectation values. Within the AdS/CFT correspondence, we translate these boundary expectations directly into bulk kinematic variables. These findings suggest that spread complexity manifests as the energy measured by a bulk observer, with its rate corresponding to the radial momentum.

Figures (3)

• Figure 1: Depiction of the holographic construction of the quantum state $|\psi(t)\rangle$.
• Figure 2: The circular blue line denotes the time evolution, the purple line, which is perpendicular to the hyperbolic boundary denoted by the black circle, is the hyperbolic geodesic connecting the starting and ending points.
• Figure 3: Illustration of our proposal: A specific observer interacts with the dual particle and performs a measurement. The spread complexity corresponds to the measured energy, while the complexity rate is identified with the radial momentum observed by that observer.