Does Boundary Distinguish Complexities?

TL;DR

This paper investigates whether the presence of a boundary in conformal field theories affects the complexity of quantum states, comparing field-theoretic and holographic approaches, and finds that boundaries generally do not distinguish complexity.

Contribution

It provides a comparative analysis of complexity in BCFTs using path-integral optimization and holographic models, showing boundary effects are not distinctive in complexity measures.

Findings

Complexity increments have similar divergence structures across models.

Boundary effects do not generally distinguish complexity.

CA complexity in AdS3/BCFT2 shows a different divergence structure.

Abstract

Recently, Chapman et al. argued that holographic complexities for defects distinguish action from volume. Motivated by their work, we study complexity of quantum states in conformal field theory with boundary. In generic two-dimensional BCFT, we work on the path-integral optimization which gives one of field-theoretic definitions for the complexity. We also perform holographic computations of the complexity in Takayanagi's AdS/BCFT model following by the "complexity $=$ volume" conjecture and "complexity $=$ action" conjecture. We find that increments of the complexity due to the boundary show the same divergent structures in these models except for the CA complexity in the AdS$_3$/BCFT$_2$ model as the argument by Chapman et al. Thus, we conclude that boundary does not distinguish the complexities in general.