Does Boundary Distinguish Complexities?

TL;DR

This work probes whether a BCFT boundary can differentiate holographic notions of quantum-state complexity defined by CV and CA, via path-integral optimization and Takayanagi’s AdS/BCFT construction. The authors compute boundary-induced increases in complexity using the boundary Liouville action in BCFT$_2$, yielding a universal term $\left.\Delta C_{\text{L}}^{\text{bdy}}\right|_{\text{univ}}=\frac{c}{6\pi}\alpha$, with $\alpha=\frac{\mu_B L}{\sqrt{1-\mu_B^2 L^2}}$, and relate this to boundary entropy $S_{\text{bdy}}=\frac{c}{12}\log\left(\frac{1+\mu_B L}{1-\mu_B L}\right)$. In the holographic dual, the CV and CA complexities yield boundary contributions that share the same UV divergence structure in general dimensions, with the notable exception of CA in AdS$_3$/BCFT$_2$, where the defect-induced difference aligns with previous defect-CFT results. Specifically, in $d=2$ the CV and path-integral complexities exhibit a universal logarithmic boundary term, while CA yields a finite universal term $\left.\Delta C_{\text{A}}^{\text{bdy}}\right|_{\text{univ}}=\frac{L}{4\pi G_N}(\sqrt{1+\alpha^2}-1)$ and no logarithmic divergence, diminishing the contrast between CV and CA in this case. Overall, the boundary does not generically distinguish CV from CA, except in the AdS$_3$/BCFT$_2$ CA case, and the results connect boundary complexity to boundary entropy via the $g$-theorem, with broader implications for BCFT/DCFT and higher-dimensional extensions.

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.

Figures (3)

• Figure 1: (Left) The setup of the path integral for vacuum wave functional in BCFT. The boundary $\partial \mathcal{M}_{1}$ is located at $x=0$ and the state is realized at $\partial \mathcal{M}_{0} = \{x>0, z = \epsilon \sim 0 \}$. (Right) The setup after the path-integral optimization. The boundary $\partial \mathcal{M}_{1}$ is tilted.
• Figure 2: The entanglement entropy associated to the subsystem $A$ ($0 \leq x \leq l$) is given by the length of the arc $\gamma_{A}$ anchored on the boundary surface $\partial \mathcal{M}_{1}$ ($x = - \alpha z$) and the edge of $A$.
• Figure 3: The WDW patch which is the causal development of the Cauchy slice $t = 0$. The bulk region $\mathcal{B}_{\text{WDW}}$ is surrounded by a portion $\mathcal{Q}_{\text{WDW}}$ of the brane $\mathcal{Q}$, null surfaces $N_{1,2}$ and the timelike surface $S_\epsilon$ at $z=\epsilon$. The red lines are joints $J_{\text{n},1,2} = N_{1,2} \cap \mathcal{Q}_{\text{WDW}}$, $J_{\text{s},\epsilon_\pm} = N_{1,2} \cap S_\epsilon$ and $\mathcal{J}_{\text{t},\epsilon} = \mathcal{Q}_{\text{WDW}} \cap S_\epsilon$. The other timelike surface $S_\infty$ at $z=z_\infty$, the other spacelike joints $J_{\text{s},\infty_\pm}$ and the other timelike joint $\mathcal{J}_{\text{t},\infty}$ are not depicted.