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Holographic Complexity for disentangled states

Tokiro Numasawa

TL;DR

This work probes holographic complexity in AdS/BCFT setups with end-of-the-world branes by examining the maximal volume $C_V$ and the Wheeler-DeWitt action $C_A$ for BTZ-like geometries. It shows that when the horizon size $r_H$ approaches the cutoff $r_0$ (or the ETW brane contacts the cutoff), holographic entanglement entropy $S_A$ vanishes and the volume $C_V$ similarly tends to zero, signaling a product-state reference in the CV picture. In contrast, the CA results depend on the chosen WdW regularization: Regularization 1 yields $C_A o 0$ in the entangled-to-disentangled limit, while Regularization 2 generally produces nonvanishing or divergent $C_A$ with tension-dependent terms, revealing subtleties in defining holographic complexity for BCFTs. The analysis emphasizes the pivotal roles of ETW-brane tension, boundary entropy, and regularization in interpreting holographic complexity and points to future work on time dependence and quantum corrections in these setups.

Abstract

In this paper we consider the maximal volume and the action, which are conjectured to be gravity duals of the complexity, in the black hole geometries with end of the world branes. These geometries are duals of boundary states in CFTs which have small real space entanglement. When we raise the black hole temperature while keeping the cutoff radius, black hole horizons or end of the world branes come in contact with the cutoff surface. In this limit, holographic entanglement entropy reduces to 0. We studied the behavior of the volume and the action. We found that the volume reduces to 0 in this limit. The behavior of the action depends on their regularization. We study the implication of these results to the reference state of the holographic complexity both in the complexity = volume or the complexity = action conjectures.

Holographic Complexity for disentangled states

TL;DR

This work probes holographic complexity in AdS/BCFT setups with end-of-the-world branes by examining the maximal volume and the Wheeler-DeWitt action for BTZ-like geometries. It shows that when the horizon size approaches the cutoff (or the ETW brane contacts the cutoff), holographic entanglement entropy vanishes and the volume similarly tends to zero, signaling a product-state reference in the CV picture. In contrast, the CA results depend on the chosen WdW regularization: Regularization 1 yields in the entangled-to-disentangled limit, while Regularization 2 generally produces nonvanishing or divergent with tension-dependent terms, revealing subtleties in defining holographic complexity for BCFTs. The analysis emphasizes the pivotal roles of ETW-brane tension, boundary entropy, and regularization in interpreting holographic complexity and points to future work on time dependence and quantum corrections in these setups.

Abstract

In this paper we consider the maximal volume and the action, which are conjectured to be gravity duals of the complexity, in the black hole geometries with end of the world branes. These geometries are duals of boundary states in CFTs which have small real space entanglement. When we raise the black hole temperature while keeping the cutoff radius, black hole horizons or end of the world branes come in contact with the cutoff surface. In this limit, holographic entanglement entropy reduces to 0. We studied the behavior of the volume and the action. We found that the volume reduces to 0 in this limit. The behavior of the action depends on their regularization. We study the implication of these results to the reference state of the holographic complexity both in the complexity = volume or the complexity = action conjectures.

Paper Structure

This paper contains 27 sections, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Entanglement structure of boundary states
  3. single sided CFT case
  4. two sided CFT case
  5. Holographic dual of EPR pairs
  6. Holographic dual of Boundary states
  7. Volume and Action for duals of local EPR pairs
  8. Volume cases
  9. Action cases
  10. Regularization 1
  11. Regularization 2
  12. Volume and action for $T=0$ boundary states
  13. Volume and Action for Boundary state with non-zero boundary entropy
  14. Volume cases
  15. Action cases
  16. ...and 12 more sections

Figures (5)

  • Figure 1: The left picture is the configuration of the ETW brane with positive tension. The middle picture is the configuration of the tensionless ETW brane, which is obtained as the orbifold of eternal black holes. The right picture is the configuration of the ETW brane with negative tension
  • Figure 2: Two regularizations of the WdW patch. The left picture is the first regularization, in which the WdW patch ends on the cutoff surface. The right picture is the second regularization, in which the WdW patch end on the asymptotic AdS boundary.
  • Figure 3: The left picture describes the configuration of WdW patch for $r_H>Tl_{AdS}r_0$. The right picture is the one for $r_H<Tl_{AdS}r_0$.
  • Figure 4: The left picture describes the configuration of the WdW patch for $T>0$. The right picture is the configuration of the WdW patch for $T<0$. $T=0$ cases can be considered as special limits of both cases.
  • Figure 5: The left picture is the configuration for regularization 1 with positive tension ETW branes for $r_H > Tl_{AdS}r_0$. The calculation for this configuration is also applicable for negative tension cases. The right picture is the configuration for regularization 1 with positive tension ETW branes for $r_H < Tl_{AdS}r_0$. If we decrease the temperature further, we will see the Hawking-Page transition.