Holographic Quantum Circuits from Splitting/Joining Local Quenches

TL;DR

The work addresses how local quenches in 2d CFTs—local operator, splitting, and joining—drive entanglement and how these processes differ between free and holographic theories. It leverages entanglement density to systematically capture the evolution, derives both CFT and holographic EE results, and reveals two distinct logarithmic growth regimes tied to the bulk geodesics and AdS/BCFT boundary surfaces. The authors show that holographic geodesics that end on boundary surfaces $Q$ encode splitting/joining effects and connect these geometries to MERA-like tensor networks, proposing a gravity dual for quantum circuits. The results illuminate nonlocal initial entanglement in holographic quenches, clarify the role of boundary entropy, and provide a framework for interpreting time-evolving entanglement through ED and geometric constructs with potential applications to discretized gravity duals and quantum information transport. The paper thus bridges conformal field theory, holography, and tensor-network perspectives to model and understand local quenches as holographic quantum circuits.

Abstract

We study three different types of local quenches (local operator, splitting and joining) in both the free fermion and holographic CFTs in two dimensions. We show that the computation of a quantity called entanglement density, provides a systematic method to capture essential properties of local quenches. This allows us to clearly understand the differences between the free and holographic CFTs as well as the distinctions between three local quenches. We also analyze holographic geometries of splitting/joining local quenches using the AdS/BCFT prescription. We show that they are essentially described by time evolutions of boundary surfaces in the bulk AdS. We find that the logarithmic time evolution of entanglement entropy arises from the region behind the Poincare horizon as well as the evolutions of boundary surfaces. In the CFT side, our analysis of entanglement density suggests such a logarithmic growth is due to initial non-local quantum entanglement just after the quench. Finally, by combining our results, we propose a new class of gravity duals, which are analogous to quantum circuits or tensor networks such as MERA, based on the AdS/BCFT construction.

Figures (28)

• Figure 1: The three local quenches are sketched: the local operator quench (left), the splitting local quench (middle), and the joining local quench (right) in two dimensional CFTs. The red points are locations where the energy density is very large.
• Figure 2: A sketch of AdS/BCFT analysis for AdS$_3$. A holographic CFT on $M$ (with the boundary $\partial M$) is dual to gravity on $N$. The boundary of $N$ consists of the surface $Q$ and $M$. The right picture shows the calculation of holographic entanglement entropy. The blue curve gives the connected geodesic contribution and the green ones are the disconnected geodesics which end on the boundary surface $Q$. The correct holographic entanglement entropy is given by the one with a smaller length.
• Figure 3: The behavior of entanglement density (ED) under the global quenches. The left graph describes the ED $\Delta n(l,t)$ as a function of $l$ at $t=3$ (we chose $\alpha=0.1$). The right 3D plot shows the behavior of ED $\Delta n(l,t)$ as a function of the time $t$ (horizontal axis) and the subsystem size $l$ (depth axis), where we set $\alpha=0.5$.
• Figure 4: The integration range $\int dl d\xi$ of the ED (red regions) and causality range (brown shaded region). The left picture shows the region of integration which computes $S_A$ for the subsystems $A=[a,b]$. The middle one shows the same one when $A$ is the half line: $a=0$ and $b=\infty$. The brown shaded region in the right picture describe the range of $(\xi,l)$ where the local excitation at $x=t=0$ can make any physical influence assuming causality. In all these three pictures, the purple dot at $(\xi,l)=(0,2t)$ represents the peak of ED due to the local quench. In the right picture, the blue curve near this dot (given by $l=2\sqrt{\xi^2+t^2}$) describes the delta functional peaks which are peculiar to holographic local quenches. Note that they are outside of the causality region. The calculations of $S_A$ for $a\ll t\ll b$ correspond to integrating the red region in the right picture. We note the left part of the blue curve gradually gets into the red region, which gives the logarithmic growth of EE.
• Figure 5: The profile of entanglement density for the 2nd Renyi entropy under the local operator quench in $c=1$ free scalar CFT at the time $t=0$ (left) and $t=1$ (right) with $\alpha=0.1$, trigger by the operator $O=e^{i\phi}+e^{-i\phi}$. The horizontal and depth coordinate correspond to $\xi$ and $l$, respectively.