$T\bar{T}$ Deformation and the Complexity=Volume Conjecture

TL;DR

The paper addresses how holographic complexity can be grounded in a quantum-field-theory construction by using the reversible $T\bar{T}$ deformation as a canonical unitary circuit for state preparation. It argues that the subregion complexity $C(\rho_A)$ is captured by the evolving bulk volume of the entanglement wedge, formalized as $C(\rho_A) \propto \int d[r]\,\text{Area}(r)$, and shows that this naturally reproduces the Complexity=Volume (CV) proposal for AdS spacetimes, with a field-theory measure $d[\mu]$ connecting to the bulk cutoff via $\mu=\frac{16\pi G}{r_c^2}$. A complementary picture interprets bulk geometry as the history of the quantum circuit, establishing a duality between gravity and the boundary circuit in a many-body Hilbert space and highlighting the emergent nature of spacetime. The work provides a field-theory grounded justification for CV, emphasizes the reversibility of $T\bar{T}$ as central to this picture, and suggests broad implications for simulating QFT complexity and understanding bulk emergence.

Abstract

Complexity in quantum physics measures how difficult a state can be reached from a reference state and more precisely it is the number of fundamental unitary gates we have to operate to transform the reference state to the state we are considering. In the holographic context, based on several explicit calculations and arguments, it is conjectured that certain bulk volume calculates the boundary field theory subregion complexity. In this paper, we will show that the $T\bar{T}$ deformation shows a strong signal of the correctness of this complexity equals volume conjecture. A bonus is a way to look at the $T\bar{T}$ deformation, by its reversibility, as operating a unitary quantum circuit which prepares states in quantum field theory.

Figures (2)

• Figure 1: A time slice of $AdS_{3}$. The shaded region is the entanglement wedge of boundary subsystem A and the blue curve $\Sigma$ is its RT surface Ryu:2006bv.
• Figure 2: We do $T\bar{T}$ deformation the opposite way starting from IR to UV. For the subsystem A we start from the gold slice tangent to the RT surface $\Sigma$ at point P and opposite $T\bar{T}$ deformation constructs the subsystem A from P. An intermediate slice along the way is shown in orange which intersects $\Sigma$ at $P_{i}$ and $P'_{i}$.