Complexity of Formation in Holography

TL;DR

This paper investigates the holographic complexity of formation using the complexity=action duality, revealing linear growth with entropy in higher dimensions and a constant value in two dimensions, thus deepening understanding of quantum complexity in holography.

Contribution

It applies the complexity=action duality to compute the complexity of formation, providing new insights into its behavior across different dimensions and comparing with the complexity=volume approach.

Findings

Complexity of formation grows linearly with entropy for $d>2$.

In $d=2$, the complexity of formation is a constant, independent of temperature.

Results are compared and contrasted with the complexity=volume duality.

Abstract

It was recently conjectured that the quantum complexity of a holographic boundary state can be computed by evaluating the gravitational action on a bulk region known as the Wheeler-DeWitt patch. We apply this complexity=action duality to evaluate the `complexity of formation' (arXiv:1509.07876, arXiv:1512.04993), i.e., the additional complexity arising in preparing the entangled thermofield double state with two copies of the boundary CFT compared to preparing the individual vacuum states of the two copies. We find that for boundary dimensions $d>2$, the difference in the complexities grows linearly with the thermal entropy at high temperatures. For the special case $d=2$, the complexity of formation is a fixed constant, independent of the temperature. We compare these results to those found using the complexity=volume duality.