Complexity of Formation in Holography

TL;DR

The paper tests the complexity=action conjecture by computing the complexity of formation for the thermofield double state through the Wheeler-DeWitt patch in AdS black holes and comparing to two copies of vacuum AdS. It shows a universal leading behavior ΔC ≈ k_d S at high temperature for d>2, with d=2 yielding a temperature-independent constant, and finds results consistent with, but distinct from, the complexity=volume picture. The analysis covers planar, spherical, and hyperbolic horizons, discusses ambiguities in null boundary terms, and interprets findings in light of MERA tensor networks and potential extensions to more general gravity theories. Overall, the results support a close connection between holographic complexity and entanglement/thermodynamic quantities, while highlighting dependence on the chosen holographic prescription and geometry.

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.

Figures (24)

• Figure 1: Penrose diagram for black holes in more than three bulk dimensions ($d>2$). We define surfaces of constant $r$ to regulate the action both near the asymptotic boundary ($r=r_\textrm{\tiny max}$) and near the past and future singularities ($r=\epsilon_0$). We identify the Wheeler-DeWitt patch as the area of the bulk bounded by the four null sheets which originate from the boundary at $t=0$. The joints between the null sheets and the regulating surfaces are indicated by red dots.
• Figure 2: Penrose diagrams of the Wheeler-DeWitt patch in vacuum AdS for the different values $k=\{+1, 0, -1\}$.
• Figure 3: The top of the WDW patch for black holes in $d>2$. The GHY surface term evaluated on the regulator surface at $r=\epsilon_0$ makes a finite contribution to the action.
• Figure 4: Complexity of formation for the different geometries in four boundary (five bulk) dimensions: large hyperbolic (blue), small hyperbolic (orange), planar (dashed green) and spherical (dot-dashed red). In the inset, a larger range of horizon radii is presented demonstrating that the leading behavior at large $r_h$ is the same for the three different horizon geometries. The two vertical dashed lines are: (1) $r_h=L/\sqrt{2}$, where the (small) hyperbolic black holes become extremal; (2) $r_h=L$, where the Hawking-Page phase transition takes place for the spherical black holes (planar and hyperbolic black holes do not admit a similar transition).
• Figure 5: Complexity of formation for the different geometries in three boundary (four bulk) dimensions: large hyperbolic (blue), small hyperbolic (orange), planar (dashed green) and spherical (dot-dashed red). In the inset, a larger range of horizon radii is presented demonstrating that the leading behavior at large $r_h$ is the same for the three different horizon geometries. The two vertical dashed lines are: (1) $r_h=L/\sqrt{3}$, where the (small) hyperbolic black holes become extremal; (2) $r_h=L$, where the Hawking-Page phase transition takes place for the spherical black holes (planar and hyperbolic black holes do not admit a similar transition).