Holographic Complexity Equals Bulk Action?

We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti–de Sitter spacetime, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.

Figure 1

The two-sided eternal AdS black hole (left) and a one-sided AdS black hole that forms from a collapsing shockwave (right). The two-sided AdS black hole is dual to an entangled (thermofield double) state of two CFTs that live on the left and right boundaries; the one-sided black hole is dual to a single CFT. Our complexity-action conjecture relates the complexity of the CFT state to the action of the Wheeler-DeWitt patch (shown shaded).

Figure 2

For a charged AdS black hole, the Wheeler-DeWitt patch does not extend all the way to the singularity, and instead ends when the ingoing light sheets self-intersect just outside the inner horizon at $r_{-}$.