Addendum to Computational Complexity and Black Hole Horizons

TL;DR

Addendum extends the conjectured complexity–ERB duality to shockwave geometries in (2+1)-D BTZ, providing quantitative links between ERB length and circuit complexity growth. It shows that precursor-induced complexity contributes to ERB length, yielding $d \approx 2 t_W + t_L$ and, in single- and multi-shock cases, $d/l_{ads} \approx (t_R+t_L)/l_{ads}$, consistent with the complexity growth picture. The paper also analyzes a Gedanken experiment where Alice measures a complete set of left-side observables, revealing a GHZ tripartite entanglement among the two black holes and memory that challenges simple bipartite signaling intuitions. It concludes that while Alice cannot signal via memory alone, a sufficiently complex precursor acting on L and M could erase the measurement and enable signaling to Bob, resolving apparent paradoxes and linking observer perspectives through tripartite entanglement.

Abstract

In this addendum to [arXiv:1402.5674] two points are discussed. In the first additional evidence is provided for a dual connection between the geometric length of an Einstein-Rosen bridge and the computational complexity of the quantum state of the dual CFT's. The relation between growth of complexity and Page's ``Extreme Cosmic Censorship" principle is also remarked on. The second point involves a gedanken experiment in which Alice measures a complete set of commuting observables at her end of an Einstein-Rosen bridge is discussed. An apparent paradox is resolved by appealing to the properties of GHZ tripartite entanglement.

Figures (1)

• Figure 1: ERB for a GHZ-entangled system of three black holes. The core of the ERB must have properties that cannot be described by classical geometry.