Holographic Interpretation of Relative State Complexity

Alexander Yosifov,Aditya Iyer,Lachezar Filipov

doi:10.3390/universe6080100

Abstract

We investigate a large-N conformal field theory (CFT) in a high-energy pure state coupled to a small auxiliary system of M weakly-interacting degrees of freedom, and argue the relative state complexity of the auxiliary system is holographically dual to an effective low-energy notion of computational cost in the bulk, i.e., to the minimal depth of the quantum circuit required to decode its gravitational dual. In light of this, using Nielsen’s approach, a new measure of quantum chaos in terms of the evolution of circuit complexity is proposed. It suggests that studying the evolution of circuit complexity of the auxiliary system can teach us about the chaotic properties of the large-N CFT. This new diagnostic for quantum chaos has important implications for the interior dynamics of evaporating black holes as it implies the radiated Hawking cloud is pseudorandom.

Highlights

The anti-de Sitter/conformal field theory (AdS/CFT) correspondence relates a d-dimensional boundary CFT to a bulk gravity theory in (d + 1)-dimensional asymptotically AdS spacetime.The existing bulk/boundary dictionary points to a deep relation between the concepts of quantum complexity, quantum information, chaos, and gravity—all of which interplay in black holes.Quantum complexity is usually defined in the literature as the minimum number of gates required to prepare a quantum state, assuming a universal gate set{ gi }
In an attempt to extend the bulk/boundary dictionary and shed light onto the elusive nature of quantum chaos in strongly-coupled quantum systems, our goal in this paper is to propose, within AdS/CFT, a holographic interpretation of relative state complexity as a decoding task in the bulk, and demonstrate how it can be used as a novel measure of quantum chaos
Employing Nielsen’s complexity geometry framework, we investigate the evolution of relative state complexity of a small auxiliary system of M weakly interacting degrees of freedom coupled to a large-N CFT in a high-energy pure state, gravitationally dual to one-sided AdS black hole

Summary

Introduction

The anti-de Sitter/conformal field theory (AdS/CFT) correspondence relates a d-dimensional boundary CFT to a bulk gravity theory in (d + 1)-dimensional asymptotically AdS spacetime. The semiclassical black hole structure, makes examining the interior highly non-trivial Different measures such as out-of-time-order correlators (OTOC) [10] and random matrix theory (RMT) [11] exist, more recently, the incorporation of quantum information-theoretic tools in holography together with the established protocol for assigning computational costs to trajectories on the unitary manifold [7,8,9] have opened the possibility of probing the highly chaotic black hole interior from a new angle. Note that for a time interval ∆t ∈ [0, t], where ∆t ∆τ we can divide ∆t into j steps, where each is of order ∆τ

Complexity and Chaos

Complexity

Geometric Interpretation

Sachdev–Ye–Kitaev

Conclusions