[Paper Review] Information scrambling in chaotic systems with dissipation

Yongliang Zhang, Yichen Huang+1 more|arXiv (Cornell University)|Feb 13, 2018

Neural Networks and Reservoir Computing4 citations

TL;DR

This paper investigates information scrambling in chaotic quantum systems under dissipation by introducing a corrected out-of-time-ordered correlator (OTOC) to isolate structural effects from overall decay. Using a dissipative Ising spin chain, it shows that despite information leakage, a finite 'information light cone' persists due to dissipation-induced re-structuring, leading to a modified Lieb-Robinson bound with a width scaling as √(ε a v_LR / Γ).

ABSTRACT

Chaotic dynamics in closed local quantum systems scrambles quantum information, which is manifested quantitatively in the decay of the out-of-time-ordered correlators (OTOC) of local operators. How is information scrambling affected when the system is coupled to the environment and suffers from dissipation? In this paper, we address this question by defining a dissipative version of OTOC and numerically study its behavior in a prototypical chaotic quantum chain in the presence of dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the `information light cone' can only reach a finite distance even when the effect of overall decay is removed. Based on this observation we conjecture a modified version of the Lieb-Robinson bound in dissipative systems.

Motivation & Objective

• To understand how dissipation affects quantum information scrambling in chaotic systems, particularly the persistence of the information light cone.
• To distinguish between overall signal decay due to leakage and structural changes in the light cone caused by dissipation.
• To develop a corrected OTOC formalism that isolates the structural effects of dissipation from decoherence.
• To derive a modified Lieb-Robinson bound applicable to open, dissipative quantum systems.
• To numerically validate the existence of a finite light cone in dissipative chaotic spin chains under amplitude, phase, and depolarizing damping.

Proposed method

• Introduces a dissipative version of the OTOC using a corrected correlator that separates structural dynamics from overall decay.
• Models a chaotic Ising chain with transverse and longitudinal fields under open boundary conditions and applies three types of dissipation: amplitude damping, phase damping, and phase depolarizing.
• Uses superoperator formalism to describe the time evolution of operators under the dissipative channel, with the adjoint propagator 𝒱†_b(t) mapping initial operators to evolved ones.
• Applies the triangle inequality and operator norm bounds to compare the commutator [𝒱†_1(t,0)·B, A] in the dissipative channel to the unitary case, establishing stability under small dissipation.
• Defines a normalized Frobenius norm ‖·‖_F to assess the robustness of the light cone signal in the thermodynamic limit.
• Derives a lower bound on the light cone width as √(ε a v_LR / Γ), valid for sufficiently small dissipation rate Γ ≪ ε a v_LR / ξ².

Experimental results

Research questions

• RQ1 Can the information light cone structure persist in chaotic systems when coupled to a dissipative environment?
• RQ2 To what extent does dissipation alter the spatial and temporal structure of information spreading beyond simple signal decay?
• RQ3 Can a corrected OTOC be defined that isolates the structural features of scrambling from decoherence effects?
• RQ4 Does the presence of dissipation lead to a modified Lieb-Robinson bound with a finite light cone width?
• RQ5 How does the width of the light cone scale with the dissipation rate Γ in open chaotic systems?

Key findings

• Dissipation leads to both overall decay of OTOC due to information leakage and structural changes in the light cone, preventing infinite propagation.
• The light cone in the dissipative system is finite in extent, with a width bounded from below by √(ε a v_LR / Γ), even when overall decay is removed.
• The corrected OTOC, defined via the normalized Frobenius norm of the commutator, successfully reveals the light cone structure in the presence of dissipation.
• For phase damping and phase depolarizing channels, the corrected OTOC matches the form 2(1 - F(t,A,B)/F(t,I,B)) = ‖[𝒱†_b(t)·B, A]‖_F² / ‖𝒱†_b(t)·B‖_F², enabling direct comparison with unitary dynamics.
• The light cone is detectable in the time range 3ξ/v_LR ≲ t ≤ √(ε a / (v_LR Γ)), where ξ is the dissipation correlation length and v_LR is the Lieb-Robinson velocity.
• The modified Lieb-Robinson bound is conjectured to hold, with the light cone width scaling as √(1/Γ), indicating that dissipation limits the effective speed of information propagation.