The Limits of Quantum Information Scrambling

TL;DR

This paper establishes mathematical bounds for quantum information scrambling rates, showing their dependence on initial states and operator properties, with validation through a spin-star model.

Contribution

It derives bounds for scrambling rates using the Maligranda inequality and analyzes their behavior under different initial states and operator configurations.

Findings

Bounds coincide for unitary-Hermitian operators.

Scrambling rate depends on initial state and operator size.

Bounds can diverge from the actual scrambling rate in multi-qubit systems.

Abstract

Quantum Information scrambling (QI-scrambling) is a pivotal area of inquiry within the study of quantum many-body systems. This research derives mathematical upper and lower bounds for the scrambling rate by applying the Maligranda inequality. Our results indicate that the upper bounds, lower bounds, and scrambling rates coincide precisely when local operators exhibit to be unitary-Hermitian. Crucially, the convergence or divergence of these upper and lower bounds relative to the scrambling rate is contingent upon the system's initial state. The spin-star model to validate this theoretical framework is investigated, considering thermal and pure initial states. The implantation of the ancilla or external qubit aligns the scrambling rate with the established bounds. The upper and lower bounds may diverge from the scrambling rate based on the system's initial state when both local operators are multi-quit systems. The scrambling rate found grows with the increase of the qubit number in local operators.