The Limits of Quantum Information Scrambling

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 Hermitian and unitary operators. Crucially, the convergence or divergence of these upper and lower bounds relative to the scrambling rate is contingent upon the system's initial state. To validate our theoretical framework, we investigated the spin-star model, considering both thermal and pure initial states. Furthermore, three distinct scenarios for local operators were examined, namely single-qubit, one multi-qubit, and both multi-qubit configurations. 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-qubit systems. We noticed that the scrambling rate increased as the number of qubits in local operators increased.