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arXiv:2408.10684 · Journal of Physics A Mathematical and Theoretical

The limits of quantum information scrambling

Mahmoud Youssef Abd‐Rabbou, Ahmed A. Zahia, Atta Ur Rahman, Cong‐Feng Qiao

Published Jun 5, 2025• 9 citations
arXivHTMLPDF

Abstract

Abstract Quantum information 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.

DOI
10.1088/1751-8121/ade1b9

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Abstract Quantum information 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.
Abstract Quantum information scrambling refers to the loss of local recoverability of quantum information, which has found widespread attention from high energy physics to quantum computing. In the present analysis we propose a possible starting point for the development of a comprehensive framework for the thermodynamics of scrambling. To this end, we prove that the growth of entanglement as quantified by the mutual information is lower bounded by the time-dependent change of out-of-time-ordered correlator. We further show that the rate of increase of the mutual information can be upper bounded by the sum of local entropy productions, and the exchange entropy arising from the flow of information between separate partitions of a quantum system. Our results are illustrated for the ion trap system, that was recently used to verify information scrambling in an experiment, and for the Sachdev–Ye–Kitaev model.
Scrambling, the delocalization of initially localized quantum information, is commonly characterized by the out-of-time ordered correlator (OTOC). Employing the OTOC-Renyi-2 entropy theorem we derive a quantum speed limit for the OTOC, which sets an lower bound for the rate with which information can be scrambled. This bound becomes particularly tractable by describing the scrambling of information in a closed quantum system as an effective decoherence process of an open system interacting with an environment. We prove that decay of the OTOC can be bounded by the strength of the system-environment coupling and two-point environmental correlation functions. We validate our analytic bound numerically using the non-integrable transverse field Ising model. Our results provide a universal and model-agnostic quantitative framework for understanding the dynamical limits of information spreading across quantum many-body physics, condensed matter, and engineered quantum platforms.
Abstract Different fields of physics characterize differently how much two quantum operations disagree: quantum information theory features uncertainty relations cast in terms of entropies. The higher an uncertainty bound, the less compatible the operations. In condensed matter and high-energy physics, initially localized, far-apart operators come to disagree as entanglement spreads through a quantum many-body system. This spread, called “scrambling,” is quantified with the out-of-time-ordered correlator (OTOC). We unite these two measures of operation disagreement by proving entropic uncertainty relations for scrambling. The uncertainty bound depends on the quasiprobability (the nonclassical generalization of a probability) known to average to the OTOC. The quasiprobability strengthens the uncertainty bound, we find, when a spin chain scrambles in numerical simulations. Hence our entropic uncertainty relations reflect the same incompatibility as scrambling, uniting two fields’ notions of quantum-operation disagreement.
Quantum information scrambling provides diagnostics about how information spreads when an isolated quantum system evolves in time. Here, the authors study the temperature dependence of scrambling in locally gapped systems. Using tensor networks, they numerically evaluate out-of-time-ordered commutators in a large, strongly interacting spin chain as a function of temperature. They find that scrambling crucially depends on the ordering of operators around the thermal circle, exposing subtleties in information propagation at finite temperature. For a regulated operator ordering, the butterfly velocity decreases with lowering the temperature, as seen in the figure. The low temperature behavior is derived from a field-theoretical analysis.
The study of information scrambling in many-body systems has sharpened our understanding of quantum chaos, complexity, and gravity. Here, we extend the framework for exploring information scrambling to infinite-dimensional continuous variable (CV) systems. Unlike their discrete variable cousins, continuous variable systems exhibit two complementary domains of information scrambling: (i) scrambling in the phase space of a single mode and (ii) scrambling across multiple modes of a many-body system. Moreover, for each of these domains, we identify two distinct types of scrambling; genuine scrambling, where an initial operator localized in phase space spreads out, and quasiscrambling, where a local ensemble of operators distorts but the overall phase space volume remains fixed. To characterize these behaviors, we introduce a CV out-of-time-order correlation (OTOC) function based upon displacement operators and offer a number of results regarding the CV analog for unitary designs. Finally, we investigate operator spreading and entanglement growth in random local Gaussian circuits; to explain the observed behavior, we propose a simple hydrodynamical model that relates the butterfly velocity, the growth exponent, and the diffusion constant. Experimental realizations of continuous variable scrambling as well as its characterization using CV OTOCs will be discussed.
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.