Lieb-Robinson and the butterfly effect (1603.09298v2)

Abstract: As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound---the Lieb-Robinson bound---and the butterfly effect in strongly-coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with the "butterfly" velocity $v_B$. Similarly, the Lieb-Robinson velocity places a state independent ballistic upper bound on the size of time evolved operators in non-relativistic lattice models. Here, we argue that $v_B$ is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with free particle computations to understand the role of strong coupling. We find that, depending on the way length and time scale, $v_B$ acquires a temperature dependence and decreases towards the IR. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling.

• The paper establishes that the quantum butterfly velocity acts as a state-dependent Lieb-Robinson speed, offering a novel metric for quantum information propagation.
• It uses holographic and field-theoretical methods to show that operator growth is temperature-dependent and bounded by chaos dynamics.
• The findings provide actionable insights for designing experiments in quantum setups, potentially advancing quantum communication and computation.

Analytical Examination of the Interplay Between Lieb-Robinson Bound and the Butterfly Effect in Quantum Many-Body Dynamics

Daniel A. Roberts and Brian Swingle's paper, "Lieb-Robinson and the butterfly effect," presents a rigorous exploration of the bounds on quantum information dynamics in systems characterized by numerous degrees of freedom. The authors scrutinize the connection between two seemingly distinct concepts in quantum many-body systems: the Lieb-Robinson bound and the quantum butterfly effect, particularly under circumstances typified by strong coupling. This work hones in on the ballistic growth of local operators, quantified by the Lieb-Robinson velocity ($v_{LR}$) and the butterfly velocity ($v_B$).

Technical Synopsis

Central to the discourse is the investigation of the Lieb-Robinson bound, which traditionally outlines the spatial-temporal domain in which quantum information can potentially spread within a non-relativistic lattice system. This effect appears as a non-relativistic analog to light cones in classical relativistic dynamics. Conversely, the butterfly effect is conceptualized with the butterfly velocity ($v_B$), which defines the rate of quantum chaos propagation across the system. A pivotal argument presented is that $v_B$ can be interpreted as a state-dependent effective Lieb-Robinson velocity, which the authors substantiate using diverse quantum field theories and holographic comparisons, contrasting these outcomes with those from free particle models.

Empirical Framework and Theoretical Insights

Through exploratory studies using holography, the paper identifies a temperature-dependent renormalization of $v_B$, specifying that as temperature decreases, $v_B$ decreases towards the infrared (IR) regime. This nuanced comprehension of $v_B$ and its dependence on IR properties introduces a novel perspective to traditional views, providing a relatable metric for understanding the quantum chaos-induced spread of information in complex many-body systems.

Numerical results reveal the variability of $v_B$ across different conditions, indicating its sensitivity to thermodynamic parameters such as temperature and universality class denoted by critical exponents $z$ and $\theta$. Significantly, the results propose that at low energies, $v_B$ places a cap on the speed of information transmission, potentially more restrictive than $v_{LR}$, highlighting $v_B$ as a practical boundary for quantum communication.

Implications and Speculation on Future Trajectories

The implications of this research are manifold. Practically, the phenomenological insight provided by $v_B$ offers an effective and versatile metric to gauge the spatial-temporal spread of perturbations in a quantum framework. Moreover, theoretical analyses concerning the evolution and growth of operators furnish new opportunities to explore the mechanics of information flow beyond traditionally understood bounds.

The exploration sets the stage for subsequent experimental endeavors, particularly those employing cold atom setups or similar platforms where quantum systems are delicately engineered to mirror the theoretical models discussed. Additionally, the framework established can be utilized to further probe the nuanced relationships between various velocity parameters and the extent of signaling possible within quantum realms under different coupling scenarios.

The paper advances the broader goal of deciphering quantum dynamics through a nuanced understanding of bounds, highlighting $v_B$ as a consequential metric in the paper of quantum information science. It invites further discourse on how these advances might be experimentally validated and leveraged to enhance communications and computational capabilities using quantum technologies.