A bound on chaos (1503.01409v1)

Abstract: We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent $\lambda_L \le 2 \pi k_B T/\hbar$. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

• The paper introduces a conjectured upper bound on the Lyapunov exponent, limiting chaos growth by using out-of-time-order correlators.
• It employs analytic properties and hyperbolic metric techniques to constrain the rate of information scrambling in quantum systems.
• The findings have significant implications for quantum gravity and holographic dualities, paving the way for experimental verifications.

A Bound on Chaos

The paper by Juan Maldacena, Stephen H. Shenker, and Douglas Stanford presents a conjecture on a fundamental limit to the growth of chaos in thermal quantum systems, characterized by the Lyapunov exponent. The authors propose that in systems with a large number of degrees of freedom, chaos cannot develop faster than a certain exponential rate, given by $\lambda_L \le 2 \pi k_B T/\hbar$. To arrive at this conjecture, they propose a methodology grounded on out-of-time-order correlation functions (OTOC), which capture the effects of initial perturbations on future states and help diagnose chaos in quantum systems.

Key Concepts and Results

• Out-of-Time-Order Correlation Functions (OTOCs): This function, denoted as $C(t)$, is pivotal for understanding chaos. It is directly linked to the commutator of operators separated in time and reflects the sensitivity of a quantum system to initial conditions, analogous to the classical butterfly effect.
• The Scrambling Time ($t_*$): This is the timescale over which the commutator—or the OTOC as an indicator of chaos—becomes large, effectively disrupting the initial state due to interactions in the system. The scrambling time exhibits a distinct hierarchy relative to other timescales such as the dissipation time ($t_d$), especially in systems with many degrees of freedom.
• Lyapunov Exponent ($\lambda_L$): This exponent quantifies the rate of divergence between quantum or semiclassical trajectories and serves as a measure of chaos. The conjectured bound on $\lambda_L$ forms the core result of the paper, ensuring that chaos does not develop faster than dictated by AdS/CFT results in thermal quantum systems.
• Analyticity and Factorization: The authors utilize conditions on analyticity and factorization of OTOCs to derive bounds on their behavior. The mathematical analysis involves mapping the OTOC to the unit circle and relying on the hyperbolic metric's properties to constrain the growth of chaos.

Implications and Future Directions

The conjecture provides a theoretical constraint on chaotic dynamics within a broad class of quantum systems. By proposing a bound rooted in thermodynamics and consistent with holographic principles, the authors lay out a universal limit akin to other inequalities in quantum field theory, such as the viscosity bound.

The implications for quantum information and computation are substantial, particularly in understanding how information propagates and is scrambled within quantum systems. It points to deeper insights into quantum gravity and the holographic principle, suggesting that Einstein gravity systems might naturally saturate this bound.

Future research could explore various domains:

• Testing the Bound in Different Systems: Extending the analysis to additional quantum systems, particularly those without gravity duals.
• String Theory and Holographic Duals: Understanding the role of stringy corrections and higher-dimensional analogs in modifying the proposed chaos bound.
• Experimental Confirmation: Using cold atom systems or other quantum simulators to examine and test these theoretical bounds on chaos in controlled settings.

Overall, this paper offers a profound theoretical advance in characterizing and bounding the dynamics of chaos in high-dimensional quantum systems, with potential ramifications across quantum physics and cosmology.