Attempt At A Definition of Quantum Chaos
Introduction
Quantum chaos is broadly defined as the study of quantum systems whose classical counterparts exhibit chaotic dynamics. However, due to the linearity and unitarity of the Schrödinger equation, chaos in the classical sense—defined by sensitivity to initial conditions and exponential divergence of nearby trajectories—does not straightforwardly translate into quantum mechanics.
Instead, quantum chaos is characterized by statistical properties of quantum spectra, the growth of operator complexity, and the scrambling of information.
Random Matrix Theory (RMT)
Early contributions from Bohigas, Giannoni, and Schmit (1984) showed that energy levels of certain quantum systems exhibit random matrix statistics, depending on the underlying symmetries—a phenomenon now considered a hallmark of quantum chaotic behavior. These results are framed within the Bohigas–Giannoni–Schmit (BGS) conjecture, which posits that quantum systems whose classical analogs are chaotic display energy level statistics identical to those of Gaussian ensembles of random matrix theory (RMT).
Cut-of-Time-Ordered Correlators (OTOC)
One of the central developments in the modern theory of quantum chaos is the use of out-of-time-ordered correlators (OTOCs) to measure information scrambling. OTOCs, typically of the form
⟨[W(t),V(0)]2⟩
quantify the sensitivity of a system's operators to perturbations in time, analogously to classical Lyapunov exponents. Maldacena, Shenker, and Stanford (2016) proposed a quantum analog of the Lyapunov exponent, showing that for thermal quantum systems with a holographic dual, this quantum Lyapunov exponent is bounded by
λL≤2πkBT/ℏ.
Systems that saturate this bound, such as the Sachdev–Ye–Kitaev (SYK) model, are considered maximally chaotic and exhibit dynamics closely related to black holes in AdS spacetime.
Thus, quantum chaos is not merely a shadow of classical behavior but a manifestation of fundamental principles governing information dynamics in many-body quantum systems.
The Connections
Furthermore, quantum chaos is deeply connected to quantum complexity, thermalization, and ergodicity.
In ergodic systems, the eigenstate thermalization hypothesis (ETH) explains how isolated quantum systems equilibrate, even in the absence of an external bath (Deutsch, 1991; Srednicki, 1994).
ETH posits that individual energy eigenstates encode thermal properties, and systems obeying ETH often display chaotic features such as RMT spectra and OTOC growth. In this context, quantum chaos plays a crucial role in understanding how closed quantum systems can exhibit irreversible behavior and approach thermodynamic equilibrium.
It also lies at the foundation of topics such as many-body localization (MBL), where deviations from chaos result in preserved quantum memory.
Conclusion
Quantum chaos is not a marginal phenomenon but a unifying framework for studying the deep interplay between randomness, entanglement, computation, and thermodynamics in quantum theory.
Sources
Bohigas, O., Giannoni, M. J., & Schmit, C. (1984). Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett., 52, 1.
Maldacena, J., Shenker, S. H., & Stanford, D. (2016). A bound on chaos. Journal of High Energy Physics, 2016, 106. [arXiv:1503.01409]
Srednicki, M. (1994). Chaos and quantum thermalization. Phys. Rev. E, 50, 888.
Deutsch, J. M. (1991). Quantum statistical mechanics in a closed system. Phys. Rev. A, 43, 2046.
Kitaev, A. (2015). A simple model of quantum holography (talks at KITP). [unpublished lectures]
Haake, F. (2010). Quantum Signatures of Chaos, 3rd ed. Springer Series in Synergetics.
