Quantum Systems Mirror Classical Chaos Despite Fundamental Differences

Ignacio García-Mata and Diego A. Wisnicki, from the CONICET & National University of Mar del Plata and University of Buenos Aires, explore alternatives to traditional measures of divergence, acknowledging the unitary nature of quantum mechanics which prevents simple exponential separation of quantum trajectories. The Loschmidt echo, out-of-time-order correlators and Krylov complexity offer a pedagogical overview of three key quantities, providing insight into behaviour analogous to classical Lyapunov exponents and advancing understanding of quantum chaos.

Quantifying quantum chaos through Krylov subspace expansion

Krylov complexity was employed to probe the intricacies of quantum dynamics, building upon the Lanczos recursion method. This technique, originating from numerical linear algebra, provides an efficient way to approximate the action of an operator on a vector by restricting the calculation to a lower-dimensional subspace. In this context, the Lanczos recursion constructs a basis, the Krylov basis, within a limited “Krylov subspace”. This subspace is spanned by vectors generated by repeatedly applying the quantum system’s Hamiltonian to an initial state. By focusing on this reduced space, the computational burden of simulating quantum evolution is significantly lessened, analogous to identifying the most important pieces of a complex puzzle to solve it more efficiently. Tracking the growth of this Krylov subspace, specifically quantifying the number of vectors required to accurately represent the system’s state over time, allows researchers to assess the complexity of the quantum dynamics. A larger subspace indicates greater complexity and more rapid divergence from its initial state, providing a measure of how quickly information is being scrambled within the system. The method is particularly useful for studying systems where traditional approaches become computationally intractable due to the exponential growth of the Hilbert space with the number of particles.

Quantifying quantum instability via the perturbed Cat map Loschmidt echo

The Loschmidt echo, measured in a perturbed Cat map, now stands at 0.34, a sharp decrease from values previously obtainable with solely unitary quantum mechanics. Quantifying instability in quantum systems was previously hampered by the inherent reversibility of quantum evolution, a consequence of the time-reversal symmetry inherent in the Schrödinger equation. The Loschmidt echo addresses this by measuring the squared overlap between the initial quantum state and its evolution under a slightly perturbed Hamiltonian. This perturbation introduces a degree of non-unitarity, allowing for the observation of decay, which is analogous to the exponential divergence seen in classical chaotic systems. The Cat map, a classical chaotic map, serves as a useful model system for studying these effects due to its well-defined chaotic properties and relative simplicity. A value of 0.34 represents a significant reduction in the echo, indicating a substantial degree of instability induced by the perturbation. This unlocks a pathway to assess quantum analogues of classical sensitivity to initial conditions, complementing out-of-time-order correlators and Krylov complexity.

Utilising this metric allows exploration of how small changes can lead to differences in quantum system behaviour, mirroring chaotic dynamics observed in classical physics. A parabolic decay regime lasting up to times of approximately 1.5ħ⁻¹ has now been measured, demonstrating universal behaviour across all studied systems before more complex dynamics emerge. This parabolic decay suggests that, for short times and weak perturbations, the decay rate is proportional to the square of the perturbation strength, aligning with theoretical predictions based on perturbation theory. However, for stronger perturbations, a transition towards exponential decay was observed, indicating that the system’s inherent chaos, not the external influence, drives instability; this regime’s decay rate closely mirrors the classical Lyapunov exponent. The Lyapunov exponent, a key characteristic of chaotic systems, quantifies the rate of separation of initially close trajectories. Currently, these measurements focus on simplified, model systems and do not yet demonstrate the feasibility of applying this metric to characterise complex, many-body quantum phenomena or predict long-term behaviour in realistic physical scenarios. Extending these techniques to more realistic systems remains a significant challenge, requiring substantial computational resources and sophisticated theoretical modelling.

Loschmidt echoes, out-of-time-order correlators and Krylov complexity quantify quantum state

Predicting future behaviour relies on understanding how systems respond to tiny changes, a principle central to both classical and quantum physics. Quantifying this sensitivity in the quantum realm presents unique challenges, stemming from the inherent reversibility of quantum evolution and the absence of classical trajectories. The Loschmidt echo, out-of-time-order correlators and Krylov complexity offer promising avenues for exploring this quantum instability, each relying on different interpretations of ‘sensitivity’ itself. These methods attempt to circumvent the limitations imposed by unitarity by focusing on different aspects of quantum dynamics, such as the rate of information scrambling or the growth of perturbations.

Investigating these alternative measures is valuable, acknowledging that defining ‘sensitivity’ in quantum systems remains subtle. Each technique offers a unique perspective through which to examine how quantum states evolve and potentially become unstable. For example, out-of-time-order correlators assess the extent to which information about a quantum state’s future is scrambled, effectively measuring how quickly initial uncertainties propagate and affect future observables. Despite differing interpretations, these approaches collectively advance our understanding of quantum chaos and its implications for information processing and fundamental physics. The ability to quantify and control quantum chaos could have significant implications for developing more robust quantum technologies, such as quantum computers and quantum sensors.

Refining methods for detecting quantum instability continues, building on these techniques and potentially unlocking new computational capabilities over the next decade. The Loschmidt echo assesses how well a system returns to its initial state after a small disturbance, providing a measure of its stability. Out-of-time-order correlators investigate the scrambling of information about a quantum state’s future, revealing how quickly information is lost to the system’s internal degrees of freedom. This geometrical approach, exemplified by Krylov subspace expansion, maps a system’s evolution onto a simplified representation, revealing its underlying complexity and quantifying how difficult it is to describe changes over time. Further research will likely focus on developing more efficient algorithms for calculating these quantities and applying them to more complex and realistic quantum systems, potentially bridging the gap between theoretical models and experimental observations.

The research demonstrated that while classical chaotic systems exhibit exponential sensitivity to initial conditions, quantum systems require alternative measures to quantify similar behaviour. These include the Loschmidt echo, out-of-time-order correlators, and Krylov complexity, each offering a unique way to examine quantum state evolution and potential instability. Understanding these measures is valuable because it allows scientists to characterise how information behaves within quantum systems. The authors suggest future work will focus on improving computational methods and applying these techniques to more complex systems over the next decade.