Chaotic Quantum Systems Possess Surprisingly Fewer Dimensions Than Expected

Koji Hashimoto of Kyoto University and colleagues from Saitama University have identified a link between quantum chaos and the geometry of probability distributions. They reveal that the effective dimension of the Wasserstein space decreases as a quantum system exhibits increasing chaotic behaviour. Applying optimal transport methods to a quantum coupled harmonic oscillator, the team found that exponential growth in out-of-time ordered correlations, or quantum scrambling, creates a folding structure within this space, potentially explaining the observed dimensional reduction. The Wasserstein distance accurately identifies the Lyapunov exponent at a critical point, and branching structures signal the presence of quantum scar states. This new approach offers a key diagnostic set of tools for understanding quantum chaos, scrambling, scars, and Lyapunov exponents, while also supporting the idea that the Wasserstein space may represent an emergent holographic space relevant to black hole physics.

Quantum scrambling reduces Wasserstein space dimensionality and reveals quantum scars

The effective dimension of the Wasserstein space, a mathematical construct providing a metric for comparing probability distributions representing quantum states, decreased from an initial value of six to 2.3 following increases in chaoticity within a quantum coupled harmonic oscillator system. This reduction surpasses previous limitations, as earlier methods struggled to pinpoint the origin of this dimensional change. Quantum scrambling, characterised by the rapid delocalisation of quantum information, now clarifies a key driver of this phenomenon. Exponential growth in out-of-time ordered correlations (OTOCs), a measure of quantum scrambling, induces a folding structure within the Wasserstein space, potentially explaining the observed reduction and supporting the idea of an emergent holographic space. OTOCs quantify the rate at which initial quantum states become entangled with later-time observables, and their exponential growth is a hallmark of chaotic systems. The Wasserstein space, in this context, provides a geometrical framework for understanding how these correlations manifest as changes in the distribution of quantum states.

A branching structure within this space reliably signals the presence of quantum scar states, unusual stable configurations within chaotic systems, offering a new diagnostic for quantum phenomena. These scars represent persistent eigenstates that survive even in the presence of strong chaos, defying the expectation of complete randomisation. Optimal transport techniques and Husimi Q-representations allowed precise measurement of this dimensional reduction. The approach reveals the underlying structure of the Wasserstein space and enables analysis of its dimensionality, linking quantum systems to concepts from gravity and black holes. Stable configurations within chaotic systems, quantum scar states, were reliably identified by the branching structure, offering a new way to diagnose these unusual quantum phenomena. The ability to identify these scars is significant, as they can influence the dynamics of the system and provide insights into the interplay between order and chaos.

Optimal transport of Husimi representations reveals dimensionality of quantum chaos

Sinkhorn-regularized optimal transport proved important in mapping the complex field of quantum states onto a geometrically interpretable space. It effectively finds the most efficient way to transform one probability distribution into another, much like determining the shortest route between cities on a map, minimising the ‘cost’ of moving probability mass. The Sinkhorn regularisation addresses the computational challenges associated with optimal transport by adding an entropic penalty, ensuring a smooth and stable solution. Husimi Q-representations were first employed to represent the quantum states, a method for visualising the probability of a quantum particle being in a specific state, like creating a heat map of its location. This representation transforms the quantum wavefunction into a classical probability distribution on phase space, allowing for the application of classical tools like optimal transport. The Husimi Q-representation effectively smears out the quantum wavefunction, providing a more manageable representation for analysis.

Construction of an ‘embedding geometry’ was then possible using a technique called the Gram-spectrum method, revealing the underlying structure of the Wasserstein space. The Gram-spectrum method calculates the eigenvalues of the Gram matrix constructed from the pairwise Wasserstein distances between quantum states, providing a measure of the intrinsic dimensionality of the space. A quantum coupled harmonic oscillator system was investigated to explore the relationship between chaos and dimensionality. This system, while relatively simple, exhibits characteristics of more complex chaotic systems, making it an ideal testbed for this methodology. This approach tested the manifold hypothesis, suggesting that complex data can be represented in lower dimensions, potentially mirroring the emergence of spacetime in holographic scenarios. The holographic principle posits that the information contained within a volume of space can be encoded on its boundary, suggesting a deep connection between quantum information and gravity.

Although this work offers a new way to diagnose quantum chaos and strengthens the idea of an emergent holographic space, it remains confined to the behaviour of a single, albeit well-studied, quantum system. The authors acknowledge that demonstrating these findings hold true across a wider variety of quantum systems, particularly those exhibiting different types of chaotic behaviour, is a vital next step. Investigating systems with varying degrees of coupling strength and different potential landscapes will be crucial for establishing the generality of these results. Mapping the behaviour of energy levels within the system using optimal transport revealed a link between increasing chaos and a reduction in the effective dimensionality of the space. The Lyapunov exponent, a measure of the rate of separation of nearby trajectories in phase space, was accurately identified using the Wasserstein distance, providing a quantitative link between geometry and chaos.

Establishing a geometrical link between quantum chaos and probability distributions offers a new approach to understanding fundamental physics. This dimensional reduction, alongside the identification of folding structures and branching patterns indicative of stable ‘quantum scar’ states, provides a novel diagnostic for quantifying quantum phenomena like scrambling and Lyapunov exponents, key for characterising chaotic systems. The resulting ‘Wasserstein space’ represents the relationships between quantum states, offering a powerful tool for analysing and understanding the complex behaviour of quantum systems. Further research could explore the application of this methodology to more complex systems, potentially shedding light on the emergence of spacetime and the nature of quantum gravity.

The research demonstrated that the effective dimension of the space describing quantum states decreases as a quantum system becomes more chaotic. This is significant because it provides a new geometrical method for diagnosing and quantifying quantum chaos, scrambling, and the presence of quantum scar states. By applying optimal transport to a quantum coupled harmonic oscillator, researchers accurately captured the Lyapunov exponent and observed a folding structure within the resulting Wasserstein space. The authors plan to extend this work to a wider range of quantum systems to confirm the generality of these findings and further investigate the connection to holographic scenarios.