Wasserstein Space of Quantum Chaos

Abstract: We find that the effective dimension of the Wasserstein space of energy eigenstates decreases as a quantum system becomes more chaotic. To demonstrate this, we study a quantum coupled harmonic oscillator system using Husimi Q-representations, to which Sinkhorn-regularized optimal transport is applied to construct an embedding geometry via the Gram-spectrum method. We also demonstrate that exponential OTOC growth, referred to here as quantum scrambling even in the absence of chaos, induces a folding structure in the emergent Wasserstein space, which may underlie the chaotic reduction of the Wasserstein dimension. At the separatrix (the scrambling point) of the inverted harmonic oscillator, the Wasserstein distance correctly captures the Lyapunov exponent. Furthermore, we discover that a branching structure in the Wasserstein space signals quantum scar states within the chaotic sea of phase space. Our optimal transport approach thus provides a new diagnostic for quantum chaos, quantum scrambling, quantum scars, and quantum Lyapunov exponents. The observed chaotic dimensional reduction also supports the recent conjecture [arXiv:2604.17649] that the Wasserstein space serves as an emergent holographic space through the manifold hypothesis, since chaoticity is a characteristic signature of black holes in holography.

• The paper introduces Wasserstein geometry and Sinkhorn-regularized optimal transport to embed quantum states and quantify chaos.
• The paper employs Husimi Q-representations to reveal clear signatures of scrambling, dimensional reduction, and quantum scars in phase-space embeddings.
• The paper demonstrates that Wasserstein metrics, through consistent scaling of Lyapunov exponents, connect classical chaos diagnostics with quantum information measures.

Wasserstein Geometry as a Diagnostic of Quantum Chaos

Introduction

"Wasserstein Space of Quantum Chaos" (2605.20995) investigates the interplay between optimal transport theory and quantum chaos, proposing Wasserstein geometry as an emergent diagnostic framework for quantum chaotic systems. The analysis leverages Husimi Q-representations and Sinkhorn-regularized optimal transport to embed quantum energy eigenstates into a metric space, thereby quantifying structural and dynamical properties such as scrambling, dimensional reduction, quantum scars, and Lyapunov exponents. The paper situates its findings within the manifold hypothesis and the broader context of holography, providing quantitative evidence for the role of quantum chaos in dimensional reduction—a phenomenon conjectured to underpin emergent holographic spacetimes.

Optimal Transport and Wasserstein Metric in Quantum Phase Space

The central methodological innovation is the application of optimal transport, specifically the 1-Wasserstein (and 2-Wasserstein) metric, to quantum states represented in phase space via Husimi Q distributions. The Sinkhorn regularization paradigm allows efficient computation of transport costs between discrete Husimi Q-representations, facilitating the construction of a pairwise distance matrix among energy eigenstates. Embedding this matrix into a Euclidean point cloud via classical multidimensional scaling (MDS) provides a geometric realization of quantum state relationships, wherein effective dimension and structural features encode physical properties.

The approach is validated for both 1- and 2-Wasserstein metrics, with robust qualitative and quantitative preservation of the main physical results across metric variants and numerically convergent parameter settings, as elaborated in the extensive error analysis.

Quantum Scrambling and Folding in Wasserstein Space

Through the analysis of models exhibiting quantum scrambling (e.g., the double-well potential and partly flattened oscillator), the paper demonstrates a direct correlation between exponential growth in OTOCs and a folding structure in Wasserstein space. Such folding emerges at energy levels corresponding to the separatrix—the saddle-point dividing distinct dynamical regimes—underscoring the connection between scrambling (even absent chaos) and geometric features of Wasserstein embeddings. In particular, the state located at the top of the potential hill in the double-well model exhibits maximal Wasserstein distance from adjacent states, manifesting as a geometric fold. This is attributed to singular localization in phase space, which imposes a high transport cost under the Wasserstein metric.

The folding effect echoes the Sieber-Richter mechanism in classical chaos, linking periodic orbit recombinations to spectral statistics, and suggesting classical separatrix-induced mechanisms carry over to quantum geometric diagnostics.

Dimensional Reduction and Manifold Hypothesis in Quantum Chaos

The coupled harmonic oscillator model serves as a prototypical quantum chaotic system wherein the chaoticity parameter $g$ governs non-integrability. Strong numerical results are presented: As $g$ increases, the effective dimension of Wasserstein space, as measured by the spectral decay of the centered Gram matrix eigenvalues derived from the distance matrix, monotonically decreases. This trend is quantified by exponential fitting (parameters $A$, $B$) of eigenvalue spectra, with the decay parameter $B$ serving as a resolution-independent probe of chaos-induced geometric compression.

Notably, at high $g$, the Wasserstein embedding demonstrates not only reduced dimension but homogeneity among high-energy eigenstates—adjacent states are metrically nearly indistinguishable, reflecting the typicality induced by chaos. This empirical dimensional reduction substantiates the manifold hypothesis, positing that data in high-dimensional quantum Hilbert space effectively resides on a low-dimensional manifold, and ties quantum chaos to the emergence of holographic geometry.

Branch Structure and Quantum Scars in Wasserstein Space

In the strongly coupled oscillator, the Wasserstein embedding reveals a branching structure (the "Wasserstein branch"), corresponding to a subset of energy eigenstates (e.g., $n=5,7,9,11$) that are mutually proximate but isolated from the chaotic sea. These branch states coincide with quantum scars—non-thermal, highly localized states associated with classical periodic orbits. Comparative analysis of Husimi Q-representations, probability densities, and classical Poincaré sections demonstrates that these branch states exhibit localization near regular islands or axial periodic orbits.

The Wasserstein branch thus serves as a nontrivial diagnostic for quantum scars, aligning with known scenarios of weak ergodicity breaking and persistent nonintegrability. The branching persists at higher energies despite the dominance of chaos, reinforcing its interpretational value as an atypical, robust structure within the quantum state landscape.

Quantum Lyapunov Exponents via Optimal Transport

The paper provides an analytic treatment of a time-evolving Gaussian state in the inverted harmonic oscillator, verifying that the Wasserstein metric (both W1 and W2), the microcanonical OTOC, and the half-probability contour length all yield identical exponential scaling, thus capturing the quantum Lyapunov exponent ($\lambda_Q$). This unifies classical-like sensitivity diagnostics with modern quantum information measures under optimal transport theory, confirming that Wasserstein distance offers a consistent quantitative tool for detecting and measuring quantum chaos.

Implications and Future Directions

The findings have both practical and theoretical significance. Wasserstein geometry provides a unifying, metric-based language for emergent features of quantum chaos, including dimensional reduction, scrambling, scars, and Lyapunov exponents. The observed relations support the conjecture that chaotic boundary quantum systems have emergent low-dimensional Wasserstein spaces, as required for holographic duality via the manifold hypothesis. The diagnostic sensitivity to quantum scars further implies non-thermal states manifest as distinct branches, potentially resisting thermalization even in chaotic regimes.

Future developments should address:

• Systematic extension to higher excitations and broader symmetry sectors
• Application to nontrivial quantum field theories and genuinely holographic systems
• Integration with complexity-theoretic measures (e.g., Krylov complexity) and machine learning techniques for automated detection of geometric signatures
• Probing the robustness and universality of dimensional reduction across different models and parameter spaces

Conclusion

"Wasserstein Space of Quantum Chaos" (2605.20995) introduces and rigorously analyzes optimal transport-based geometric diagnostics for quantum chaotic systems, demonstrating dimensional reduction, scrambling-induced foldings, quantum scar branches, and precise quantification of Lyapunov exponents in phase-space embeddings. The results substantiate the manifold hypothesis as a dynamical consequence of chaos and provide a versatile framework connecting quantum chaos with emergent holographic geometry. The research invites further exploration into the interplay between metric geometry, quantum information, and fundamental holographic principles.