Wasserstein Space of Quantum Chaos

TL;DR

This paper introduces a novel approach using Wasserstein space and optimal transport to diagnose quantum chaos, scrambling, and scars, revealing dimensional reduction linked to chaotic behavior.

Contribution

It demonstrates that the Wasserstein space of energy eigenstates shrinks with increased chaos and introduces a new geometric diagnostic for quantum chaos phenomena.

Findings

Wasserstein space dimension decreases as quantum chaos increases.

Exponential OTOC growth induces a folding structure in Wasserstein space.

Wasserstein distance captures Lyapunov exponents at the separatrix.

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.